Let $a, b$ and $c$ be positive real numbers such that $\min \{a b, b c, c a\} \geqslant 1$. Prove that  $$ \sqrt[3]{\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)} \leqslant\left(\frac{a+b+c}{3}\right)^{2}+1 $$

Solution: We first show the following.  - Claim. For any positive real numbers $x, y$ with $x y \geqslant 1$, we have  $$ \left(x^{2}+1\right)\left(y^{2}+1\right) \leqslant\left(\left(\frac{x+y}{2}\right)^{2}+1\right)^{2} $$  Proof. Note that $x y \geqslant 1$ implies $\left(\frac{x+y}{2}\right)^{2}-1 \geqslant x y-1 \geqslant 0$. We find that $\left(x^{2}+1\right)\left(y^{2}+1\right)=(x y-1)^{2}+(x+y)^{2} \leqslant\left(\left(\frac{x+y}{2}\right)^{2}-1\right)^{2}+(x+y)^{2}=\left(\left(\frac{x+y}{2}\right)^{2}+1\right)^{2}$.  Without loss of generality, assume $a \geqslant b \geqslant c$. This implies $a \geqslant 1$. Let $d=\frac{a+b+c}{3}$. Note that  $$ a d=\frac{a(a+b+c)}{3} \geqslant \frac{1+1+1}{3}=1 . $$  Then we can apply (2) to the pair $(a, d)$ and the pair $(b, c)$. We get  $$ \left(a^{2}+1\right)\left(d^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right) \leqslant\left(\left(\frac{a+d}{2}\right)^{2}+1\right)^{2}\left(\left(\frac{b+c}{2}\right)^{2}+1\right)^{2} $$  Next, from  $$ \frac{a+d}{2} \cdot \frac{b+c}{2} \geqslant \sqrt{a d} \cdot \sqrt{b c} \geqslant 1 $$  we can apply (2) again to the pair $\left(\frac{a+d}{2}, \frac{b+c}{2}\right)$. Together with (3), we have  $$ \left(a^{2}+1\right)\left(d^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right) \leqslant\left(\left(\frac{a+b+c+d}{4}\right)^{2}+1\right)^{4}=\left(d^{2}+1\right)^{4} $$  Therefore, $\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right) \leqslant\left(d^{2}+1\right)^{3}$, and (1) follows by taking cube root of both sides.  Comment. After justifying the Claim, one may also obtain (1) by mixing variables. Indeed, the function involved is clearly continuous, and hence it suffices to check that the condition $x y \geqslant 1$ is preserved under each mixing step. This is true since whenever $a b, b c, c a \geqslant 1$, we have  $$ \frac{a+b}{2} \cdot \frac{a+b}{2} \geqslant a b \geqslant 1 \quad \text { and } \quad \frac{a+b}{2} \cdot c \geqslant \frac{1+1}{2}=1 . $$


Solution: Let $f(x)=\ln \left(1+x^{2}\right)$. Then the inequality (1) to be shown is equivalent to  $$ \frac{f(a)+f(b)+f(c)}{3} \leqslant f\left(\frac{a+b+c}{3}\right), $$  while (2) becomes  $$ \frac{f(x)+f(y)}{2} \leqslant f\left(\frac{x+y}{2}\right) $$  for $x y \geqslant 1$. Without loss of generality, assume $a \geqslant b \geqslant c$. From the Claim in Solution 1, we have  $$ \frac{f(a)+f(b)+f(c)}{3} \leqslant \frac{f(a)+2 f\left(\frac{b+c}{2}\right)}{3} . $$  Note that $a \geqslant 1$ and $\frac{b+c}{2} \geqslant \sqrt{b c} \geqslant 1$. Since  $$ f^{\prime \prime}(x)=\frac{2\left(1-x^{2}\right)}{\left(1+x^{2}\right)^{2}} $$  we know that $f$ is concave on $[1, \infty)$. Then we can apply Jensen's Theorem to get  $$ \frac{f(a)+2 f\left(\frac{b+c}{2}\right)}{3} \leqslant f\left(\frac{a+2 \cdot \frac{b+c}{2}}{3}\right)=f\left(\frac{a+b+c}{3}\right) . $$  This completes the proof.


Find the smallest real constant $C$ such that for any positive real numbers $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ (not necessarily distinct), one can always choose distinct subscripts $i, j, k$ and $l$ such that  $$ \left|\frac{a_{i}}{a_{j}}-\frac{a_{k}}{a_{l}}\right| \leqslant C $$

Solution: We first show that $C \leqslant \frac{1}{2}$. For any positive real numbers $a_{1} \leqslant a_{2} \leqslant a_{3} \leqslant a_{4} \leqslant a_{5}$, consider the five fractions  $$ \frac{a_{1}}{a_{2}}, \frac{a_{3}}{a_{4}}, \frac{a_{1}}{a_{5}}, \frac{a_{2}}{a_{3}}, \frac{a_{4}}{a_{5}} . $$  Each of them lies in the interval $(0,1]$. Therefore, by the Pigeonhole Principle, at least three of them must lie in $\left(0, \frac{1}{2}\right]$ or lie in $\left(\frac{1}{2}, 1\right]$ simultaneously. In particular, there must be two consecutive terms in (2) which belong to an interval of length $\frac{1}{2}$ (here, we regard $\frac{a_{1}}{a_{2}}$ and $\frac{a_{4}}{a_{5}}$ as consecutive). In other words, the difference of these two fractions is less than $\frac{1}{2}$. As the indices involved in these two fractions are distinct, we can choose them to be $i, j, k, l$ and conclude that $C \leqslant \frac{1}{2}$.  Next, we show that $C=\frac{1}{2}$ is best possible. Consider the numbers $1,2,2,2, n$ where $n$ is a large real number. The fractions formed by two of these numbers in ascending order are $\frac{1}{n}, \frac{2}{n}, \frac{1}{2}, \frac{2}{2}, \frac{2}{1}, \frac{n}{2}, \frac{n}{1}$. Since the indices $i, j, k, l$ are distinct, $\frac{1}{n}$ and $\frac{2}{n}$ cannot be chosen simultaneously. Therefore the minimum value of the left-hand side of (1) is $\frac{1}{2}-\frac{2}{n}$. When $n$ tends to infinity, this value approaches $\frac{1}{2}$, and so $C$ cannot be less than $\frac{1}{2}$.  These conclude that $C=\frac{1}{2}$ is the smallest possible choice. Comment. The conclusion still holds if $a_{1}, a_{2}, \ldots, a_{5}$ are pairwise distinct, since in the construction, we may replace the 2's by real numbers sufficiently close to 2 .  There are two possible simplifications for this problem: (i) the answer $C=\frac{1}{2}$ is given to the contestants; or (ii) simply ask the contestants to prove the inequality (1) for $C=\frac{1}{2}$.


Find all integers $n \geqslant 3$ with the following property: for all real numbers $a_{1}, a_{2}, \ldots, a_{n}$ and $b_{1}, b_{2}, \ldots, b_{n}$ satisfying $\left|a_{k}\right|+\left|b_{k}\right|=1$ for $1 \leqslant k \leqslant n$, there exist $x_{1}, x_{2}, \ldots, x_{n}$, each of which is either -1 or 1 , such that  $$ \left|\sum_{k=1}^{n} x_{k} a_{k}\right|+\left|\sum_{k=1}^{n} x_{k} b_{k}\right| \leqslant 1 $$

Solution: For any even integer $n \geqslant 4$, we consider the case  $$ a_{1}=a_{2}=\cdots=a_{n-1}=b_{n}=0 \quad \text { and } \quad b_{1}=b_{2}=\cdots=b_{n-1}=a_{n}=1 $$  The condition $\left|a_{k}\right|+\left|b_{k}\right|=1$ is satisfied for each $1 \leqslant k \leqslant n$. No matter how we choose each $x_{k}$, both sums $\sum_{k=1}^{n} x_{k} a_{k}$ and $\sum_{k=1}^{n} x_{k} b_{k}$ are odd integers. This implies $\left|\sum_{k=1}^{n} x_{k} a_{k}\right| \geqslant 1$ and $\left|\sum_{k=1}^{n} x_{k} b_{k}\right| \geqslant 1$, which shows (1) cannot hold.  For any odd integer $n \geqslant 3$, we may assume without loss of generality $b_{k} \geqslant 0$ for $1 \leqslant k \leqslant n$ (this can be done by flipping the pair $\left(a_{k}, b_{k}\right)$ to $\left(-a_{k},-b_{k}\right)$ and $x_{k}$ to $-x_{k}$ if necessary) and $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{m} \geqslant 0>a_{m+1} \geqslant \cdots \geqslant a_{n}$. We claim that the choice $x_{k}=(-1)^{k+1}$ for $1 \leqslant k \leqslant n$ will work. Define  $$ s=\sum_{k=1}^{m} x_{k} a_{k} \quad \text { and } \quad t=-\sum_{k=m+1}^{n} x_{k} a_{k} . $$  Note that  $$ s=\left(a_{1}-a_{2}\right)+\left(a_{3}-a_{4}\right)+\cdots \geqslant 0 $$  by the assumption $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{m}$ (when $m$ is odd, there is a single term $a_{m}$ at the end, which is also positive). Next, we have  $$ s=a_{1}-\left(a_{2}-a_{3}\right)-\left(a_{4}-a_{5}\right)-\cdots \leqslant a_{1} \leqslant 1 $$  Similarly,  $$ t=\left(-a_{n}+a_{n-1}\right)+\left(-a_{n-2}+a_{n-3}\right)+\cdots \geqslant 0 $$  and  $$ t=-a_{n}+\left(a_{n-1}-a_{n-2}\right)+\left(a_{n-3}-a_{n-4}\right)+\cdots \leqslant-a_{n} \leqslant 1 . $$  From the condition, we have $a_{k}+b_{k}=1$ for $1 \leqslant k \leqslant m$ and $-a_{k}+b_{k}=1$ for $m+1 \leqslant k \leqslant n$. It follows that $\sum_{k=1}^{n} x_{k} a_{k}=s-t$ and $\sum_{k=1}^{n} x_{k} b_{k}=1-s-t$. Hence it remains to prove  $$ |s-t|+|1-s-t| \leqslant 1 $$  under the constraint $0 \leqslant s, t \leqslant 1$. By symmetry, we may assume $s \geqslant t$. If $1-s-t \geqslant 0$, then we have  $$ |s-t|+|1-s-t|=s-t+1-s-t=1-2 t \leqslant 1 $$  If $1-s-t \leqslant 0$, then we have  $$ |s-t|+|1-s-t|=s-t-1+s+t=2 s-1 \leqslant 1 $$  Hence, the inequality is true in both cases. These show $n$ can be any odd integer greater than or equal to 3 .


Solution: The even case can be handled in the same way as   Firstly, for $n=3$, we may assume without loss of generality $a_{1} \geqslant a_{2} \geqslant a_{3} \geqslant 0$ and $b_{1}=a_{1}-1$ (if $b_{1}=1-a_{1}$, we may replace each $b_{k}$ by $-b_{k}$ ).  - Case 1. $b_{2}=a_{2}-1$ and $b_{3}=a_{3}-1$, in which case we take $\left(x_{1}, x_{2}, x_{3}\right)=(1,-1,1)$.  Let $c=a_{1}-a_{2}+a_{3}$ so that $0 \leqslant c \leqslant 1$. Then $\left|b_{1}-b_{2}+b_{3}\right|=\left|a_{1}-a_{2}+a_{3}-1\right|=1-c$ and hence $|c|+\left|b_{1}-b_{2}+b_{3}\right|=1$.  - Case 2. $b_{2}=1-a_{2}$ and $b_{3}=1-a_{3}$, in which case we take $\left(x_{1}, x_{2}, x_{3}\right)=(1,-1,1)$.  Let $c=a_{1}-a_{2}+a_{3}$ so that $0 \leqslant c \leqslant 1$. Since $a_{3} \leqslant a_{2}$ and $a_{1} \leqslant 1$, we have  $$ c-1 \leqslant b_{1}-b_{2}+b_{3}=a_{1}+a_{2}-a_{3}-1 \leqslant 1-c $$  This gives $\left|b_{1}-b_{2}+b_{3}\right| \leqslant 1-c$ and hence $|c|+\left|b_{1}-b_{2}+b_{3}\right| \leqslant 1$.  - Case 3. $b_{2}=a_{2}-1$ and $b_{3}=1-a_{3}$, in which case we take $\left(x_{1}, x_{2}, x_{3}\right)=(-1,1,1)$.  Let $c=-a_{1}+a_{2}+a_{3}$. If $c \geqslant 0$, then $a_{3} \leqslant 1$ and $a_{2} \leqslant a_{1}$ imply  $$ c-1 \leqslant-b_{1}+b_{2}+b_{3}=-a_{1}+a_{2}-a_{3}+1 \leqslant 1-c $$  If $c<0$, then $a_{1} \leqslant a_{2}+1$ and $a_{3} \geqslant 0$ imply  $$ -c-1 \leqslant-b_{1}+b_{2}+b_{3}=-a_{1}+a_{2}-a_{3}+1 \leqslant 1+c $$  In both cases, we get $\left|-b_{1}+b_{2}+b_{3}\right| \leqslant 1-|c|$ and hence $|c|+\left|-b_{1}+b_{2}+b_{3}\right| \leqslant 1$.  - Case 4. $b_{2}=1-a_{2}$ and $b_{3}=a_{3}-1$, in which case we take $\left(x_{1}, x_{2}, x_{3}\right)=(-1,1,1)$.  Let $c=-a_{1}+a_{2}+a_{3}$. If $c \geqslant 0$, then $a_{2} \leqslant 1$ and $a_{3} \leqslant a_{1}$ imply  $$ c-1 \leqslant-b_{1}+b_{2}+b_{3}=-a_{1}-a_{2}+a_{3}+1 \leqslant 1-c $$  If $c<0$, then $a_{1} \leqslant a_{3}+1$ and $a_{2} \geqslant 0$ imply  $$ -c-1 \leqslant-b_{1}+b_{2}+b_{3}=-a_{1}-a_{2}+a_{3}+1 \leqslant 1+c $$  In both cases, we get $\left|-b_{1}+b_{2}+b_{3}\right| \leqslant 1-|c|$ and hence $|c|+\left|-b_{1}+b_{2}+b_{3}\right| \leqslant 1$. We have found $x_{1}, x_{2}, x_{3}$ satisfying (1) in each case for $n=3$. Now, let $n \geqslant 5$ be odd and suppose the result holds for any smaller odd cases. Again we may assume $a_{k} \geqslant 0$ for each $1 \leqslant k \leqslant n$. By the Pigeonhole Principle, there are at least three indices $k$ for which $b_{k}=a_{k}-1$ or $b_{k}=1-a_{k}$. Without loss of generality, suppose $b_{k}=a_{k}-1$ for $k=1,2,3$. Again by the Pigeonhole Principle, as $a_{1}, a_{2}, a_{3}$ lies between 0 and 1 , the difference of two of them is at most $\frac{1}{2}$. By changing indices if necessary, we may assume $0 \leqslant d=a_{1}-a_{2} \leqslant \frac{1}{2}$.  By the inductive hypothesis, we can choose $x_{3}, x_{4}, \ldots, x_{n}$ such that $a^{\prime}=\sum_{k=3}^{n} x_{k} a_{k}$ and $b^{\prime}=\sum_{k=3}^{n} x_{k} b_{k}$ satisfy $\left|a^{\prime}\right|+\left|b^{\prime}\right| \leqslant 1$. We may further assume $a^{\prime} \geqslant 0$.  - Case 1. $b^{\prime} \geqslant 0$, in which case we take $\left(x_{1}, x_{2}\right)=(-1,1)$.  We have $\left|-a_{1}+a_{2}+a^{\prime}\right|+\left|-\left(a_{1}-1\right)+\left(a_{2}-1\right)+b^{\prime}\right|=\left|-d+a^{\prime}\right|+\left|-d+b^{\prime}\right| \leqslant$ $\max \left\{a^{\prime}+b^{\prime}-2 d, a^{\prime}-b^{\prime}, b^{\prime}-a^{\prime}, 2 d-a^{\prime}-b^{\prime}\right\} \leqslant 1$ since $0 \leqslant a^{\prime}, b^{\prime}, a^{\prime}+b^{\prime} \leqslant 1$ and $0 \leqslant d \leqslant \frac{1}{2}$.  - Case 2. $0>b^{\prime} \geqslant-a^{\prime}$, in which case we take $\left(x_{1}, x_{2}\right)=(-1,1)$.  We have $\left|-a_{1}+a_{2}+a^{\prime}\right|+\left|-\left(a_{1}-1\right)+\left(a_{2}-1\right)+b^{\prime}\right|=\left|-d+a^{\prime}\right|+\left|-d+b^{\prime}\right|$. If $-d+a^{\prime} \geqslant 0$, this equals $a^{\prime}-b^{\prime}=\left|a^{\prime}\right|+\left|b^{\prime}\right| \leqslant 1$. If $-d+a^{\prime}<0$, this equals $2 d-a^{\prime}-b^{\prime} \leqslant 2 d \leqslant 1$.  - Case 3. $b^{\prime}<-a^{\prime}$, in which case we take $\left(x_{1}, x_{2}\right)=(1,-1)$.  We have $\left|a_{1}-a_{2}+a^{\prime}\right|+\left|\left(a_{1}-1\right)-\left(a_{2}-1\right)+b^{\prime}\right|=\left|d+a^{\prime}\right|+\left|d+b^{\prime}\right|$. If $d+b^{\prime} \geqslant 0$, this equals $2 d+a^{\prime}+b^{\prime}<2 d \leqslant 1$. If $d+b^{\prime}<0$, this equals $a^{\prime}-b^{\prime}=\left|a^{\prime}\right|+\left|b^{\prime}\right| \leqslant 1$.  Therefore, we have found $x_{1}, x_{2}, \ldots, x_{n}$ satisfying (1) in each case. By induction, the property holds for all odd integers $n \geqslant 3$.


Denote by $\mathbb{R}^{+}$the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$such that  $$ x f\left(x^{2}\right) f(f(y))+f(y f(x))=f(x y)\left(f\left(f\left(x^{2}\right)\right)+f\left(f\left(y^{2}\right)\right)\right) $$  for all positive real numbers $x$ and $y$.

Solution: Taking $x=y=1$ in (1), we get $f(1) f(f(1))+f(f(1))=2 f(1) f(f(1))$ and hence $f(1)=1$. Putting $x=1$ in (1), we have $f(f(y))+f(y)=f(y)\left(1+f\left(f\left(y^{2}\right)\right)\right)$ so that  $$ f(f(y))=f(y) f\left(f\left(y^{2}\right)\right) $$  Putting $y=1$ in (1), we get $x f\left(x^{2}\right)+f(f(x))=f(x)\left(f\left(f\left(x^{2}\right)\right)+1\right)$. Using (9), this gives  $$ x f\left(x^{2}\right)=f(x) $$  Replace $y$ by $\frac{1}{x}$ in (1). Then we have  $$ x f\left(x^{2}\right) f\left(f\left(\frac{1}{x}\right)\right)+f\left(\frac{f(x)}{x}\right)=f\left(f\left(x^{2}\right)\right)+f\left(f\left(\frac{1}{x^{2}}\right)\right) . $$  The relation (10) shows $f\left(\frac{f(x)}{x}\right)=f\left(f\left(x^{2}\right)\right)$. Also, using (9) with $y=\frac{1}{x}$ and using (10) again, the last equation reduces to  $$ f(x) f\left(\frac{1}{x}\right)=1 $$  Replace $x$ by $\frac{1}{x}$ and $y$ by $\frac{1}{y}$ in (1) and apply (11). We get  $$ \frac{1}{x f\left(x^{2}\right) f(f(y))}+\frac{1}{f(y f(x))}=\frac{1}{f(x y)}\left(\frac{1}{f\left(f\left(x^{2}\right)\right)}+\frac{1}{f\left(f\left(y^{2}\right)\right)}\right) . $$  Clearing denominators, we can use (1) to simplify the numerators and obtain  $$ f(x y)^{2} f\left(f\left(x^{2}\right)\right) f\left(f\left(y^{2}\right)\right)=x f\left(x^{2}\right) f(f(y)) f(y f(x)) $$  Using (9) and (10), this is the same as  $$ f(x y)^{2} f(f(x))=f(x)^{2} f(y) f(y f(x)) $$  Substitute $y=f(x)$ in (12) and apply (10) (with $x$ replaced by $f(x)$ ). We have  $$ f(x f(x))^{2}=f(x) f(f(x)) $$  Taking $y=x$ in (12), squaring both sides, and using (10) and (13), we find that  $$ f(f(x))=x^{4} f(x)^{3} $$  Finally, we combine (9), (10) and (14) to get  $$ y^{4} f(y)^{3} \stackrel{(14)}{=} f(f(y)) \stackrel{(9)}{=} f(y) f\left(f\left(y^{2}\right)\right) \stackrel{(14)}{=} f(y) y^{8} f\left(y^{2}\right)^{3} \stackrel{(10)}{=} y^{5} f(y)^{4}, $$  which implies $f(y)=\frac{1}{y}$. This is a solution by the checking in Solution 1.


(a) Prove that for every positive integer $n$, there exists a fraction $\frac{a}{b}$ where $a$ and $b$ are integers satisfying $0<b \leqslant \sqrt{n}+1$ and $\sqrt{n} \leqslant \frac{a}{b} \leqslant \sqrt{n+1}$. (b) Prove that there are infinitely many positive integers $n$ such that there is no fraction $\frac{a}{b}$ where $a$ and $b$ are integers satisfying $0<b \leqslant \sqrt{n}$ and $\sqrt{n} \leqslant \frac{a}{b} \leqslant \sqrt{n+1}$.

Solution: (a) Let $r$ be the unique positive integer for which $r^{2} \leqslant n<(r+1)^{2}$. Write $n=r^{2}+s$. Then we have $0 \leqslant s \leqslant 2 r$. We discuss in two cases according to the parity of $s$.  - Case 1. $s$ is even.  Consider the number $\left(r+\frac{s}{2 r}\right)^{2}=r^{2}+s+\left(\frac{s}{2 r}\right)^{2}$. We find that  $$ n=r^{2}+s \leqslant r^{2}+s+\left(\frac{s}{2 r}\right)^{2} \leqslant r^{2}+s+1=n+1 $$  It follows that  $$ \sqrt{n} \leqslant r+\frac{s}{2 r} \leqslant \sqrt{n+1} $$  Since $s$ is even, we can choose the fraction $r+\frac{s}{2 r}=\frac{r^{2}+(s / 2)}{r}$ since $r \leqslant \sqrt{n}$.  - Case 2. $s$ is odd.  Consider the number $\left(r+1-\frac{2 r+1-s}{2(r+1)}\right)^{2}=(r+1)^{2}-(2 r+1-s)+\left(\frac{2 r+1-s}{2(r+1)}\right)^{2}$. We find that  $$ \begin{aligned} n=r^{2}+s=(r+1)^{2}-(2 r+1-s) & \leqslant(r+1)^{2}-(2 r+1-s)+\left(\frac{2 r+1-s}{2(r+1)}\right)^{2} \\ & \leqslant(r+1)^{2}-(2 r+1-s)+1=n+1 \end{aligned} $$  It follows that  $$ \sqrt{n} \leqslant r+1-\frac{2 r+1-s}{2(r+1)} \leqslant \sqrt{n+1} $$  Since $s$ is odd, we can choose the fraction $(r+1)-\frac{2 r+1-s}{2(r+1)}=\frac{(r+1)^{2}-r+((s-1) / 2)}{r+1}$ since $r+1 \leqslant \sqrt{n}+1$. (b) We show that for every positive integer $r$, there is no fraction $\frac{a}{b}$ with $b \leqslant \sqrt{r^{2}+1}$ such that $\sqrt{r^{2}+1} \leqslant \frac{a}{b} \leqslant \sqrt{r^{2}+2}$. Suppose on the contrary that such a fraction exists. Since $b \leqslant \sqrt{r^{2}+1}<r+1$ and $b$ is an integer, we have $b \leqslant r$. Hence,  $$ (b r)^{2}<b^{2}\left(r^{2}+1\right) \leqslant a^{2} \leqslant b^{2}\left(r^{2}+2\right) \leqslant b^{2} r^{2}+2 b r<(b r+1)^{2} $$  This shows the square number $a^{2}$ is strictly bounded between the two consecutive squares $(b r)^{2}$ and $(b r+1)^{2}$, which is impossible. Hence, we have found infinitely many $n=r^{2}+1$ for which there is no fraction of the desired form.


The equation  $$ (x-1)(x-2) \cdots(x-2016)=(x-1)(x-2) \cdots(x-2016) $$  is written on the board. One tries to erase some linear factors from both sides so that each side still has at least one factor, and the resulting equation has no real roots. Find the least number of linear factors one needs to erase to achieve this.

Solution: Since there are 2016 common linear factors on both sides, we need to erase at least 2016 factors. We claim that the equation has no real roots if we erase all factors $(x-k)$ on the left-hand side with $k \equiv 2,3(\bmod 4)$, and all factors $(x-m)$ on the right-hand side with $m \equiv 0,1(\bmod 4)$. Therefore, it suffices to show that no real number $x$ satisfies  $$ \prod_{j=0}^{503}(x-4 j-1)(x-4 j-4)=\prod_{j=0}^{503}(x-4 j-2)(x-4 j-3) $$  - Case 1. $x=1,2, \ldots, 2016$.  In this case, one side of (1) is zero while the other side is not. This shows $x$ cannot satisfy (1).  - Case 2. $4 k+1<x<4 k+2$ or $4 k+3<x<4 k+4$ for some $k=0,1, \ldots, 503$.  For $j=0,1, \ldots, 503$ with $j \neq k$, the product $(x-4 j-1)(x-4 j-4)$ is positive. For $j=k$, the product $(x-4 k-1)(x-4 k-4)$ is negative. This shows the left-hand side of (1) is negative. On the other hand, each product $(x-4 j-2)(x-4 j-3)$ on the right-hand side of (1) is positive. This yields a contradiction.  - Case 3. $x<1$ or $x>2016$ or $4 k<x<4 k+1$ for some $k=1,2, \ldots, 503$.  The equation (1) can be rewritten as  $$ 1=\prod_{j=0}^{503} \frac{(x-4 j-1)(x-4 j-4)}{(x-4 j-2)(x-4 j-3)}=\prod_{j=0}^{503}\left(1-\frac{2}{(x-4 j-2)(x-4 j-3)}\right) $$  Note that $(x-4 j-2)(x-4 j-3)>2$ for $0 \leqslant j \leqslant 503$ in this case. So each term in the product lies strictly between 0 and 1 , and the whole product must be less than 1 , which is impossible.  - Case 4. $4 k+2<x<4 k+3$ for some $k=0,1, \ldots, 503$.  This time we rewrite (1) as  $$ \begin{aligned} 1 & =\frac{x-1}{x-2} \cdot \frac{x-2016}{x-2015} \prod_{j=1}^{503} \frac{(x-4 j)(x-4 j-1)}{(x-4 j+1)(x-4 j-2)} \\ & =\frac{x-1}{x-2} \cdot \frac{x-2016}{x-2015} \prod_{j=1}^{503}\left(1+\frac{2}{(x-4 j+1)(x-4 j-2)}\right) \end{aligned} $$  Clearly, $\frac{x-1}{x-2}$ and $\frac{x-2016}{x-2015}$ are both greater than 1 . For the range of $x$ in this case, each term in the product is also greater than 1 . Then the right-hand side must be greater than 1 and hence a contradiction arises.  From the four cases, we conclude that (1) has no real roots. Hence, the minimum number of linear factors to be erased is 2016.  Comment. We discuss the general case when 2016 is replaced by a positive integer $n$. The above solution works equally well when $n$ is divisible by 4 .  If $n \equiv 2(\bmod 4)$, one may leave $l(x)=(x-1)(x-2) \cdots\left(x-\frac{n}{2}\right)$ on the left-hand side and $r(x)=\left(x-\frac{n}{2}-1\right)\left(x-\frac{n}{2}-2\right) \cdots(x-n)$ on the right-hand side. One checks that for $x<\frac{n+1}{2}$, we have $|l(x)|<|r(x)|$, while for $x>\frac{n+1}{2}$, we have $|l(x)|>|r(x)|$.  If $n \equiv 3(\bmod 4)$, one may leave $l(x)=(x-1)(x-2) \cdots\left(x-\frac{n+1}{2}\right)$ on the left-hand side and $r(x)=\left(x-\frac{n+3}{2}\right)\left(x-\frac{x+5}{2}\right) \cdots(x-n)$ on the right-hand side. For $x<1$ or $\frac{n+1}{2}<x<\frac{n+3}{2}$, we have $l(x)>0>r(x)$. For $1<x<\frac{n+1}{2}$, we have $|l(x)|<|r(x)|$. For $x>\frac{n+3}{2}$, we have $|l(x)|>|r(x)|$.  If $n \equiv 1(\bmod 4)$, as the proposer mentioned, the situation is a bit more out of control. Since the construction for $n-1 \equiv 0(\bmod 4)$ works, the answer can be either $n$ or $n-1$. For $n=5$, we can leave the products $(x-1)(x-2)(x-3)(x-4)$ and $(x-5)$. For $n=9$, the only example that works is $l(x)=(x-1)(x-2)(x-9)$ and $r(x)=(x-3)(x-4) \cdots(x-8)$, while there seems to be no such partition for $n=13$.


Denote by $\mathbb{R}$ the set of all real numbers. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(0) \neq 0$ and  $$ f(x+y)^{2}=2 f(x) f(y)+\max \left\{f\left(x^{2}\right)+f\left(y^{2}\right), f\left(x^{2}+y^{2}\right)\right\} $$  for all real numbers $x$ and $y$.

Solution: Taking $x=y=0$ in (1), we get $f(0)^{2}=2 f(0)^{2}+\max \{2 f(0), f(0)\}$. If $f(0)>0$, then $f(0)^{2}+2 f(0)=0$ gives no positive solution. If $f(0)<0$, then $f(0)^{2}+f(0)=0$ gives $f(0)=-1$. Putting $y=0$ in (1), we have $f(x)^{2}=-2 f(x)+f\left(x^{2}\right)$, which is the same as $(f(x)+1)^{2}=f\left(x^{2}\right)+1$. Let $g(x)=f(x)+1$. Then for any $x \in \mathbb{R}$, we have  $$ g\left(x^{2}\right)=g(x)^{2} \geqslant 0 $$  From (1), we find that $f(x+y)^{2} \geqslant 2 f(x) f(y)+f\left(x^{2}\right)+f\left(y^{2}\right)$. In terms of $g$, this becomes $(g(x+y)-1)^{2} \geqslant 2(g(x)-1)(g(y)-1)+g\left(x^{2}\right)+g\left(y^{2}\right)-2$. Using (2), this means  $$ (g(x+y)-1)^{2} \geqslant(g(x)+g(y)-1)^{2}-1 $$  Putting $x=1$ in (2), we get $g(1)=0$ or 1 . The two cases are handled separately.  - Case 1. $g(1)=0$, which is the same as $f(1)=-1$.  We put $x=-1$ and $y=0$ in (1). This gives $f(-1)^{2}=-2 f(-1)-1$, which forces $f(-1)=-1$. Next, we take $x=-1$ and $y=1$ in (1) to get $1=2+\max \{-2, f(2)\}$. This clearly implies $1=2+f(2)$ and hence $f(2)=-1$, that is, $g(2)=0$. From (2), we can prove inductively that $g\left(2^{2^{n}}\right)=g(2)^{2^{n}}=0$ for any $n \in \mathbb{N}$. Substitute $y=2^{2^{n}}-x$ in (3). We obtain  $$ \left(g(x)+g\left(2^{2^{n}}-x\right)-1\right)^{2} \leqslant\left(g\left(2^{2^{n}}\right)-1\right)^{2}+1=2 $$  For any fixed $x \geqslant 0$, we consider $n$ to be sufficiently large so that $2^{2^{n}}-x>0$. From (2), this implies $g\left(2^{2^{n}}-x\right) \geqslant 0$ so that $g(x) \leqslant 1+\sqrt{2}$. Using (2) again, we get  $$ g(x)^{2^{n}}=g\left(x^{2^{n}}\right) \leqslant 1+\sqrt{2} $$  for any $n \in \mathbb{N}$. Therefore, $|g(x)| \leqslant 1$ for any $x \geqslant 0$. If there exists $a \in \mathbb{R}$ for which $g(a) \neq 0$, then for sufficiently large $n$ we must have $g\left(\left(a^{2}\right)^{\frac{1}{2^{n}}}\right)=g\left(a^{2}\right)^{\frac{1}{2^{n}}}>\frac{1}{2}$. By taking $x=-y=-\left(a^{2}\right)^{\frac{1}{2^{n}}}$ in (1), we obtain  $$ \begin{aligned} 1 & =2 f(x) f(-x)+\max \left\{2 f\left(x^{2}\right), f\left(2 x^{2}\right)\right\} \\ & =2(g(x)-1)(g(-x)-1)+\max \left\{2\left(g\left(x^{2}\right)-1\right), g\left(2 x^{2}\right)-1\right\} \\ & \leqslant 2\left(-\frac{1}{2}\right)\left(-\frac{1}{2}\right)+0=\frac{1}{2} \end{aligned} $$  since $|g(-x)|=|g(x)| \in\left(\frac{1}{2}, 1\right]$ by (2) and the choice of $x$, and since $g(z) \leqslant 1$ for $z \geqslant 0$. This yields a contradiction and hence $g(x)=0$ must hold for any $x$. This means $f(x)=-1$ for any $x \in \mathbb{R}$, which clearly satisfies (1).  - Case 2. $g(1)=1$, which is the same as $f(1)=0$.  We put $x=-1$ and $y=1$ in (1) to get $1=\max \{0, f(2)\}$. This clearly implies $f(2)=1$ and hence $g(2)=2$. Setting $x=2 n$ and $y=2$ in (3), we have  $$ (g(2 n+2)-1)^{2} \geqslant(g(2 n)+1)^{2}-1 $$  By induction on $n$, it is easy to prove that $g(2 n) \geqslant n+1$ for all $n \in \mathbb{N}$. For any real number $a>1$, we choose a large $n \in \mathbb{N}$ and take $k$ to be the positive integer such that $2 k \leqslant a^{2^{n}}<2 k+2$. From (2) and (3), we have  $$ \left(g(a)^{2^{n}}-1\right)^{2}+1=\left(g\left(a^{2^{n}}\right)-1\right)^{2}+1 \geqslant\left(g(2 k)+g\left(a^{2^{n}}-2 k\right)-1\right)^{2} \geqslant k^{2}>\frac{1}{4}\left(a^{2^{n}}-2\right)^{2} $$  since $g\left(a^{2^{n}}-2 k\right) \geqslant 0$. For large $n$, this clearly implies $g(a)^{2^{n}}>1$. Thus,  $$ \left(g(a)^{2^{n}}\right)^{2}>\left(g(a)^{2^{n}}-1\right)^{2}+1>\frac{1}{4}\left(a^{2^{n}}-2\right)^{2} $$  This yields  $$ g(a)^{2^{n}}>\frac{1}{2}\left(a^{2^{n}}-2\right) $$  Note that  $$ \frac{a^{2^{n}}}{a^{2^{n}}-2}=1+\frac{2}{a^{2^{n}}-2} \leqslant\left(1+\frac{2}{2^{n}\left(a^{2^{n}}-2\right)}\right)^{2^{n}} $$  by binomial expansion. This can be rewritten as  $$ \left(a^{2^{n}}-2\right)^{\frac{1}{2^{n}}} \geqslant \frac{a}{1+\frac{2}{2^{n}\left(a^{2^{n}}-2\right)}} $$  Together with (4), we conclude $g(a) \geqslant a$ by taking $n$ sufficiently large. Consider $x=n a$ and $y=a>1$ in (3). This gives $(g((n+1) a)-1)^{2} \geqslant(g(n a)+g(a)-1)^{2}-1$. By induction on $n$, it is easy to show $g(n a) \geqslant(n-1)(g(a)-1)+a$ for any $n \in \mathbb{N}$. We choose a large $n \in \mathbb{N}$ and take $k$ to be the positive integer such that $k a \leqslant 2^{2^{n}}<(k+1) a$. Using (2) and (3), we have $2^{2^{n+1}}>\left(2^{2^{n}}-1\right)^{2}+1=\left(g\left(2^{2^{n}}\right)-1\right)^{2}+1 \geqslant\left(g\left(2^{2^{n}}-k a\right)+g(k a)-1\right)^{2} \geqslant((k-1)(g(a)-1)+a-1)^{2}$, from which it follows that  $$ 2^{2^{n}} \geqslant(k-1)(g(a)-1)+a-1>\frac{2^{2^{n}}}{a}(g(a)-1)-2(g(a)-1)+a-1 $$  holds for sufficiently large $n$. Hence, we must have $\frac{g(a)-1}{a} \leqslant 1$, which implies $g(a) \leqslant a+1$ for any $a>1$. Then for large $n \in \mathbb{N}$, from (3) and (2) we have  $$ 4 a^{2^{n+1}}=\left(2 a^{2^{n}}\right)^{2} \geqslant\left(g\left(2 a^{2^{n}}\right)-1\right)^{2} \geqslant\left(2 g\left(a^{2^{n}}\right)-1\right)^{2}-1=\left(2 g(a)^{2^{n}}-1\right)^{2}-1 $$  This implies  $$ 2 a^{2^{n}}>\frac{1}{2}\left(1+\sqrt{4 a^{2^{n+1}}+1}\right) \geqslant g(a)^{2^{n}} $$  When $n$ tends to infinity, this forces $g(a) \leqslant a$. Together with $g(a) \geqslant a$, we get $g(a)=a$ for all real numbers $a>1$, that is, $f(a)=a-1$ for all $a>1$.  Finally, for any $x \in \mathbb{R}$, we choose $y$ sufficiently large in (1) so that $y, x+y>1$. This gives $(x+y-1)^{2}=2 f(x)(y-1)+\max \left\{f\left(x^{2}\right)+y^{2}-1, x^{2}+y^{2}-1\right\}$, which can be rewritten as  $$ 2(x-1-f(x)) y=-x^{2}+2 x-2-2 f(x)+\max \left\{f\left(x^{2}\right), x^{2}\right\} $$  As the right-hand side is fixed, this can only hold for all large $y$ when $f(x)=x-1$. We now check that this function satisfies (1). Indeed, we have  $$ \begin{aligned} f(x+y)^{2} & =(x+y-1)^{2}=2(x-1)(y-1)+\left(x^{2}+y^{2}-1\right) \\ & =2 f(x) f(y)+\max \left\{f\left(x^{2}\right)+f\left(y^{2}\right), f\left(x^{2}+y^{2}\right)\right\} . \end{aligned} $$


Solution: Taking $x=y=0$ in (1), we get $f(0)^{2}=2 f(0)^{2}+\max \{2 f(0), f(0)\}$. If $f(0)>0$, then $f(0)^{2}+2 f(0)=0$ gives no positive solution. If $f(0)<0$, then $f(0)^{2}+f(0)=0$ gives $f(0)=-1$. Putting $y=0$ in (1), we have  $$ f(x)^{2}=-2 f(x)+f\left(x^{2}\right) $$  Replace $x$ by $-x$ in (5) and compare with (5) again. We get $f(x)^{2}+2 f(x)=f(-x)^{2}+2 f(-x)$, which implies  $$ f(x)=f(-x) \quad \text { or } \quad f(x)+f(-x)=-2 $$  Taking $x=y$ and $x=-y$ respectively in (1) and comparing the two equations obtained, we have  $$ f(2 x)^{2}-2 f(x)^{2}=1-2 f(x) f(-x) . $$  Combining (6) and (7) to eliminate $f(-x)$, we find that $f(2 x)$ can be $\pm 1$ (when $f(x)=f(-x)$ ) or $\pm(2 f(x)+1)$ (when $f(x)+f(-x)=-2$ ).  We prove the following.  - Claim. $f(x)+f(-x)=-2$ for any $x \in \mathbb{R}$.  Proof. Suppose there exists $a \in \mathbb{R}$ such that $f(a)+f(-a) \neq-2$. Then $f(a)=f(-a) \neq-1$ and we may assume $a>0$. We first show that $f(a) \neq 1$. Suppose $f(a)=1$. Consider $y=a$ in (7). We get $f(2 a)^{2}=1$. Taking $x=y=a$ in (1), we have $1=2+\max \left\{2 f\left(a^{2}\right), f\left(2 a^{2}\right)\right\}$. From (5), $f\left(a^{2}\right)=3$ so that $1 \geqslant 2+6$. This is impossible, and thus $f(a) \neq 1$.  As $f(a) \neq \pm 1$, we have $f(a)= \pm\left(2 f\left(\frac{a}{2}\right)+1\right)$. Similarly, $f(-a)= \pm\left(2 f\left(-\frac{a}{2}\right)+1\right)$. These two expressions are equal since $f(a)=f(-a)$. If $f\left(\frac{a}{2}\right)=f\left(-\frac{a}{2}\right)$, then the above argument works when we replace $a$ by $\frac{a}{2}$. In particular, we have $f(a)^{2}=f\left(2 \cdot \frac{a}{2}\right)^{2}=1$, which is a contradiction. Therefore, (6) forces $f\left(\frac{a}{2}\right)+f\left(-\frac{a}{2}\right)=-2$. Then we get  $$ \pm\left(2 f\left(\frac{a}{2}\right)+1\right)= \pm\left(-2 f\left(\frac{a}{2}\right)-3\right) . $$  For any choices of the two signs, we either get a contradiction or $f\left(\frac{a}{2}\right)=-1$, in which case $f\left(\frac{a}{2}\right)=f\left(-\frac{a}{2}\right)$ and hence $f(a)= \pm 1$ again. Therefore, there is no such real number $a$ and the Claim follows.  Replace $x$ and $y$ by $-x$ and $-y$ in (1) respectively and compare with (1). We get  $$ f(x+y)^{2}-2 f(x) f(y)=f(-x-y)^{2}-2 f(-x) f(-y) $$  Using the Claim, this simplifies to $f(x+y)=f(x)+f(y)+1$. In addition, (5) can be rewritten as $(f(x)+1)^{2}=f\left(x^{2}\right)+1$. Therefore, the function $g$ defined by $g(x)=f(x)+1$ satisfies $g(x+y)=g(x)+g(y)$ and $g(x)^{2}=g\left(x^{2}\right)$. The latter relation shows $g(y)$ is nonnegative for $y \geqslant 0$. For such a function satisfying the Cauchy Equation $g(x+y)=g(x)+g(y)$, it must be monotonic increasing and hence $g(x)=c x$ for some constant $c$.  From $(c x)^{2}=g(x)^{2}=g\left(x^{2}\right)=c x^{2}$, we get $c=0$ or 1 , which corresponds to the two functions $f(x)=-1$ and $f(x)=x-1$ respectively, both of which are solutions to (1) as checked in Solution 1.


Solution: As in   $$ (f(x)+1)^{2}=f\left(x^{2}\right)+1 $$  and  $$ f(x)=f(-x) \quad \text { or } \quad f(x)+f(-x)=-2 $$  for any $x \in \mathbb{R}$. We shall show that one of the statements in (9) holds for all $x \in \mathbb{R}$. Suppose $f(a)=f(-a)$ but $f(a)+f(-a) \neq-2$, while $f(b) \neq f(-b)$ but $f(b)+f(-b)=-2$. Clearly, $a, b \neq 0$ and $f(a), f(b) \neq-1$.  Taking $y=a$ and $y=-a$ in (1) respectively and comparing the two equations obtained, we have $f(x+a)^{2}=f(x-a)^{2}$, that is, $f(x+a)= \pm f(x-a)$. This implies $f(x+2 a)= \pm f(x)$ for all $x \in \mathbb{R}$. Putting $x=b$ and $x=-2 a-b$ respectively, we find $f(2 a+b)= \pm f(b)$ and $f(-2 a-b)= \pm f(-b)= \pm(-2-f(b))$. Since $f(b) \neq-1$, the term $\pm(-2-f(b))$ is distinct from $\pm f(b)$ in any case. So $f(2 a+b) \neq f(-2 a-b)$. From (9), we must have $f(2 a+b)+f(-2 a-b)=-2$. Note that we also have $f(b)+f(-b)=-2$ where $|f(b)|,|f(-b)|$ are equal to $|f(2 a+b)|,|f(-2 a-b)|$ respectively. The only possible case is $f(2 a+b)=f(b)$ and $f(-2 a-b)=f(-b)$.  Applying the argument to $-a$ instead of $a$ and using induction, we have $f(2 k a+b)=f(b)$ and $f(2 k a-b)=f(-b)$ for any integer $k$. Note that $f(b)+f(-b)=-2$ and $f(b) \neq-1$ imply one of $f(b), f(-b)$ is less than -1 . Without loss of generality, assume $f(b)<-1$. We consider $x=\sqrt{2 k a+b}$ in (8) for sufficiently large $k$ so that  $$ (f(x)+1)^{2}=f(2 k a+b)+1=f(b)+1<0 $$  yields a contradiction. Therefore, one of the statements in (9) must hold for all $x \in \mathbb{R}$.  - Case 1. $f(x)=f(-x)$ for any $x \in \mathbb{R}$.  For any $a \in \mathbb{R}$, setting $x=y=\frac{a}{2}$ and $x=-y=\frac{a}{2}$ in (1) respectively and comparing these, we obtain $f(a)^{2}=f(0)^{2}=1$, which means $f(a)= \pm 1$ for all $a \in \mathbb{R}$. If $f(a)=1$ for some $a$, we may assume $a>0$ since $f(a)=f(-a)$. Taking $x=y=\sqrt{a}$ in (1), we get  $$ f(2 \sqrt{a})^{2}=2 f(\sqrt{a})^{2}+\max \{2, f(2 a)\}=2 f(\sqrt{a})^{2}+2 $$  Note that the left-hand side is $\pm 1$ while the right-hand side is an even integer. This is a contradiction. Therefore, $f(x)=-1$ for all $x \in \mathbb{R}$, which is clearly a solution.  - Case 2. $f(x)+f(-x)=-2$ for any $x \in \mathbb{R}$.  This case can be handled in the same way as in Solution 2, which yields another solution $f(x)=x-1$.


Determine the largest real number $a$ such that for all $n \geqslant 1$ and for all real numbers $x_{0}, x_{1}, \ldots, x_{n}$ satisfying $0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}$, we have  $$ \frac{1}{x_{1}-x_{0}}+\frac{1}{x_{2}-x_{1}}+\cdots+\frac{1}{x_{n}-x_{n-1}} \geqslant a\left(\frac{2}{x_{1}}+\frac{3}{x_{2}}+\cdots+\frac{n+1}{x_{n}}\right) . $$

Solution: We first show that $a=\frac{4}{9}$ is admissible. For each $2 \leqslant k \leqslant n$, by the CauchySchwarz Inequality, we have  $$ \left(x_{k-1}+\left(x_{k}-x_{k-1}\right)\right)\left(\frac{(k-1)^{2}}{x_{k-1}}+\frac{3^{2}}{x_{k}-x_{k-1}}\right) \geqslant(k-1+3)^{2}, $$  which can be rewritten as  $$ \frac{9}{x_{k}-x_{k-1}} \geqslant \frac{(k+2)^{2}}{x_{k}}-\frac{(k-1)^{2}}{x_{k-1}} $$  Summing (2) over $k=2,3, \ldots, n$ and adding $\frac{9}{x_{1}}$ to both sides, we have  $$ 9 \sum_{k=1}^{n} \frac{1}{x_{k}-x_{k-1}} \geqslant 4 \sum_{k=1}^{n} \frac{k+1}{x_{k}}+\frac{n^{2}}{x_{n}}>4 \sum_{k=1}^{n} \frac{k+1}{x_{k}} . $$  This shows (1) holds for $a=\frac{4}{9}$. Next, we show that $a=\frac{4}{9}$ is the optimal choice. Consider the sequence defined by $x_{0}=0$ and $x_{k}=x_{k-1}+k(k+1)$ for $k \geqslant 1$, that is, $x_{k}=\frac{1}{3} k(k+1)(k+2)$. Then the left-hand side of (1) equals  $$ \sum_{k=1}^{n} \frac{1}{k(k+1)}=\sum_{k=1}^{n}\left(\frac{1}{k}-\frac{1}{k+1}\right)=1-\frac{1}{n+1} $$  while the right-hand side equals  $$ a \sum_{k=1}^{n} \frac{k+1}{x_{k}}=3 a \sum_{k=1}^{n} \frac{1}{k(k+2)}=\frac{3}{2} a \sum_{k=1}^{n}\left(\frac{1}{k}-\frac{1}{k+2}\right)=\frac{3}{2}\left(1+\frac{1}{2}-\frac{1}{n+1}-\frac{1}{n+2}\right) a . $$  When $n$ tends to infinity, the left-hand side tends to 1 while the right-hand side tends to $\frac{9}{4} a$. Therefore $a$ has to be at most $\frac{4}{9}$.  Hence the largest value of $a$ is $\frac{4}{9}$.


Solution: We shall give an alternative method to establish (1) with $a=\frac{4}{9}$. We define $y_{k}=x_{k}-x_{k-1}>0$ for $1 \leqslant k \leqslant n$. By the Cauchy-Schwarz Inequality, for $1 \leqslant k \leqslant n$, we have  $$ \left(y_{1}+y_{2}+\cdots+y_{k}\right)\left(\sum_{j=1}^{k} \frac{1}{y_{j}}\binom{j+1}{2}^{2}\right) \geqslant\left(\binom{2}{2}+\binom{3}{2}+\cdots+\binom{k+1}{2}\right)^{2}=\binom{k+2}{3}^{2} $$  This can be rewritten as  $$ \frac{k+1}{y_{1}+y_{2}+\cdots+y_{k}} \leqslant \frac{36}{k^{2}(k+1)(k+2)^{2}}\left(\sum_{j=1}^{k} \frac{1}{y_{j}}\binom{j+1}{2}^{2}\right) . $$  Summing (3) over $k=1,2, \ldots, n$, we get  $$ \frac{2}{y_{1}}+\frac{3}{y_{1}+y_{2}}+\cdots+\frac{n+1}{y_{1}+y_{2}+\cdots+y_{n}} \leqslant \frac{c_{1}}{y_{1}}+\frac{c_{2}}{y_{2}}+\cdots+\frac{c_{n}}{y_{n}} $$  where for $1 \leqslant m \leqslant n$,  $$ \begin{aligned} c_{m} & =36\binom{m+1}{2}^{2} \sum_{k=m}^{n} \frac{1}{k^{2}(k+1)(k+2)^{2}} \\ & =\frac{9 m^{2}(m+1)^{2}}{4} \sum_{k=m}^{n}\left(\frac{1}{k^{2}(k+1)^{2}}-\frac{1}{(k+1)^{2}(k+2)^{2}}\right) \\ & =\frac{9 m^{2}(m+1)^{2}}{4}\left(\frac{1}{m^{2}(m+1)^{2}}-\frac{1}{(n+1)^{2}(n+2)^{2}}\right)<\frac{9}{4} . \end{aligned} $$  From (4), the inequality (1) holds for $a=\frac{4}{9}$. This is also the upper bound as can be verified in the same way as Solution 1.


The leader of an IMO team chooses positive integers $n$ and $k$ with $n>k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader's in exactly $k$ positions. (For example, if $n=3$ and $k=1$, and if the leader chooses 101, the deputy leader would write down 001, 111 and 100.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader's string. What is the minimum number of guesses (in terms of $n$ and $k$ ) needed to guarantee the correct answer?

Solution: Let $X$ be the binary string chosen by the leader and let $X^{\prime}$ be the binary string of length $n$ every digit of which is different from that of $X$. The strings written by the deputy leader are the same as those in the case when the leader's string is $X^{\prime}$ and $k$ is changed to $n-k$. In view of this, we may assume $k \geqslant \frac{n}{2}$. Also, for the particular case $k=\frac{n}{2}$, this argument shows that the strings $X$ and $X^{\prime}$ cannot be distinguished, and hence in that case the contestant has to guess at least twice.  It remains to show that the number of guesses claimed suffices. Consider any string $Y$ which differs from $X$ in $m$ digits where $0<m<2 k$. Without loss of generality, assume the first $m$ digits of $X$ and $Y$ are distinct. Let $Z$ be the binary string obtained from $X$ by changing its first $k$ digits. Then $Z$ is written by the deputy leader. Note that $Z$ differs from $Y$ by $|m-k|$ digits where $|m-k|<k$ since $0<m<2 k$. From this observation, the contestant must know that $Y$ is not the desired string.  As we have assumed $k \geqslant \frac{n}{2}$, when $n<2 k$, every string $Y \neq X$ differs from $X$ in fewer than $2 k$ digits. When $n=2 k$, every string except $X$ and $X^{\prime}$ differs from $X$ in fewer than $2 k$ digits. Hence, the answer is as claimed.


Solution: Firstly, assume $n \neq 2 k$. Without loss of generality suppose the first digit of the leader's string is 1 . Then among the $\binom{n}{k}$ strings written by the deputy leader, $\binom{n-1}{k}$ will begin with 1 and $\binom{n-1}{k-1}$ will begin with 0 . Since $n \neq 2 k$, we have $k+(k-1) \neq n-1$ and so $\binom{n-1}{k} \neq\binom{ n-1}{k-1}$. Thus, by counting the number of strings written by the deputy leader that start with 0 and 1 , the contestant can tell the first digit of the leader's string. The same can be done on the other digits, so 1 guess suffices when $n \neq 2 k$.  Secondly, for the case $n=2$ and $k=1$, the answer is clearly 2 . For the remaining cases where $n=2 k>2$, the deputy leader would write down the same strings if the leader's string $X$ is replaced by $X^{\prime}$ obtained by changing each digit of $X$. This shows at least 2 guesses are needed. We shall show that 2 guesses suffice in this case. Suppose the first two digits of the leader's string are the same. Then among the strings written by the deputy leader, the prefices 01 and 10 will occur $\binom{2 k-2}{k-1}$ times each, while the prefices 00 and 11 will occur $\binom{2 k-2}{k}$ times each. The two numbers are interchanged if the first two digits of the leader's string are different. Since $\binom{2 k-2}{k-1} \neq\binom{ 2 k-2}{k}$, the contestant can tell whether the first two digits of the leader's string are the same or not. He can work out the relation of the first digit and the other digits in the same way and reduce the leader's string to only 2 possibilities. The proof is complete.


Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:  - each cell contains a distinct divisor; - the sums of all rows are equal; and - the sums of all columns are equal.

Solution: Suppose all positive divisors of $n$ can be arranged into a rectangular table of size $k \times l$ where the number of rows $k$ does not exceed the number of columns $l$. Let the sum of numbers in each column be $s$. Since $n$ belongs to one of the columns, we have $s \geqslant n$, where equality holds only when $n=1$.  For $j=1,2, \ldots, l$, let $d_{j}$ be the largest number in the $j$-th column. Without loss of generality, assume $d_{1}>d_{2}>\cdots>d_{l}$. Since these are divisors of $n$, we have  $$ d_{l} \leqslant \frac{n}{l} $$  As $d_{l}$ is the maximum entry of the $l$-th column, we must have  $$ d_{l} \geqslant \frac{s}{k} \geqslant \frac{n}{k} . $$  The relations (1) and (2) combine to give $\frac{n}{l} \geqslant \frac{n}{k}$, that is, $k \geqslant l$. Together with $k \leqslant l$, we conclude that $k=l$. Then all inequalities in (1) and (2) are equalities. In particular, $s=n$ and so $n=1$, in which case the conditions are clearly satisfied.


Solution: Clearly $n=1$ works. Then we assume $n>1$ and let its prime factorization be $n=p_{1}^{r_{1}} p_{2}^{r_{2}} \cdots p_{t}^{r_{t}}$. Suppose the table has $k$ rows and $l$ columns with $1<k \leqslant l$. Note that $k l$ is the number of positive divisors of $n$ and the sum of all entries is the sum of positive divisors of $n$, which we denote by $\sigma(n)$. Consider the column containing $n$. Since the column sum is $\frac{\sigma(n)}{l}$, we must have $\frac{\sigma(n)}{l}>n$. Therefore, we have  $$ \begin{aligned} \left(r_{1}+1\right)\left(r_{2}+1\right) \cdots\left(r_{t}+1\right) & =k l \leqslant l^{2}<\left(\frac{\sigma(n)}{n}\right)^{2} \\ & =\left(1+\frac{1}{p_{1}}+\cdots+\frac{1}{p_{1}^{r_{1}}}\right)^{2} \cdots\left(1+\frac{1}{p_{t}}+\cdots+\frac{1}{p_{t}^{r_{t}}}\right)^{2} \end{aligned} $$  This can be rewritten as  $$ f\left(p_{1}, r_{1}\right) f\left(p_{2}, r_{2}\right) \cdots f\left(p_{t}, r_{t}\right)<1 $$  where  $$ f(p, r)=\frac{r+1}{\left(1+\frac{1}{p}+\cdots+\frac{1}{p^{r}}\right)^{2}}=\frac{(r+1)\left(1-\frac{1}{p}\right)^{2}}{\left(1-\frac{1}{p^{r+1}}\right)^{2}} $$  Direct computation yields  $$ f(2,1)=\frac{8}{9}, \quad f(2,2)=\frac{48}{49}, \quad f(3,1)=\frac{9}{8} . $$  Also, we find that  $$ \begin{aligned} & f(2, r) \geqslant\left(1-\frac{1}{2^{r+1}}\right)^{-2}>1 \quad \text { for } r \geqslant 3 \\ & f(3, r) \geqslant \frac{4}{3}\left(1-\frac{1}{3^{r+1}}\right)^{-2}>\frac{4}{3}>\frac{9}{8} \quad \text { for } r \geqslant 2, \text { and } \\ & f(p, r) \geqslant \frac{32}{25}\left(1-\frac{1}{p^{r+1}}\right)^{-2}>\frac{32}{25}>\frac{9}{8} \quad \text { for } p \geqslant 5 \end{aligned} $$  From these values and bounds, it is clear that (3) holds only when $n=2$ or 4 . In both cases, it is easy to see that the conditions are not satisfied. Hence, the only possible $n$ is 1 .


Let $n$ be a positive integer relatively prime to 6 . We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours. $\mathbf{C 4}$. Find all positive integers $n$ for which we can fill in the entries of an $n \times n$ table with the following properties:  - each entry can be one of $I, M$ and $O$; - in each row and each column, the letters $I, M$ and $O$ occur the same number of times; and - in any diagonal whose number of entries is a multiple of three, the letters $I, M$ and $O$ occur the same number of times.

Solution: For $k=1,2,3$, let $a_{k}$ be the number of isosceles triangles whose vertices contain exactly $k$ colours. Suppose on the contrary that $a_{3}=0$. Let $b, c, d$ be the number of vertices of the three different colours respectively. We now count the number of pairs $(\triangle, E)$ where $\triangle$ is an isosceles triangle and $E$ is a side of $\triangle$ whose endpoints are of different colours.  On the one hand, since we have assumed $a_{3}=0$, each triangle in the pair must contain exactly two colours, and hence each triangle contributes twice. Thus the number of pairs is $2 a_{2}$ 。  On the other hand, if we pick any two vertices $A, B$ of distinct colours, then there are three isosceles triangles having these as vertices, two when $A B$ is not the base and one when $A B$ is the base since $n$ is odd. Note that the three triangles are all distinct as $(n, 3)=1$. In this way, we count the number of pairs to be $3(b c+c d+d b)$. However, note that $2 a_{2}$ is even while $3(b c+c d+d b)$ is odd, as each of $b, c, d$ is. This yields a contradiction and hence $a_{3} \geqslant 1$.  Comment. A slightly stronger version of this problem is to replace the condition $(n, 6)=1$ by $n$ being odd (where equilateral triangles are regarded as isosceles triangles). In that case, the only difference in the proof is that by fixing any two vertices $A, B$, one can find exactly one or three isosceles triangles having these as vertices. But since only parity is concerned in the solution, the proof goes the same way.  The condition that there is an odd number of vertices of each colour is necessary, as can be seen from the following example. Consider $n=25$ and we label the vertices $A_{0}, A_{1}, \ldots, A_{24}$. Suppose colour 1 is used for $A_{0}$, colour 2 is used for $A_{5}, A_{10}, A_{15}, A_{20}$, while colour 3 is used for the remaining vertices. Then any isosceles triangle having colours 1 and 2 must contain $A_{0}$ and one of $A_{5}, A_{10}, A_{15}, A_{20}$. Clearly, the third vertex must have index which is a multiple of 5 so it is not of colour 3 .


Let $n \geqslant 3$ be a positive integer. Find the maximum number of diagonals of a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

Solution: We consider two cases according to the parity of $n$.  - Case 1. $n$ is odd.  We first claim that no pair of diagonals is perpendicular. Suppose $A, B, C, D$ are vertices where $A B$ and $C D$ are perpendicular, and let $E$ be the vertex lying on the perpendicular bisector of $A B$. Let $E^{\prime}$ be the opposite point of $E$ on the circumcircle of the regular polygon. Since $E C=E^{\prime} D$ and $C, D, E$ are vertices of the regular polygon, $E^{\prime}$ should also belong to the polygon. This contradicts the fact that a regular polygon with an odd number of vertices does not contain opposite points on the circumcircle. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-39.jpg?height=513&width=1250&top_left_y=1061&top_left_x=493)  Therefore in the odd case we can only select diagonals which do not intersect. In the maximal case these diagonals should divide the regular $n$-gon into $n-2$ triangles, so we can select at most $n-3$ diagonals. This can be done, for example, by selecting all diagonals emanated from a particular vertex.  - Case 2. $n$ is even.  If there is no intersection, then the proof in the odd case works. Suppose there are two perpendicular diagonals selected. We consider the set $S$ of all selected diagonals parallel to one of them which intersect with some selected diagonals. Suppose $S$ contains $k$ diagonals and the number of distinct endpoints of the $k$ diagonals is $l$.  Firstly, consider the longest diagonal in one of the two directions in $S$. No other diagonal in $S$ can start from either endpoint of that diagonal, since otherwise it has to meet another longer diagonal in $S$. The same holds true for the other direction. Ignoring these two longest diagonals and their four endpoints, the remaining $k-2$ diagonals share $l-4$ endpoints where each endpoint can belong to at most two diagonals. This gives $2(l-4) \geqslant 2(k-2)$, so that $k \leqslant l-2$. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-40.jpg?height=430&width=1442&top_left_y=289&top_left_x=294)  Consider a group of consecutive vertices of the regular $n$-gon so that each of the two outermost vertices is an endpoint of a diagonal in $S$, while the interior points are not. There are $l$ such groups. We label these groups $P_{1}, P_{2}, \ldots, P_{l}$ in this order. We claim that each selected diagonal outside $S$ must connect vertices of the same group $P_{i}$. Consider any diagonal $d$ joining vertices from distinct groups $P_{i}$ and $P_{j}$. Let $d_{1}$ and $d_{2}$ be two diagonals in $S$ each having one of the outermost points of $P_{i}$ as endpoint. Then $d$ must meet either $d_{1}, d_{2}$ or a diagonal in $S$ which is perpendicular to both $d_{1}$ and $d_{2}$. In any case $d$ should belong to $S$ by definition, which is a contradiction.  Within the same group $P_{i}$, there are no perpendicular diagonals since the vertices belong to the same side of a diameter of the circumcircle. Hence there can be at most $\left|P_{i}\right|-2$ selected diagonals within $P_{i}$, including the one joining the two outermost points of $P_{i}$ when $\left|P_{i}\right|>2$. Therefore, the maximum number of diagonals selected is  $$ \sum_{i=1}^{l}\left(\left|P_{i}\right|-2\right)+k=\sum_{i=1}^{l}\left|P_{i}\right|-2 l+k=(n+l)-2 l+k=n-l+k \leqslant n-2 $$  This upper bound can be attained as follows. We take any vertex $A$ and let $A^{\prime}$ be the vertex for which $A A^{\prime}$ is a diameter of the circumcircle. If we select all diagonals emanated from $A$ together with the diagonal $d^{\prime}$ joining the two neighbouring vertices of $A^{\prime}$, then the only pair of diagonals that meet each other is $A A^{\prime}$ and $d^{\prime}$, which are perpendicular to each other. In total we can take $n-2$ diagonals. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-40.jpg?height=459&width=421&top_left_y=1798&top_left_x=803)


Solution: The constructions and the odd case are the same as   The base case $n=3$ is trivial since there is no diagonal at all. Suppose the upper bound holds for any cyclic polygon with fewer than $n$ sides. For a cyclic $n$-gon, if there is a selected diagonal which does not intersect any other selected diagonal, then this diagonal divides the $n$-gon into an $m$-gon and an $l$-gon (with $m+l=n+2$ ) so that each selected diagonal belongs to one of them. Without loss of generality, we may assume the $m$-gon lies on the same side of a diameter of $\Gamma$. Then no two selected diagonals of the $m$-gon can intersect, and hence we can select at most $m-3$ diagonals. Also, we can apply the inductive hypothesis to the $l$-gon. This shows the maximum number of selected diagonals is $(m-3)+(l-2)+1=n-2$.  It remains to consider the case when all selected diagonals meet at least one other selected diagonal. Consider a pair of selected perpendicular diagonals $d_{1}, d_{2}$. They divide the circumference of $\Gamma$ into four arcs, each of which lies on the same side of a diameter of $\Gamma$. If there are two selected diagonals intersecting each other and neither is parallel to $d_{1}$ or $d_{2}$, then their endpoints must belong to the same arc determined by $d_{1}, d_{2}$, and hence they cannot be perpendicular. This violates the condition, and hence all selected diagonals must have the same direction as one of $d_{1}, d_{2}$. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-41.jpg?height=435&width=1234&top_left_y=1036&top_left_x=494)  Take the longest selected diagonal in one of the two directions. We argue as in Solution 1 that its endpoints do not belong to any other selected diagonal. The same holds true for the longest diagonal in the other direction. Apart from these four endpoints, each of the remaining $n-4$ vertices can belong to at most two selected diagonals. Thus we can select at most $\frac{1}{2}(2(n-4)+4)=n-2$ diagonals. Then the proof follows by induction.


There are $n \geqslant 3$ islands in a city. Initially, the ferry company offers some routes between some pairs of islands so that it is impossible to divide the islands into two groups such that no two islands in different groups are connected by a ferry route.  After each year, the ferry company will close a ferry route between some two islands $X$ and $Y$. At the same time, in order to maintain its service, the company will open new routes according to the following rule: for any island which is connected by a ferry route to exactly one of $X$ and $Y$, a new route between this island and the other of $X$ and $Y$ is added.  Suppose at any moment, if we partition all islands into two nonempty groups in any way, then it is known that the ferry company will close a certain route connecting two islands from the two groups after some years. Prove that after some years there will be an island which is connected to all other islands by ferry routes.

Solution: Initially, we pick any pair of islands $A$ and $B$ which are connected by a ferry route and put $A$ in set $\mathcal{A}$ and $B$ in set $\mathcal{B}$. From the condition, without loss of generality there must be another island which is connected to $A$. We put such an island $C$ in set $\mathcal{B}$. We say that two sets of islands form a network if each island in one set is connected to each island in the other set.  Next, we shall included all islands to $\mathcal{A} \cup \mathcal{B}$ one by one. Suppose we have two sets $\mathcal{A}$ and $\mathcal{B}$ which form a network where $3 \leqslant|\mathcal{A} \cup \mathcal{B}|<n$. This relation no longer holds only when a ferry route between islands $A \in \mathcal{A}$ and $B \in \mathcal{B}$ is closed. In that case, we define $\mathcal{A}^{\prime}=\{A, B\}$, and $\mathcal{B}^{\prime}=(\mathcal{A} \cup \mathcal{B})-\{A, B\}$. Note that $\mathcal{B}^{\prime}$ is nonempty. Consider any island $C \in \mathcal{A}-\{A\}$. From the relation of $\mathcal{A}$ and $\mathcal{B}$, we know that $C$ is connected to $B$. If $C$ was not connected to $A$ before the route between $A$ and $B$ closes, then there will be a route added between $C$ and $A$ afterwards. Hence, $C$ must now be connected to both $A$ and $B$. The same holds true for any island in $\mathcal{B}-\{B\}$. Therefore, $\mathcal{A}^{\prime}$ and $\mathcal{B}^{\prime}$ form a network, and $\mathcal{A}^{\prime} \cup \mathcal{B}^{\prime}=\mathcal{A} \cup \mathcal{B}$. Hence these islands can always be partitioned into sets $\mathcal{A}$ and $\mathcal{B}$ which form a network.  As $|\mathcal{A} \cup \mathcal{B}|<n$, there are some islands which are not included in $\mathcal{A} \cup \mathcal{B}$. From the condition, after some years there must be a ferry route between an island $A$ in $\mathcal{A} \cup \mathcal{B}$ and an island $D$ outside $\mathcal{A} \cup \mathcal{B}$ which closes. Without loss of generality assume $A \in \mathcal{A}$. Then each island in $\mathcal{B}$ must then be connected to $D$, no matter it was or not before. Hence, we can put $D$ in set $\mathcal{A}$ so that the new sets $\mathcal{A}$ and $\mathcal{B}$ still form a network and the size of $\mathcal{A} \cup \mathcal{B}$ is increased by 1 . The same process can be done to increase the size of $\mathcal{A} \cup \mathcal{B}$. Eventually, all islands are included in this way so we may now assume $|\mathcal{A} \cup \mathcal{B}|=n$.  Suppose a ferry route between $A \in \mathcal{A}$ and $B \in \mathcal{B}$ is closed after some years. We put $A$ and $B$ in set $\mathcal{A}^{\prime}$ and all remaining islands in set $\mathcal{B}^{\prime}$. Then $\mathcal{A}^{\prime}$ and $\mathcal{B}^{\prime}$ form a network. This relation no longer holds only when a route between $A$, without loss of generality, and $C \in \mathcal{B}^{\prime}$ is closed. Since this must eventually occur, at that time island $B$ will be connected to all other islands and the result follows.


Let $n \geqslant 2$ be an integer. In the plane, there are $n$ segments given in such a way that any two segments have an intersection point in the interior, and no three segments intersect at a single point. Jeff places a snail at one of the endpoints of each of the segments and claps his hands $n-1$ times. Each time when he claps his hands, all the snails move along their own segments and stay at the next intersection points until the next clap. Since there are $n-1$ intersection points on each segment, all snails will reach the furthest intersection points from their starting points after $n-1$ claps. (a) Prove that if $n$ is odd then Jeff can always place the snails so that no two of them ever occupy the same intersection point. (b) Prove that if $n$ is even then there must be a moment when some two snails occupy the same intersection point no matter how Jeff places the snails.

Solution: We consider a big disk which contains all the segments. We extend each segment to a line $l_{i}$ so that each of them cuts the disk at two distinct points $A_{i}, B_{i}$. (a) For odd $n$, we travel along the circumference of the disk and mark each of the points $A_{i}$ or $B_{i}$ 'in' and 'out' alternately. Since every pair of lines intersect in the disk, there are exactly $n-1$ points between $A_{i}$ and $B_{i}$ for any fixed $1 \leqslant i \leqslant n$. As $n$ is odd, this means one of $A_{i}$ and $B_{i}$ is marked 'in' and the other is marked 'out'. Then Jeff can put a snail on the endpoint of each segment which is closer to the 'in' side of the corresponding line. We claim that the snails on $l_{i}$ and $l_{j}$ do not meet for any pairs $i, j$, hence proving part (a). ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-43.jpg?height=695&width=1500&top_left_y=1512&top_left_x=401)  Without loss of generality, we may assume the snails start at $A_{i}$ and $A_{j}$ respectively. Let $l_{i}$ intersect $l_{j}$ at $P$. Note that there is an odd number of points between arc $A_{i} A_{j}$. Each of these points belongs to a line $l_{k}$. Such a line $l_{k}$ must intersect exactly one of the segments $A_{i} P$ and $A_{j} P$, making an odd number of intersections. For the other lines, they may intersect both segments $A_{i} P$ and $A_{j} P$, or meet none of them. Therefore, the total number of intersection points on segments $A_{i} P$ and $A_{j} P$ (not counting $P$ ) is odd. However, if the snails arrive at $P$ at the same time, then there should be the same number of intersections on $A_{i} P$ and $A_{j} P$, which gives an even number of intersections. This is a contradiction so the snails do not meet each other. (b) For even $n$, we consider any way that Jeff places the snails and mark each of the points $A_{i}$ or $B_{i}$ 'in' and 'out' according to the directions travelled by the snails. In this case there must be two neighbouring points $A_{i}$ and $A_{j}$ both of which are marked 'in'. Let $P$ be the intersection of the segments $A_{i} B_{i}$ and $A_{j} B_{j}$. Then any other segment meeting one of the segments $A_{i} P$ and $A_{j} P$ must also meet the other one, and so the number of intersections on $A_{i} P$ and $A_{j} P$ are the same. This shows the snails starting from $A_{i}$ and $A_{j}$ will meet at $P$.  Comment. The conclusions do not hold for pseudosegments, as can be seen from the following examples. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-44.jpg?height=383&width=1203&top_left_y=1169&top_left_x=409)


Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2 n \times 2 n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contains two marked cells.

Solution: We first construct an example of marking $2 n$ cells satisfying the requirement. Label the rows and columns $1,2, \ldots, 2 n$ and label the cell in the $i$-th row and the $j$-th column $(i, j)$.  For $i=1,2, \ldots, n$, we mark the cells $(i, i)$ and $(i, i+1)$. We claim that the required partition exists and is unique. The two diagonals of the board divide the board into four regions. Note that the domino covering cell $(1,1)$ must be vertical. This in turn shows that each domino covering $(2,2),(3,3), \ldots,(n, n)$ is vertical. By induction, the dominoes in the left region are all vertical. By rotational symmetry, the dominoes in the bottom region are horizontal, and so on. This shows that the partition exists and is unique. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-45.jpg?height=323&width=820&top_left_y=1080&top_left_x=696)  It remains to show that this value of $k$ is the smallest possible. Assume that only $k<2 n$ cells are marked, and there exists a partition $P$ satisfying the requirement. It suffices to show there exists another desirable partition distinct from $P$. Let $d$ be the main diagonal of the board.  Construct the following graph with edges of two colours. Its vertices are the cells of the board. Connect two vertices with a red edge if they belong to the same domino of $P$. Connect two vertices with a blue edge if their reflections in $d$ are connected by a red edge. It is possible that two vertices are connected by edges of both colours. Clearly, each vertex has both red and blue degrees equal to 1 . Thus the graph splits into cycles where the colours of edges in each cycle alternate (a cycle may have length 2).  Consider any cell $c$ lying on the diagonal $d$. Its two edges are symmetrical with respect to $d$. Thus they connect $c$ to different cells. This shows $c$ belongs to a cycle $C(c)$ of length at least 4. Consider a part of this cycle $c_{0}, c_{1}, \ldots, c_{m}$ where $c_{0}=c$ and $m$ is the least positive integer such that $c_{m}$ lies on $d$. Clearly, $c_{m}$ is distinct from $c$. From the construction, the path symmetrical to this with respect to $d$ also lies in the graph, and so these paths together form $C(c)$. Hence, $C(c)$ contains exactly two cells from $d$. Then all $2 n$ cells in $d$ belong to $n$ cycles $C_{1}, C_{2}, \ldots, C_{n}$, each has length at least 4.  By the Pigeonhole Principle, there exists a cycle $C_{i}$ containing at most one of the $k$ marked cells. We modify $P$ as follows. We remove all dominoes containing the vertices of $C_{i}$, which correspond to the red edges of $C_{i}$. Then we put the dominoes corresponding to the blue edges of $C_{i}$. Since $C_{i}$ has at least 4 vertices, the resultant partition $P^{\prime}$ is different from $P$. Clearly, no domino in $P^{\prime}$ contains two marked cells as $C_{i}$ contains at most one marked cell. This shows the partition is not unique and hence $k$ cannot be less than $2 n$.


In a convex pentagon $A B C D E$, let $F$ be a point on $A C$ such that $\angle F B C=90^{\circ}$. Suppose triangles $A B F, A C D$ and $A D E$ are similar isosceles triangles with  $$ \angle F A B=\angle F B A=\angle D A C=\angle D C A=\angle E A D=\angle E D A $$  Let $M$ be the midpoint of $C F$. Point $X$ is chosen such that $A M X E$ is a parallelogram. Show that $B D, E M$ and $F X$ are concurrent.

Solution: Denote the common angle in (1) by $\theta$. As $\triangle A B F \sim \triangle A C D$, we have $\frac{A B}{A C}=\frac{A F}{A D}$ so that $\triangle A B C \sim \triangle A F D$. From $E A=E D$, we get  $$ \angle A F D=\angle A B C=90^{\circ}+\theta=180^{\circ}-\frac{1}{2} \angle A E D . $$  Hence, $F$ lies on the circle with centre $E$ and radius $E A$. In particular, $E F=E A=E D$. As $\angle E F A=\angle E A F=2 \theta=\angle B F C$, points $B, F, E$ are collinear.  As $\angle E D A=\angle M A D$, we have $E D / / A M$ and hence $E, D, X$ are collinear. As $M$ is the midpoint of $C F$ and $\angle C B F=90^{\circ}$, we get $M F=M B$. In the isosceles triangles $E F A$ and $M F B$, we have $\angle E F A=\angle M F B$ and $A F=B F$. Therefore, they are congruent to each other. Then we have $B M=A E=X M$ and $B E=B F+F E=A F+F M=A M=E X$. This shows $\triangle E M B \cong \triangle E M X$. As $F$ and $D$ lie on $E B$ and $E X$ respectively and $E F=E D$, we know that lines $B D$ and $X F$ are symmetric with respect to $E M$. It follows that the three lines are concurrent. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-47.jpg?height=860&width=1050&top_left_y=1527&top_left_x=565)


Solution: From $\angle C A D=\angle E D A$, we have $A C / / E D$. Together with $A C / / E X$, we know that $E, D, X$ are collinear. Denote the common angle in (1) by $\theta$. From $\triangle A B F \sim \triangle A C D$, we get $\frac{A B}{A C}=\frac{A F}{A D}$ so that $\triangle A B C \sim \triangle A F D$. This yields $\angle A F D=\angle A B C=90^{\circ}+\theta$ and hence $\angle F D C=90^{\circ}$, implying that $B C D F$ is cyclic. Let $\Gamma_{1}$ be its circumcircle.  Next, from $\triangle A B F \sim \triangle A D E$, we have $\frac{A B}{A D}=\frac{A F}{A E}$ so that $\triangle A B D \sim \triangle A F E$. Therefore,  $$ \angle A F E=\angle A B D=\theta+\angle F B D=\theta+\angle F C D=2 \theta=180^{\circ}-\angle B F A . $$  This implies $B, F, E$ are collinear. Note that $F$ is the incentre of triangle $D A B$. Point $E$ lies on the internal angle bisector of $\angle D B A$ and lies on the perpendicular bisector of $A D$. It follows that $E$ lies on the circumcircle $\Gamma_{2}$ of triangle $A B D$, and $E A=E F=E D$.  Also, since $C F$ is a diameter of $\Gamma_{1}$ and $M$ is the midpoint of $C F, M$ is the centre of $\Gamma_{1}$ and hence $\angle A M D=2 \theta=\angle A B D$. This shows $M$ lies on $\Gamma_{2}$. Next, $\angle M D X=\angle M A E=\angle D X M$ since $A M X E$ is a parallelogram. Hence $M D=M X$ and $X$ lies on $\Gamma_{1}$. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-48.jpg?height=925&width=1104&top_left_y=936&top_left_x=467)  We now have two ways to complete the solution.  - Method 1. From $E F=E A=X M$ and $E X / / F M, E F M X$ is an isosceles trapezoid and is cyclic. Denote its circumcircle by $\Gamma_{3}$. Since $B D, E M, F X$ are the three radical axes of $\Gamma_{1}, \Gamma_{2}, \Gamma_{3}$, they must be concurrent. - Method 2. As $\angle D M F=2 \theta=\angle B F M$, we have $D M / / E B$. Also,  $$ \angle B F D+\angle X B F=\angle B F C+\angle C F D+90^{\circ}-\angle C B X=2 \theta+\left(90^{\circ}-\theta\right)+90^{\circ}-\theta=180^{\circ} $$  implies $D F / / X B$. These show the corresponding sides of triangles $D M F$ and $B E X$ are parallel. By Desargues' Theorem, the two triangles are perspective and hence $D B, M E, F X$ meet at a point.  Comment. In Solution 2, both the Radical Axis Theorem and Desargues' Theorem could imply $D B, M E, F X$ are parallel. However, this is impossible as can be seen from the configuration. For example, it is obvious that $D B$ and $M E$ meet each other.


Solution: Let the common angle in (1) be $\theta$. From $\triangle A B F \sim \triangle A C D$, we have $\frac{A B}{A C}=\frac{A F}{A D}$ so that $\triangle A B C \sim \triangle A F D$. Then $\angle A D F=\angle A C B=90^{\circ}-2 \theta=90^{\circ}-\angle B A D$ and hence $D F \perp A B$. As $F A=F B$, this implies $\triangle D A B$ is isosceles with $D A=D B$. Then $F$ is the incentre of $\triangle D A B$.  Next, from $\angle A E D=180^{\circ}-2 \theta=180^{\circ}-\angle D B A$, points $A, B, D, E$ are concyclic. Since we also have $E A=E D$, this shows $E, F, B$ are collinear and $E A=E F=E D$. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-49.jpg?height=871&width=1045&top_left_y=822&top_left_x=567)  Note that $C$ lies on the internal angle bisector of $\angle B A D$ and lies on the external angle bisector of $\angle D B A$. It follows that it is the $A$-excentre of triangle $D A B$. As $M$ is the midpoint of $C F, M$ lies on the circumcircle of triangle $D A B$ and it is the centre of the circle passing through $D, F, B, C$. By symmetry, $D E F M$ is a rhombus. Then the midpoints of $A X, E M$ and $D F$ coincide, and it follows that $D A F X$ is a parallelogram.  Let $P$ be the intersection of $B D$ and $E M$, and $Q$ be the intersection of $A D$ and $B E$. From $\angle B A C=\angle D C A$, we know that $D C, A B, E M$ are parallel. Thus we have $\frac{D P}{P B}=\frac{C M}{M A}$. This is further equal to $\frac{A E}{B E}$ since $C M=D M=D E=A E$ and $M A=B E$. From $\triangle A E Q \sim \triangle B E A$, we find that  $$ \frac{D P}{P B}=\frac{A E}{B E}=\frac{A Q}{B A}=\frac{Q F}{F B} $$  by the Angle Bisector Theorem. This implies $Q D / / F P$ and hence $F, P, X$ are collinear, as desired.


Let $A B C$ be a triangle with circumcircle $\Gamma$ and incentre $I$. Let $M$ be the midpoint of side $B C$. Denote by $D$ the foot of perpendicular from $I$ to side $B C$. The line through $I$ perpendicular to $A I$ meets sides $A B$ and $A C$ at $F$ and $E$ respectively. Suppose the circumcircle of triangle $A E F$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $X D$ and $A M$ meet on $\Gamma$.

Solution: Let $A M$ meet $\Gamma$ again at $Y$ and $X Y$ meet $B C$ at $D^{\prime}$. It suffices to show $D^{\prime}=D$. We shall apply the following fact.  - Claim. For any cyclic quadrilateral $P Q R S$ whose diagonals meet at $T$, we have  $$ \frac{Q T}{T S}=\frac{P Q \cdot Q R}{P S \cdot S R} $$  Proof. We use $\left[W_{1} W_{2} W_{3}\right]$ to denote the area of $W_{1} W_{2} W_{3}$. Then  $$ \frac{Q T}{T S}=\frac{[P Q R]}{[P S R]}=\frac{\frac{1}{2} P Q \cdot Q R \sin \angle P Q R}{\frac{1}{2} P S \cdot S R \sin \angle P S R}=\frac{P Q \cdot Q R}{P S \cdot S R} $$  Applying the Claim to $A B Y C$ and $X B Y C$ respectively, we have $1=\frac{B M}{M C}=\frac{A B \cdot B Y}{A C \cdot C Y}$ and $\frac{B D^{\prime}}{D^{\prime} C}=\frac{X B \cdot B Y}{X C \cdot C Y}$. These combine to give  $$ \frac{B D^{\prime}}{C D^{\prime}}=\frac{X B}{X C} \cdot \frac{B Y}{C Y}=\frac{X B}{X C} \cdot \frac{A C}{A B} $$  Next, we use directed angles to find that $\measuredangle X B F=\measuredangle X B A=\measuredangle X C A=\measuredangle X C E$ and $\measuredangle X F B=\measuredangle X F A=\measuredangle X E A=\measuredangle X E C$. This shows triangles $X B F$ and $X C E$ are directly similar. In particular, we have  $$ \frac{X B}{X C}=\frac{B F}{C E} $$  In the following, we give two ways to continue the proof.  - Method 1. Here is a geometrical method. As $\angle F I B=\angle A I B-90^{\circ}=\frac{1}{2} \angle A C B=\angle I C B$ and $\angle F B I=\angle I B C$, the triangles $F B I$ and $I B C$ are similar. Analogously, triangles $E I C$ and $I B C$ are also similar. Hence, we get  $$ \frac{F B}{I B}=\frac{B I}{B C} \quad \text { and } \quad \frac{E C}{I C}=\frac{I C}{B C} $$  ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-51.jpg?height=1009&width=977&top_left_y=260&top_left_x=623)  Next, construct a line parallel to $B C$ and tangent to the incircle. Suppose it meets sides $A B$ and $A C$ at $B_{1}$ and $C_{1}$ respectively. Let the incircle touch $A B$ and $A C$ at $B_{2}$ and $C_{2}$ respectively. By homothety, the line $B_{1} I$ is parallel to the external angle bisector of $\angle A B C$, and hence $\angle B_{1} I B=90^{\circ}$. Since $\angle B B_{2} I=90^{\circ}$, we get $B B_{2} \cdot B B_{1}=B I^{2}$, and similarly $C C_{2} \cdot C C_{1}=C I^{2}$. Hence,  $$ \frac{B I^{2}}{C I^{2}}=\frac{B B_{2} \cdot B B_{1}}{C C_{2} \cdot C C_{1}}=\frac{B B_{1}}{C C_{1}} \cdot \frac{B D}{C D}=\frac{A B}{A C} \cdot \frac{B D}{C D} $$  Combining (1), (2), (3) and (4), we conclude  $$ \frac{B D^{\prime}}{C D^{\prime}}=\frac{X B}{X C} \cdot \frac{A C}{A B}=\frac{B F}{C E} \cdot \frac{A C}{A B}=\frac{B I^{2}}{C I^{2}} \cdot \frac{A C}{A B}=\frac{B D}{C D} $$  so that $D^{\prime}=D$. The result then follows.  - Method 2. We continue the proof of Solution 1 using trigonometry. Let $\beta=\frac{1}{2} \angle A B C$ and $\gamma=\frac{1}{2} \angle A C B$. Observe that $\angle F I B=\angle A I B-90^{\circ}=\gamma$. Hence, $\frac{B F}{F I}=\frac{\sin \angle F I B}{\sin \angle I B F}=\frac{\sin \gamma}{\sin \beta}$. Similarly, $\frac{C E}{E I}=\frac{\sin \beta}{\sin \gamma}$. As $F I=E I$, we get  $$ \frac{B F}{C E}=\frac{B F}{F I} \cdot \frac{E I}{C E}=\left(\frac{\sin \gamma}{\sin \beta}\right)^{2} $$  Together with (1) and (2), we find that  $$ \frac{B D^{\prime}}{C D^{\prime}}=\frac{A C}{A B} \cdot\left(\frac{\sin \gamma}{\sin \beta}\right)^{2}=\frac{\sin 2 \beta}{\sin 2 \gamma} \cdot\left(\frac{\sin \gamma}{\sin \beta}\right)^{2}=\frac{\tan \gamma}{\tan \beta}=\frac{I D / C D}{I D / B D}=\frac{B D}{C D} $$  This shows $D^{\prime}=D$ and the result follows.


Solution: Let $\omega_{A}$ be the $A$-mixtilinear incircle of triangle $A B C$. From the properties of mixtilinear incircles, $\omega_{A}$ touches sides $A B$ and $A C$ at $F$ and $E$ respectively. Suppose $\omega_{A}$ is tangent to $\Gamma$ at $T$. Let $A M$ meet $\Gamma$ again at $Y$, and let $D_{1}, T_{1}$ be the reflections of $D$ and $T$ with respect to the perpendicular bisector of $B C$ respectively. It is well-known that $\angle B A T=\angle D_{1} A C$ so that $A, D_{1}, T_{1}$ are collinear. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-52.jpg?height=852&width=1094&top_left_y=902&top_left_x=472)  We then show that $X, M, T_{1}$ are collinear. Let $R$ be the radical centre of $\omega_{A}, \Gamma$ and the circumcircle of triangle $A E F$. Then $R$ lies on $A X, E F$ and the tangent at $T$ to $\Gamma$. Let $A T$ meet $\omega_{A}$ again at $S$ and meet $E F$ at $P$. Obviously, $S F T E$ is a harmonic quadrilateral. Projecting from $T$, the pencil $(R, P ; F, E)$ is harmonic. We further project the pencil onto $\Gamma$ from $A$, so that $X B T C$ is a harmonic quadrilateral. As $T T_{1} / / B C$, the projection from $T_{1}$ onto $B C$ maps $T$ to a point at infinity, and hence maps $X$ to the midpoint of $B C$, which is $M$. This shows $X, M, T_{1}$ are collinear.  We have two ways to finish the proof.  - Method 1. Note that both $A Y$ and $X T_{1}$ are chords of $\Gamma$ passing through the midpoint $M$ of the chord $B C$. By the Butterfly Theorem, $X Y$ and $A T_{1}$ cut $B C$ at a pair of symmetric points with respect to $M$, and hence $X, D, Y$ are collinear. The proof is thus complete. - Method 2. Here, we finish the proof without using the Butterfly Theorem. As $D T T_{1} D_{1}$ is an isosceles trapezoid, we have  $$ \measuredangle Y T D=\measuredangle Y T T_{1}+\measuredangle T_{1} T D=\measuredangle Y A T_{1}+\measuredangle A D_{1} D=\measuredangle Y M D $$  so that $D, T, Y, M$ are concyclic. As $X, M, T_{1}$ are collinear, we have  $$ \measuredangle A Y D=\measuredangle M T D=\measuredangle D_{1} T_{1} M=\measuredangle A T_{1} X=\measuredangle A Y X $$  This shows $X, D, Y$ are collinear.


Let $B=(-1,0)$ and $C=(1,0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be nice if (i) there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $T Q$ lies entirely in $S$; and (ii) for any triangle $P_{1} P_{2} P_{3}$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1,2,3\}$ for which triangles $A B C$ and $P_{\sigma(1)} P_{\sigma(2)} P_{\sigma(3)}$ are similar.  Prove that there exist two distinct nice subsets $S$ and $S^{\prime}$ of the set $\{(x, y): x \geqslant 0, y \geqslant 0\}$ such that if $A \in S$ and $A^{\prime} \in S^{\prime}$ are the unique choices of points in (ii), then the product $B A \cdot B A^{\prime}$ is a constant independent of the triangle $P_{1} P_{2} P_{3}$.

Solution: If in the similarity of $\triangle A B C$ and $\triangle P_{\sigma(1)} P_{\sigma(2)} P_{\sigma(3)}, B C$ corresponds to the longest side of $\triangle P_{1} P_{2} P_{3}$, then we have $B C \geqslant A B \geqslant A C$. The condition $B C \geqslant A B$ is equivalent to $(x+1)^{2}+y^{2} \leqslant 4$, while $A B \geqslant A C$ is trivially satisfied for any point in the first quadrant. Then we first define  $$ S=\left\{(x, y):(x+1)^{2}+y^{2} \leqslant 4, x \geqslant 0, y \geqslant 0\right\} $$  Note that $S$ is the intersection of a disk and the first quadrant, so it is bounded and convex, and we can choose any $T \in S$ to meet condition (i). For any point $A$ in $S$, the relation $B C \geqslant A B \geqslant A C$ always holds. Therefore, the point $A$ in (ii) is uniquely determined, while its existence is guaranteed by the above construction. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-54.jpg?height=657&width=858&top_left_y=1450&top_left_x=601)  Next, if in the similarity of $\triangle A^{\prime} B C$ and $\triangle P_{\sigma(1)} P_{\sigma(2)} P_{\sigma(3)}, B C$ corresponds to the second longest side of $\triangle P_{1} P_{2} P_{3}$, then we have $A^{\prime} B \geqslant B C \geqslant A^{\prime} C$. The two inequalities are equivalent to $(x+1)^{2}+y^{2} \geqslant 4$ and $(x-1)^{2}+y^{2} \leqslant 4$ respectively. Then we define  $$ S^{\prime}=\left\{(x, y):(x+1)^{2}+y^{2} \geqslant 4,(x-1)^{2}+y^{2} \leqslant 4, x \geqslant 0, y \geqslant 0\right\} . $$  The boundedness condition is satisfied while (ii) can be argued as in the previous case. For (i), note that $S^{\prime}$ contains points inside the disk $(x-1)^{2}+y^{2} \leqslant 4$ and outside the disk $(x+1)^{2}+y^{2} \geqslant 4$. This shows we can take $T^{\prime}=(1,2)$ in (i), which is the topmost point of the circle $(x-1)^{2}+y^{2}=4$.  It remains to check that the product $B A \cdot B A^{\prime}$ is a constant. Suppose we are given a triangle $P_{1} P_{2} P_{3}$ with $P_{1} P_{2} \geqslant P_{2} P_{3} \geqslant P_{3} P_{1}$. By the similarity, we have  $$ B A=B C \cdot \frac{P_{2} P_{3}}{P_{1} P_{2}} \quad \text { and } \quad B A^{\prime}=B C \cdot \frac{P_{1} P_{2}}{P_{2} P_{3}} $$  Thus $B A \cdot B A^{\prime}=B C^{2}=4$, which is certainly independent of the triangle $P_{1} P_{2} P_{3}$. Comment. The original version of this problem includes the condition that the interiors of $S$ and $S^{\prime}$ are disjoint. We remove this condition since it is hard to define the interior of a point set rigorously.


Let $A B C$ be a triangle with $A B=A C \neq B C$ and let $I$ be its incentre. The line $B I$ meets $A C$ at $D$, and the line through $D$ perpendicular to $A C$ meets $A I$ at $E$. Prove that the reflection of $I$ in $A C$ lies on the circumcircle of triangle $B D E$.

Solution: ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-56.jpg?height=676&width=1134&top_left_y=513&top_left_x=436)  Let $\Gamma$ be the circle with centre $E$ passing through $B$ and $C$. Since $E D \perp A C$, the point $F$ symmetric to $C$ with respect to $D$ lies on $\Gamma$. From $\angle D C I=\angle I C B=\angle C B I$, the line $D C$ is a tangent to the circumcircle of triangle $I B C$. Let $J$ be the symmetric point of $I$ with respect to $D$. Using directed lengths, from  $$ D C \cdot D F=-D C^{2}=-D I \cdot D B=D J \cdot D B $$  the point $J$ also lies on $\Gamma$. Let $I^{\prime}$ be the reflection of $I$ in $A C$. Since $I J$ and $C F$ bisect each other, $C J F I$ is a parallelogram. From $\angle F I^{\prime} C=\angle C I F=\angle F J C$, we find that $I^{\prime}$ lies on $\Gamma$. This gives $E I^{\prime}=E B$.  Note that $A C$ is the internal angle bisector of $\angle B D I^{\prime}$. This shows $D E$ is the external angle bisector of $\angle B D I^{\prime}$ as $D E \perp A C$. Together with $E I^{\prime}=E B$, it is well-known that $E$ lies on the circumcircle of triangle $B D I^{\prime}$.


Solution: Let $I^{\prime}$ be the reflection of $I$ in $A C$ and let $S$ be the intersection of $I^{\prime} C$ and $A I$. Using directed angles, we let $\theta=\measuredangle A C I=\measuredangle I C B=\measuredangle C B I$. We have  $$ \measuredangle I^{\prime} S E=\measuredangle I^{\prime} C A+\measuredangle C A I=\theta+\left(\frac{\pi}{2}+2 \theta\right)=3 \theta+\frac{\pi}{2} $$  and  $$ \measuredangle I^{\prime} D E=\measuredangle I^{\prime} D C+\frac{\pi}{2}=\measuredangle C D I+\frac{\pi}{2}=\measuredangle D C B+\measuredangle C B D+\frac{\pi}{2}=3 \theta+\frac{\pi}{2} $$  This shows $I^{\prime}, D, E, S$ are concyclic. Next, we find $\measuredangle I^{\prime} S B=2 \measuredangle I^{\prime} S E=6 \theta$ and $\measuredangle I^{\prime} D B=2 \measuredangle C D I=6 \theta$. Therefore, $I^{\prime}, D, B, S$ are concyclic so that $I^{\prime}, D, E, B, S$ lie on the same circle. The result then follows. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-57.jpg?height=594&width=1044&top_left_y=256&top_left_x=584)  Comment. The point $S$ constructed in Solution 2 may lie on the same side as $A$ of $B C$. Also, since $S$ lies on the circumcircle of the non-degenerate triangle $B D E$, we implicitly know that $S$ is not an ideal point. Indeed, one can verify that $I^{\prime} C$ and $A I$ are parallel if and only if triangle $A B C$ is equilateral.


Solution: Let $I^{\prime}$ be the reflection of $I$ in $A C$, and let $D^{\prime}$ be the second intersection of $A I$ and the circumcircle of triangle $A B D$. Since $A D^{\prime}$ bisects $\angle B A D$, point $D^{\prime}$ is the midpoint of the arc $B D$ and $D D^{\prime}=B D^{\prime}=C D^{\prime}$. Obviously, $A, E, D^{\prime}$ lie on $A I$ in this order. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-57.jpg?height=486&width=812&top_left_y=1302&top_left_x=716)  We find that $\angle E D^{\prime} D=\angle A D^{\prime} D=\angle A B D=\angle I B C=\angle I C B$. Next, since $D^{\prime}$ is the circumcentre of triangle $B C D$, we have $\angle E D D^{\prime}=90^{\circ}-\angle D^{\prime} D C=\angle C B D=\angle I B C$. The two relations show that triangles $E D^{\prime} D$ and $I C B$ are similar. Therefore, we have  $$ \frac{B C}{C I^{\prime}}=\frac{B C}{C I}=\frac{D D^{\prime}}{D^{\prime} E}=\frac{B D^{\prime}}{D^{\prime} E} $$  Also, we get  $$ \angle B C I^{\prime}=\angle B C A+\angle A C I^{\prime}=\angle B C A+\angle I C A=\angle B C A+\angle D B C=\angle B D A=\angle B D^{\prime} E . $$  These show triangles $B C I^{\prime}$ and $B D^{\prime} E$ are similar, and hence triangles $B C D^{\prime}$ and $B I^{\prime} E$ are similar. As $B C D^{\prime}$ is isosceles, we obtain $B E=I^{\prime} E$.  As $D E$ is the external angle bisector of $\angle B D I^{\prime}$ and $E I^{\prime}=E B$, we know that $E$ lies on the circumcircle of triangle $B D I^{\prime}$.


Solution: Let $A I$ and $B I$ meet the circumcircle of triangle $A B C$ again at $A^{\prime}$ and $B^{\prime}$ respectively, and let $E^{\prime}$ be the reflection of $E$ in $A C$. From  $$ \begin{aligned} \angle B^{\prime} A E^{\prime} & =\angle B^{\prime} A D-\angle E^{\prime} A D=\frac{\angle A B C}{2}-\frac{\angle B A C}{2}=90^{\circ}-\angle B A C-\frac{\angle A B C}{2} \\ & =90^{\circ}-\angle B^{\prime} D A=\angle B^{\prime} D E^{\prime} \end{aligned} $$  points $B^{\prime}, A, D, E^{\prime}$ are concyclic. Then  $$ \angle D B^{\prime} E^{\prime}=\angle D A E^{\prime}=\frac{\angle B A C}{2}=\angle B A A^{\prime}=\angle D B^{\prime} A^{\prime} $$  and hence $B^{\prime}, E^{\prime}, A^{\prime}$ are collinear. It is well-known that $A^{\prime} B^{\prime}$ is the perpendicular bisector of $C I$, so that $C E^{\prime}=I E^{\prime}$. Let $I^{\prime}$ be the reflection of $I$ in $A C$. This implies $B E=C E=I^{\prime} E$. As $D E$ is the external angle bisector of $\angle B D I^{\prime}$ and $E I^{\prime}=E B$, we know that $E$ lies on the circumcircle of triangle $B D I^{\prime}$. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-58.jpg?height=730&width=766&top_left_y=979&top_left_x=647)


Solution: Let $F$ be the intersection of $C I$ and $A B$. Clearly, $F$ and $D$ are symmetric with respect to $A I$. Let $O$ be the circumcentre of triangle $B I F$, and let $I^{\prime}$ be the reflection of $I$ in $A C$. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-58.jpg?height=505&width=763&top_left_y=1927&top_left_x=624)  From $\angle B F O=90^{\circ}-\angle F I B=\frac{1}{2} \angle B A C=\angle B A I$, we get $E I / / F O$. Also, from the relation $\angle O I B=90^{\circ}-\angle B F I=90^{\circ}-\angle C D I=\angle I^{\prime} I D$, we know that $O, I, I^{\prime}$ are collinear.  Note that $E D / / O I$ since both are perpendicular to $A C$. Then $\angle F E I=\angle D E I=\angle O I E$. Together with $E I / / F O$, the quadrilateral $E F O I$ is an isosceles trapezoid. Therefore, we find that $\angle D I E=\angle F I E=\angle O E I$ so $O E / / I D$. Then $D E O I$ is a parallelogram. Hence, we have $D I^{\prime}=D I=E O$, which shows $D E O I^{\prime}$ is an isosceles trapezoid. In addition, $E D=O I=O B$ and $O E / / B D$ imply $E O B D$ is another isosceles trapezoid. In particular, both $D E O I^{\prime}$ and $E O B D$ are cyclic. This shows $B, D, E, I^{\prime}$ are concyclic.


Solution: Let $I^{\prime}$ be the reflection of $I$ in $A C$. Denote by $T$ and $M$ the projections from $I$ to sides $A B$ and $B C$ respectively. Since $B I$ is the perpendicular bisector of $T M$, we have  $$ D T=D M $$  Since $\angle A D E=\angle A T I=90^{\circ}$ and $\angle D A E=\angle T A I$, we have $\triangle A D E \sim \triangle A T I$. This shows $\frac{A D}{A E}=\frac{A T}{A I}=\frac{A T}{A I^{\prime}}$. Together with $\angle D A T=2 \angle D A E=\angle E A I^{\prime}$, this yields $\triangle D A T \sim \triangle E A I^{\prime}$. In particular, we have  $$ \frac{D T}{E I^{\prime}}=\frac{A T}{A I^{\prime}}=\frac{A T}{A I} $$  Obviously, the right-angled triangles $A M B$ and $A T I$ are similar. Then we get  $$ \frac{A M}{A B}=\frac{A T}{A I} $$  Next, from $\triangle A M B \sim \triangle A T I \sim \triangle A D E$, we have $\frac{A M}{A B}=\frac{A D}{A E}$ so that $\triangle A D M \sim \triangle A E B$. It follows that  $$ \frac{D M}{E B}=\frac{A M}{A B} . $$  Combining (1), (2), (3) and (4), we get $E B=E I^{\prime}$. As $D E$ is the external angle bisector of $\angle B D I^{\prime}$, we know that $E$ lies on the circumcircle of triangle $B D I^{\prime}$. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-59.jpg?height=570&width=773&top_left_y=1813&top_left_x=749)  Comment. A stronger version of this problem is to ask the contestants to prove the reflection of $I$ in $A C$ lies on the circumcircle of triangle $B D E$ if and only if $A B=A C$. Some of the above solutions can be modified to prove the converse statement to the original problem. For example, we borrow some ideas from Solution 2 to establish the converse as follows. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-60.jpg?height=595&width=1036&top_left_y=510&top_left_x=493)  Let $I^{\prime}$ be the reflection of $I$ in $A C$ and suppose $B, E, D, I^{\prime}$ lie on a circle $\Gamma$. Let $A I$ meet $\Gamma$ again at $S$. As $D E$ is the external angle bisector of $\angle B D I^{\prime}$, we have $E B=E I^{\prime}$. Using directed angles, we get  $$ \measuredangle C I^{\prime} S=\measuredangle C I^{\prime} D+\measuredangle D I^{\prime} S=\measuredangle D I C+\measuredangle D E S=\measuredangle D I C+\measuredangle D E A=\measuredangle D I C+\measuredangle D C B=0 . $$  This means $I^{\prime}, C, S$ are collinear. From this we get $\measuredangle B S E=\measuredangle E S I^{\prime}=\measuredangle E S C$ and hence $A S$ bisects both $\angle B A C$ and $\angle B S C$. Clearly, $S$ and $A$ are distinct points. It follows that $\triangle B A S \cong \triangle C A S$ and thus $A B=A C$.  As in some of the above solutions, an obvious way to prove the stronger version is to establish the following equivalence: $B E=E I^{\prime}$ if and only if $A B=A C$. In addition to the ideas used in those solutions, one can use trigonometry to express the lengths of $B E$ and $E I^{\prime}$ in terms of the side lengths of triangle $A B C$ and to establish the equivalence.


Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $A B C$. A circle $\omega$ with centre $S$ passes through $A$ and $D$, and it intersects sides $A B$ and $A C$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $B C$, and let $M$ be the midpoint of $B C$. Prove that the circumcentre of triangle $X S Y$ is equidistant from $P$ and $M$.

Solution: Denote the orthocentre and circumcentre of triangle $A B C$ by $H$ and $O$ respectively. Let $Q$ be the midpoint of $A H$ and $N$ be the nine-point centre of triangle $A B C$. It is known that $Q$ lies on the nine-point circle of triangle $A B C, N$ is the midpoint of $Q M$ and that $Q M$ is parallel to $A O$.  Let the perpendicular from $S$ to $X Y$ meet line $Q M$ at $S^{\prime}$. Let $E$ be the foot of altitude from $B$ to side $A C$. Since $Q$ and $S$ lie on the perpendicular bisector of $A D$, using directed angles, we have  $$ \begin{aligned} \measuredangle S D Q & =\measuredangle Q A S=\measuredangle X A S-\measuredangle X A Q=\left(\frac{\pi}{2}-\measuredangle A Y X\right)-\measuredangle B A P=\measuredangle C B A-\measuredangle A Y X \\ & =(\measuredangle C B A-\measuredangle A C B)-\measuredangle B C A-\measuredangle A Y X=\measuredangle P E M-(\measuredangle B C A+\measuredangle A Y X) \\ & =\measuredangle P Q M-\measuredangle(B C, X Y)=\frac{\pi}{2}-\measuredangle\left(S^{\prime} Q, B C\right)-\angle(B C, X Y)=\measuredangle S S^{\prime} Q \end{aligned} $$  This shows $D, S^{\prime}, S, Q$ are concyclic. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-61.jpg?height=773&width=955&top_left_y=1321&top_left_x=626)  Let the perpendicular from $N$ to $B C$ intersect line $S S^{\prime}$ at $O_{1}$. (Note that the two lines coincide when $S$ is the midpoint of $A O$, in which case the result is true since the circumcentre of triangle $X S Y$ must lie on this line.) It suffices to show that $O_{1}$ is the circumcentre of triangle $X S Y$ since $N$ lies on the perpendicular bisector of $P M$. From  $$ \measuredangle D S^{\prime} O_{1}=\measuredangle D Q S=\measuredangle S Q A=\angle(S Q, Q A)=\angle\left(O D, O_{1} N\right)=\measuredangle D N O_{1} $$  since $S Q / / O D$ and $Q A / / O_{1} N$, we know that $D, O_{1}, S^{\prime}, N$ are concyclic. Therefore, we get  $$ \measuredangle S D S^{\prime}=\measuredangle S Q S^{\prime}=\angle\left(S Q, Q S^{\prime}\right)=\angle\left(N D, N S^{\prime}\right)=\measuredangle D N S^{\prime} $$  so that $S D$ is a tangent to the circle through $D, O_{1}, S^{\prime}, N$. Then we have  $$ S S^{\prime} \cdot S O_{1}=S D^{2}=S X^{2} $$  Next, we show that $S$ and $S^{\prime}$ are symmetric with respect to $X Y$. By the Sine Law, we have  $$ \frac{S S^{\prime}}{\sin \angle S Q S^{\prime}}=\frac{S Q}{\sin \angle S S^{\prime} Q}=\frac{S Q}{\sin \angle S D Q}=\frac{S Q}{\sin \angle S A Q}=\frac{S A}{\sin \angle S Q A} $$  It follows that  $$ S S^{\prime}=S A \cdot \frac{\sin \angle S Q S^{\prime}}{\sin \angle S Q A}=S A \cdot \frac{\sin \angle H O A}{\sin \angle O H A}=S A \cdot \frac{A H}{A O}=S A \cdot 2 \cos A, $$  which is twice the distance from $S$ to $X Y$. Note that $S$ and $C$ lie on the same side of the perpendicular bisector of $P M$ if and only if $\angle S A C<\angle O A C$ if and only if $\angle Y X A>\angle C B A$. This shows $S$ and $O_{1}$ lie on different sides of $X Y$. As $S^{\prime}$ lies on ray $S O_{1}$, it follows that $S$ and $S^{\prime}$ cannot lie on the same side of $X Y$. Therefore, $S$ and $S^{\prime}$ are symmetric with respect to $X Y$.  Let $d$ be the diameter of the circumcircle of triangle $X S Y$. As $S S^{\prime}$ is twice the distance from $S$ to $X Y$ and $S X=S Y$, we have $S S^{\prime}=2 \frac{S X^{2}}{d}$. It follows from (1) that $d=2 S O_{1}$. As $S O_{1}$ is the perpendicular bisector of $X Y$, point $O_{1}^{d}$ is the circumcentre of triangle $X S Y$.


Solution: Denote the orthocentre and circumcentre of triangle $A B C$ by $H$ and $O$ respectively. Let $O_{1}$ be the circumcentre of triangle $X S Y$. Consider two other possible positions of $S$. We name them $S^{\prime}$ and $S^{\prime \prime}$ and define the analogous points $X^{\prime}, Y^{\prime}, O_{1}^{\prime}, X^{\prime \prime}, Y^{\prime \prime} O_{1}^{\prime \prime}$ accordingly. Note that $S, S^{\prime}, S^{\prime \prime}$ lie on the perpendicular bisector of $A D$.  As $X X^{\prime}$ and $Y Y^{\prime}$ meet at $A$ and the circumcircles of triangles $A X Y$ and $A X^{\prime} Y^{\prime}$ meet at $D$, there is a spiral similarity with centre $D$ mapping $X Y$ to $X^{\prime} Y^{\prime}$. We find that  $$ \measuredangle S X Y=\frac{\pi}{2}-\measuredangle Y A X=\frac{\pi}{2}-\measuredangle Y^{\prime} A X^{\prime}=\measuredangle S^{\prime} X^{\prime} Y^{\prime} $$  and similarly $\measuredangle S Y X=\measuredangle S^{\prime} Y^{\prime} X^{\prime}$. This shows triangles $S X Y$ and $S^{\prime} X^{\prime} Y^{\prime}$ are directly similar. Then the spiral similarity with centre $D$ takes points $S, X, Y, O_{1}$ to $S^{\prime}, X^{\prime}, Y^{\prime}, O_{1}^{\prime}$. Similarly, there is a spiral similarity with centre $D$ mapping $S, X, Y, O_{1}$ to $S^{\prime \prime}, X^{\prime \prime}, Y^{\prime \prime}, O_{1}^{\prime \prime}$. From these, we see that there is a spiral similarity taking the corresponding points $S, S^{\prime}, S^{\prime \prime}$ to points $O_{1}, O_{1}^{\prime}, O_{1}^{\prime \prime}$. In particular, $O_{1}, O_{1}^{\prime}, O_{1}^{\prime \prime}$ are collinear. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-63.jpg?height=750&width=1071&top_left_y=251&top_left_x=592)  It now suffices to show that $O_{1}$ lies on the perpendicular bisector of $P M$ for two special cases.  Firstly, we take $S$ to be the midpoint of $A H$. Then $X$ and $Y$ are the feet of altitudes from $C$ and $B$ respectively. It is well-known that the circumcircle of triangle $X S Y$ is the nine-point circle of triangle $A B C$. Then $O_{1}$ is the nine-point centre and $O_{1} P=O_{1} M$. Indeed, $P$ and $M$ also lies on the nine-point circle.  Secondly, we take $S^{\prime}$ to be the midpoint of $A O$. Then $X^{\prime}$ and $Y^{\prime}$ are the midpoints of $A B$ and $A C$ respectively. Then $X^{\prime} Y^{\prime} / / B C$. Clearly, $S^{\prime}$ lies on the perpendicular bisector of $P M$. This shows the perpendicular bisectors of $X^{\prime} Y^{\prime}$ and $P M$ coincide. Hence, we must have $O_{1}^{\prime} P=O_{1}^{\prime} M$. ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-63.jpg?height=548&width=1257&top_left_y=1610&top_left_x=481)


Let $A B C D$ be a convex quadrilateral with $\angle A B C=\angle A D C<90^{\circ}$. The internal angle bisectors of $\angle A B C$ and $\angle A D C$ meet $A C$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of $A C$ and let $\omega$ be the circumcircle of triangle $B P D$. Segments $B M$ and $D M$ intersect $\omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $X E$ and $Y F$. Prove that $P Q \perp A C$.

Solution: ![](https://cdn.mathpix.com/cropped/2024_11_18_e5e735288fb13d37ab0cg-64.jpg?height=857&width=1492&top_left_y=604&top_left_x=273)  Let $\omega_{1}$ be the circumcircle of triangle $A B C$. We first prove that $Y$ lies on $\omega_{1}$. Let $Y^{\prime}$ be the point on ray $M D$ such that $M Y^{\prime} \cdot M D=M A^{2}$. Then triangles $M A Y^{\prime}$ and $M D A$ are oppositely similar. Since $M C^{2}=M A^{2}=M Y^{\prime} \cdot M D$, triangles $M C Y^{\prime}$ and $M D C$ are also oppositely similar. Therefore, using directed angles, we have  $$ \measuredangle A Y^{\prime} C=\measuredangle A Y^{\prime} M+\measuredangle M Y^{\prime} C=\measuredangle M A D+\measuredangle D C M=\measuredangle C D A=\measuredangle A B C $$  so that $Y^{\prime}$ lies on $\omega_{1}$. Let $Z$ be the intersection point of lines $B C$ and $A D$. Since $\measuredangle P D Z=\measuredangle P B C=\measuredangle P B Z$, point $Z$ lies on $\omega$. In addition, from $\measuredangle Y^{\prime} B Z=\measuredangle Y^{\prime} B C=\measuredangle Y^{\prime} A C=\measuredangle Y^{\prime} A M=\measuredangle Y^{\prime} D Z$, we also know that $Y^{\prime}$ lies on $\omega$. Note that $\angle A D C$ is acute implies $M A \neq M D$ so $M Y^{\prime} \neq M D$. Therefore, $Y^{\prime}$ is the second intersection of $D M$ and $\omega$. Then $Y^{\prime}=Y$ and hence $Y$ lies on $\omega_{1}$.  Next, by the Angle Bisector Theorem and the similar triangles, we have  $$ \frac{F A}{F C}=\frac{A D}{C D}=\frac{A D}{A M} \cdot \frac{C M}{C D}=\frac{Y A}{Y M} \cdot \frac{Y M}{Y C}=\frac{Y A}{Y C} $$  Hence, $F Y$ is the internal angle bisector of $\angle A Y C$. Let $B^{\prime}$ be the second intersection of the internal angle bisector of $\angle C B A$ and $\omega_{1}$. Then $B^{\prime}$ is the midpoint of arc $A C$ not containing $B$. Therefore, $Y B^{\prime}$ is the external angle bisector of $\angle A Y C$, so that $B^{\prime} Y \perp F Y$.  Denote by $l$ the line through $P$ parallel to $A C$. Suppose $l$ meets line $B^{\prime} Y$ at $S$. From  $$ \begin{aligned} \measuredangle P S Y & =\measuredangle\left(A C, B^{\prime} Y\right)=\measuredangle A C Y+\measuredangle C Y B^{\prime}=\measuredangle A C Y+\measuredangle C A B^{\prime}=\measuredangle A C Y+\measuredangle B^{\prime} C A \\ & =\measuredangle B^{\prime} C Y=\measuredangle B^{\prime} B Y=\measuredangle P B Y \end{aligned} $$  the point $S$ lies on $\omega$. Similarly, the line through $X$ perpendicular to $X E$ also passes through the second intersection of $l$ and $\omega$, which is the point $S$. From $Q Y \perp Y S$ and $Q X \perp X S$, point $Q$ lies on $\omega$ and $Q S$ is a diameter of $\omega$. Therefore, $P Q \perp P S$ so that $P Q \perp A C$.