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eebb93b | 1 2 3 4 5 6 7 8 9 | {"year":2017,"label":"1","problem":"In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is color-identifiable if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$. For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.","solution":" Answer: $\\lceil n \/ t\\rceil$. To see this is an upper bound, note that one can easily construct a sports league with that many teams anyways. A quick warning: Remark (Misreading the problem). It is common to misread the problem by ignoring the word \"any\". Here is an illustration. Suppose we have two teams, MIT and Harvard; the colors of MIT are red\/grey\/black, and the colors of Harvard are red\/white. (Thus $n=4$ and $t=3$.) The assignment of MIT to grey and Harvard to red is not acceptable because red is a signature color of MIT, even though not the one assigned. \u3010 Approach by deleting teams (Gopal Goel). Initially, place all teams in a set $S$. Then we repeat the following algorithm: If there is a team all of whose signature colors are shared by some other team in $S$ already, then we delete that team. (If there is more than one such team, we pick arbitrarily.) At the end of the process, all $n$ colors are still present at least once, so at least $\\lceil n \/ t\\rceil$ teams remain. Moreover, since the algorithm is no longer possible, the remaining set $S$ is already color-identifiable. Remark (Gopal Goel). It might seem counter-intuitive that we are deleting teams from the full set when the original problem is trying to get a large set $S$. This is less strange when one thinks of it instead as \"safely deleting useless teams\". Basically, if one deletes such a team, the problem statement implies that the task must still be possible, since $g(n, t)$ does not depend on the number of teams: $n$ is the number of colors present, and deleting a useless team does not change this. It turns out that this optimization is already enough to solve the problem."}
{"year":2017,"label":"1","problem":"In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is color-identifiable if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$. For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.","solution":" Answer: $\\lceil n \/ t\\rceil$. To see this is an upper bound, note that one can easily construct a sports league with that many teams anyways. A quick warning: Remark (Misreading the problem). It is common to misread the problem by ignoring the word \"any\". Here is an illustration. Suppose we have two teams, MIT and Harvard; the colors of MIT are red\/grey\/black, and the colors of Harvard are red\/white. (Thus $n=4$ and $t=3$.) The assignment of MIT to grey and Harvard to red is not acceptable because red is a signature color of MIT, even though not the one assigned. \u3010 Approach by adding colors. For a constructive algorithmic approach, the idea is to greedy pick by color (rather than by team), taking at each step the least used color. Select the color $C_{1}$ with the fewest teams using it, and a team $T_{1}$ using it. Then delete all colors $T_{1}$ uses, and all teams which use $C_{1}$. Note that - By problem condition, this deletes at most $t$ teams total. - Any remaining color $C$ still has at least one user. Indeed, if not, then $C$ had the same set of teams as $C_{1}$ did (by minimality of $C$ ), but then it should have deleted as a color of $T_{1}$. Now repeat this algorithm with $C_{2}$ and $T_{2}$, and so on. This operations uses at most $t$ colors each time, so we select at least $\\lceil n \/ t\\rceil$ colors. As before, assume our league has teams, MIT and Harvard; the colors of MIT are red\/grey\/black, and the colors of Harvard are red\/white. (Thus $n=4$ and $t=3$.) If we start by selecting MIT and red, then it is impossible to select any more teams; but $g(n, t)=2$."}
{"year":2017,"label":"2","problem":"Let $A B C$ be an acute scalene triangle with circumcenter $O$, and let $T$ be on line $B C$ such that $\\angle T A O=90^{\\circ}$. The circle with diameter $\\overline{A T}$ intersects the circumcircle of $\\triangle B O C$ at two points $A_{1}$ and $A_{2}$, where $O A_{1}<O A_{2}$. Points $B_{1}, B_{2}, C_{1}, C_{2}$ are defined analogously. (a) Prove that $\\overline{A A_{1}}, \\overline{B B_{1}}, \\overline{C C_{1}}$ are concurrent. (b) Prove that $\\overline{A A_{2}}, \\overline{B B_{2}}, \\overline{C C_{2}}$ are concurrent on the Euler line of triangle $A B C$.","solution":" Let triangle $A B C$ have circumcircle $\\Gamma$. Let $\\triangle X Y Z$ be the tangential triangle of $\\triangle A B C$ (hence $\\Gamma$ is the incircle of $\\triangle X Y Z$ ), and denote by $\\Omega$ its circumcircle. Suppose the symmedian $\\overline{A X}$ meets $\\Gamma$ again at $D$, and let $M$ be the midpoint of $\\overline{A D}$. Finally, let $K$ be the Miquel point of quadrilateral $Z B C Y$, meaning it is the intersection of $\\Omega$ and the circumcircle of $\\triangle B O C$ (other than $X$ ).  We first claim that $M$ and $K$ are $A_{1}$ and $A_{2}$. In that case $O M<O A<O K$, so $M=A_{1}, K=A_{2}$. To see that $M=A_{1}$, note that $\\angle O M X=90^{\\circ}$, and moreover that $\\overline{T A}, \\overline{T D}$ are tangents to $\\Gamma$, whence we also have $M=\\overline{T O} \\cap \\overline{A D}$. Thus $M$ lies on both $(B O C)$ and $(A T)$. This solves part (a) of the problem: the concurrency point is the symmedian point of $\\triangle A B C$. Now, note that since $K$ is the Miquel point, $$ \\frac{Z K}{Y K}=\\frac{Z B}{Y C}=\\frac{Z A}{Y A} $$ and hence $\\overline{K A}$ is an angle bisector of $\\angle Z K Y$. Thus from $(T A ; Y Z)=-1$ we obtain $\\angle T K A=90^{\\circ}$. It remains to show $\\overline{A K}$ passes through a fixed point on the Euler line. We claim it is the exsimilicenter of $\\Gamma$ and $\\Omega$. Let $L$ be the midpoint of the $\\operatorname{arc} Y Z$ of $\\triangle X Y Z$ not containing $X$. Then we know that $K, A, L$ are collinear. Now the positive homothety sending $\\Gamma$ to $\\Omega$ maps $A$ to $L$; this proves the claim. Finally, it is well-known that the line through $O$ and the circumcenter of $\\triangle X Y Z$ coincides with the Euler line of $\\triangle A B C$; hence done. I Authorship comments. This problem was inspired by the fact that $K, A, L$ are collinear in the figure, which was produced by one of my students (Ryan Kim) in a solution to a homework problem. I realized for example that this implied that line $A K$ passed through the $X_{56}$ point of $\\triangle X Y Z$ (which lies on the Euler line of $\\triangle A B C$ ). This problem was the result of playing around with the resulting very nice picture: all the power comes from the \"magic\" point $K$."}
{"year":2017,"label":"3","problem":"Let $P, Q \\in \\mathbb{R}[x]$ be relatively prime nonconstant polynomials. Show that there can be at most three real numbers $\\lambda$ such that $P+\\lambda Q$ is the square of a polynomial.","solution":" This is true even with $\\mathbb{R}$ replaced by $\\mathbb{C}$, and it will be necessary to work in this generality. \u3010 First solution using transformations. We will prove the claim in the following form: Claim - Assume $P, Q \\in \\mathbb{C}[x]$ are relatively prime. If $\\alpha P+\\beta Q$ is a square for four different choices of the ratio $[\\alpha: \\beta]$ then $P$ and $Q$ must be constant. Call pairs $(P, Q)$ as in the claim bad; so we wish to show the only bad pairs are pairs of constant polynomials. Assume not, and take a bad pair with $\\operatorname{deg} P+\\operatorname{deg} Q$ minimal. By a suitable M\u00f6bius transformation, we may transform $(P, Q)$ so that the four ratios are $[1: 0],[0: 1],[1:-1]$ and $[1:-k]$, so we find there are polynomials $A$ and $B$ such that $$ \\begin{aligned} A^{2}-B^{2} & =C^{2} \\\\ A^{2}-k B^{2} & =D^{2} \\end{aligned} $$ where $A^{2}=P+\\lambda_{1} Q, B^{2}=P+\\lambda_{2} Q$, say. Of course $\\operatorname{gcd}(A, B)=1$. Consequently, we have $C^{2}=(A+B)(A-B)$ and $D^{2}=(A+\\mu B)(A-\\mu B)$ where $\\mu^{2}=k$. Now $\\operatorname{gcd}(A, B)=1$, so $A+B, A-B, A+\\mu B$ and $A-\\mu B$ are squares; id est $(A, B)$ is bad. This is a contradiction, since $\\operatorname{deg} A+\\operatorname{deg} B<\\operatorname{deg} P+\\operatorname{deg} Q$."}
{"year":2017,"label":"3","problem":"Let $P, Q \\in \\mathbb{R}[x]$ be relatively prime nonconstant polynomials. Show that there can be at most three real numbers $\\lambda$ such that $P+\\lambda Q$ is the square of a polynomial.","solution":" This is true even with $\\mathbb{R}$ replaced by $\\mathbb{C}$, and it will be necessary to work in this generality. \u3010 Second solution using derivatives (by Zack Chroman). We will assume without loss of generality that $\\operatorname{deg} P \\neq \\operatorname{deg} Q$; if not, then one can replace $(P, Q)$ with $(P+c Q, Q)$ for a suitable constant $c$. Then, there exist $\\lambda_{i} \\in \\mathbb{C}$ and polynomials $R_{i}$ for $i=1,2,3,4$ such that $$ \\begin{aligned} & P+\\lambda_{i} Q=R_{i}^{2} \\\\ \\Longrightarrow & P^{\\prime}+\\lambda_{i} Q^{\\prime}=2 R_{i} R_{i}^{\\prime} \\\\ & \\Longrightarrow R_{i} \\mid Q^{\\prime}\\left(P+\\lambda_{i} Q\\right)-Q\\left(P^{\\prime}+\\lambda_{i} Q^{\\prime}\\right)=Q^{\\prime} P-Q P^{\\prime} \\end{aligned} $$ On the other hand by Euclidean algorithm it follows that $R_{i}$ are relatively prime to each other. Therefore $$ R_{1} R_{2} R_{3} R_{4} \\mid Q^{\\prime} P-Q P^{\\prime} $$ However, we have $$ \\begin{aligned} \\sum_{1}^{4} \\operatorname{deg} R_{i} & \\geq \\frac{3 \\max (\\operatorname{deg} P, \\operatorname{deg} Q)+\\min (\\operatorname{deg} P, \\operatorname{deg} Q)}{2} \\\\ & \\geq \\operatorname{deg} P+\\operatorname{deg} Q>\\operatorname{deg}\\left(Q^{\\prime} P-Q P^{\\prime}\\right) \\end{aligned} $$ This can only occur if $Q^{\\prime} P-Q P^{\\prime}=0$ or $(P \/ Q)^{\\prime}=0$ by the quotient rule! But $P \/ Q$ can't be constant, the end. Remark. The result is previously known; see e.g. Lemma 1.6 of http:\/\/math.mit.edu\/ ebelmont\/ec-notes.pdf or Exercise 6.5.L(a) of Vakil's notes."}
{"year":2017,"label":"4","problem":"You are cheating at a trivia contest. For each question, you can peek at each of the $n>1$ other contestant's guesses before writing your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth 0 points. An incorrect guess is worth -2 points for other contestants, but only -1 point for you, because you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read out the next question. Show that if you are leading by $2^{n-1}$ points at any time, then you can surely win first place.","solution":" We will prove the result with $2^{n-1}$ replaced even by $2^{n-2}+1$. We first make the following reductions. First, change the weights to be $+1,-1,0$ respectively (rather than $0,-2,-1$ ); this clearly has no effect. Also, WLOG that all contestants except you initially have score zero (and that your score exceeds $2^{n-2}$ ). WLOG ignore rounds in which all answers are the same. Finally, ignore rounds in which you get the correct answer, since that leaves you at least as well off as before - in other words, we'll assume your score is always fixed, but you can pick any group of people with the same answers and ensure they lose 1 point, while some other group gains 1 point. The key observation is the following. Consider two rounds $R_{1}$ and $R_{2}$ such that: - In round $R_{1}$, some set $S$ of contestants gains a point. - In round $R_{2}$, the set $S$ of contestants all have the same answer. Then, if we copy the answers of contestants in $S$ during $R_{2}$, then the sum of the scorings in $R_{1}$ and $R_{2}$ cancel each other out. In other words we can then ignore $R_{1}$ and $R_{2}$ forever. We thus consider the following strategy. We keep a list $\\mathcal{L}$ of subsets of $\\{1, \\ldots, n\\}$, initially empty. Now do the following strategy: - On a round, suppose there exists a set $S$ of people with the same answer such that $S \\in \\mathcal{L}$. (If multiple exist, choose one arbitrarily.) Then, copy the answer of $S$, causing them to lose a point. Delete $S$ from $\\mathcal{L}$. (Importantly, we do not add any new sets to $\\mathcal{L}$.) - Otherwise, copy any set $T$ of contestants, selecting $|T| \\geq n \/ 2$ if possible. Let $S$ be the set of contestants who answer correctly (if any), and add $S$ to the list $\\mathcal{L}$. Note that $|S| \\leq n \/ 2$, since $S$ is disjoint from $T$. By construction, $\\mathcal{L}$ has no duplicate sets. So the score of any contestant $c$ is bounded above by the number of times that $c$ appears among sets in $\\mathcal{L}$. The number of such sets is clearly at most $\\frac{1}{2} \\cdot 2^{n-1}$. So, if you lead by $2^{n-2}+1$ then you ensure victory. This completes the proof! Remark. Several remarks are in order. First, we comment on the bound $2^{n-2}+1$ itself. The most natural solution using only the list idea gives an upper bound of $\\left(2^{n}-2\\right)+1$, which is the number of nonempty proper subsets of $\\{1, \\ldots, n\\}$. Then, there are two optimizations one can observe: - In fact we can improve to the number of times any particular contestant $c$ appears in some set, rather than the total number of sets. - When adding new sets $S$ to $\\mathcal{L}$, one can ensure $|S| \\leq n \/ 2$. Either observation alone improves the bound from $2^{n}-1$ to $2^{n-1}$, but both together give the bound $2^{n-2}+1$. Additionally, when $n$ is odd the calculation of subsets actually gives $2^{n-2}-\\frac{1}{2}\\binom{n-1}{\\frac{n-1}{2}}+1$. This gives the best possible value at both $n=2$ and $n=3$. It seems likely some further improvements are possible, and the true bound is suspected to be polynomial in $n$. 1. The exponential bound $2^{n}$ suggests looking at subsets. 2. The $n=2$ case suggests the idea of \"repeated rounds\". (I think this $n=2$ case is actually really good.) 3. The \"two distinct answers\" case suggests looking at rounds as partitions (even though the WLOG does not work, at least not without further thought). 4. There's something weird about this problem: it's a finite bound over unbounded time. This is a hint to not worry excessively about the actual scores, which turn out to be almost irrelevant."}
{"year":2017,"label":"5","problem":"Let $A B C$ be a triangle with altitude $\\overline{A E}$. The $A$-excircle touches $\\overline{B C}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $D G$ and $D F$ such that quadrilateral $E V A N$ is a rhombus.","solution":" Let $I$ denote the incenter, $J$ the $A$-excenter, and $L$ the midpoint of $\\overline{A E}$. Denote by $\\overline{I Y}$, $\\overline{I Z}$ the tangents from $I$ to the $A$-excircle. Note that lines $\\overline{B C}, \\overline{G F}, \\overline{Y Z}$ then concur at $H$ (unless $A B=A C$, but this case is obvious), as it's the radical center of cyclic hexagon $B I C Y J Z$, the circumcircle and the $A$-excircle.  Now let $\\overline{H D}$ and $\\overline{H T}$ be the tangents from $H$ to the $A$-excircle. It follows that $\\overline{D T}$ is the symmedian of $\\triangle D Z Y$, hence passes through $I=\\overline{Y Y} \\cap \\overline{Z Z}$. Moreover, it's well known that $\\overline{D I}$ passes through $L$, the midpoint of the $A$-altitude (for example by homothety). Finally, $(D T ; F G)=-1$, hence project through $D$ onto the line through $L$ parallel to $\\overline{B C}$ to obtain $(\\infty L ; V N)=-1$ as desired. \u3010 Authorship comments. This is a joint proposal with Danielle Wang (mostly by her). The formulation given was that the tangents to the $A$-excircle at $F$ and $G$ was on line $\\overline{D I}$; I solved this formulation using the radical axis argument above. I then got the idea to involve the point $L$, already knowing it was on $\\overline{D I}$. Observing the harmonic quadrilateral, I took perspectivity through $M$ onto the line through $L$ parallel to $\\overline{B C}$ (before this I had tried to use the $A$-altitude with little luck). This yields the rhombus in the problem."}
{"year":2017,"label":"6","problem":"Prove that there are infinitely many triples $(a, b, p)$ of integers, with $p$ prime and $0<a \\leq b<p$, for which $p^{5}$ divides $(a+b)^{p}-a^{p}-b^{p}$.","solution":" The key claim is that if $p \\equiv 1(\\bmod 3)$, then $$ p\\left(x^{2}+x y+y^{2}\\right)^{2} \\text { divides }(x+y)^{p}-x^{p}-y^{p} $$ as polynomials in $x$ and $y$. Since it's known that one can select $a$ and $b$ such that $p^{2} \\mid a^{2}+a b+b^{2}$, the conclusion follows. (The theory of quadratic forms tells us we can do it with $p^{2}=a^{2}+a b+b^{2}$; Thue's lemma lets us do it by solving $x^{2}+x+1 \\equiv 0\\left(\\bmod p^{2}\\right)$.) To prove this, it is the same to show that $$ \\left(x^{2}+x+1\\right)^{2} \\text { divides } F(x):=(x+1)^{p}-x^{p}-1 $$ since the binomial coefficients $\\binom{p}{k}$ are clearly divisible by $p$. Let $\\zeta$ be a third root of unity. Then $F(\\zeta)=(1+\\zeta)^{p}-\\zeta^{p}-1=-\\zeta^{2}-\\zeta-1=0$. Moreover, $F^{\\prime}(x)=p(x+1)^{p-1}-p x^{p-1}$, so $F^{\\prime}(\\zeta)=p-p=0$. Hence $\\zeta$ is a double root of $F$ as needed. (Incidentally, $p=2017$ works!) $$ (x+1)^{7}-x^{7}-1=7 x(x+1)\\left(x^{4}+2 x^{3}+3 x^{2}+2 x+1\\right) $$ The key is now to notice that the last factor is $\\left(x^{2}+x+1\\right)^{2}$, which suggests the entire solution. In fact, even if $p \\equiv 2(\\bmod 3)$, the polynomial $x^{2}+x+1$ still divides $(x+1)^{p}-x^{p}-1$. So even the $p=5$ case can motivate the main idea."}
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