APMO/md/en-apmo2006_sol.md

Problem 1. Let $n$ be a positive integer. Find the largest nonnegative real number $f(n)$ (depending on $n$ ) with the following property: whenever $a_{1}, a_{2}, \ldots, a_{n}$ are real numbers such that $a_{1}+a_{2}+\cdots+a_{n}$ is an integer, there exists some $i$ such that $\left|a_{i}-\frac{1}{2}\right| \geq f(n)$. (Solution) The answer is

First, assume that $n$ is even. If $a_{i}=\frac{1}{2}$ for all $i$, then the sum $a_{1}+a_{2}+\cdots+a_{n}$ is an integer. Since $\left|a_{i}-\frac{1}{2}\right|=0$ for all $i$, we may conclude $f(n)=0$ for any even $n$.

Now assume that $n$ is odd. Suppose that $\left|a_{i}-\frac{1}{2}\right|<\frac{1}{2 n}$ for all $1 \leq i \leq n$. Then, since $\sum_{i=1}^{n} a_{i}$ is an integer,

a contradiction. Thus $\left|a_{i}-\frac{1}{2}\right| \geq \frac{1}{2 n}$ for some $i$, as required. On the other hand, putting $n=2 m+1$ and $a_{i}=\frac{m}{2 m+1}$ for all $i$ gives $\sum a_{i}=m$, while

$$\mid a_i-\frac{1}{2}\mid =\frac{1}{2}-\frac{m}{2m+1}=\frac{1}{2(2m+1)}=\frac{1}{2n}\backslash left|a\_\{i\}-\backslash frac\{1\}\{2\}\backslash right|=\backslash frac\{1\}\{2\}-\backslash frac\{m\}\{2\; m+1\}=\backslash frac\{1\}\{2(2\; m+1)\}=\backslash frac\{1\}\{2\; n\}$$

for all $i$. Therefore, $f(n)=\frac{1}{2 n}$ is the best possible for any odd $n$.

Problem 2. Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden mean $\tau=\frac{1+\sqrt{5}}{2}$. Here, an integral power of $\tau$ is of the form $\tau^{i}$, where $i$ is an integer (not necessarily positive). (Solution) We will prove this statement by induction using the equality

$${\tau }^2=\tau +1\backslash tau^\{2\}=\backslash tau+1$$

If $n=1$, then $1=\tau^{0}$. Suppose that $n-1$ can be written as a finite sum of integral powers of $\tau$, say

$$n-1=\sum _{i=-k}^k{a}_i{\tau }^{i}n-1=\backslash sum\_\{i=-k\}^\{k\}\; a\_\{i\}\; \backslash tau^\{i\}$$

where $a_{i} \in{0,1}$ and $n \geq 2$. We will write (1) as

$$n-1=a_k\cdots a_1{a}_0\cdot a_{-1}{a}_{-2}\cdots a_{-k}n-1=a\_\{k\}\; \backslash cdots\; a\_\{1\}\; a\_\{0\}\; \backslash cdot\; a\_\{-1\}\; a\_\{-2\}\; \backslash cdots\; a\_\{-k\}$$

For example,

$$1=1.0=0.11=0.1011=0.1010111=1.0=0.11=0.1011=0.101011$$

Firstly, we will prove that we may assume that in (2) we have $a_{i} a_{i+1}=0$ for all $i$ with $-k \leq i \leq k-1$. Indeed, if we have several occurrences of 11 , then we take the leftmost such occurrence. Since we may assume that it is preceded by a 0 , we can replace 011 with 100 using the identity $\tau^{i+1}+\tau^{i}=\tau^{i+2}$. By doing so repeatedly, if necessary, we will eliminate all occurrences of two 1's standing together. Now we have the representation

$$n-1=\sum _{i=-K}^K{b}_i{\tau }^{i}n-1=\backslash sum\_\{i=-K\}^\{K\}\; b\_\{i\}\; \backslash tau^\{i\}$$

where $b_{i} \in{0,1}$ and $b_{i} b_{i+1}=0$. If $b_{0}=0$ in (3), then we just add $1=\tau^{0}$ to both sides of (3) and we are done. Suppose now that there is 1 in the unit position of (3), that is $b_{0}=1$. If there are two 0 's to the right of it, i.e.

$$n-1=\cdots 1.00\cdots n-1=\backslash cdots\; 1.00\; \backslash cdots$$

then we can replace 1.00 with 0.11 because $1=\tau^{-1}+\tau^{-2}$, and we are done because we obtain 0 in the unit position. Thus we may assume that

$$n-1=\cdots 1.010\cdots n-1=\backslash cdots\; 1.010\; \backslash cdots$$

Again, if we have $n-1=\cdots 1.0100 \cdots$, we may rewrite it as

$$n-1=\cdots 1.0100\cdots =\cdots 1.0011\cdots =\cdots 0.1111\cdots n-1=\backslash cdots\; 1.0100\; \backslash cdots=\backslash cdots\; 1.0011\; \backslash cdots=\backslash cdots\; 0.1111\; \backslash cdots$$

and obtain 0 in the unit position. Therefore, we may assume that

$$n-1=\cdots 1.01010\cdots n-1=\backslash cdots\; 1.01010\; \backslash cdots$$

Since the number of 1's is finite, eventually we will obtain an occurrence of 100 at the end, i.e.

$$n-1=\cdots 1.01010\cdots 100n-1=\backslash cdots\; 1.01010\; \backslash cdots\; 100$$

Then we can shift all 1's to the right to obtain 0 in the unit position, i.e.

$$n-1=\cdots 0.11\cdots 11n-1=\backslash cdots\; 0.11\; \backslash cdots\; 11$$

and we are done.

Problem 3. Let $p \geq 5$ be a prime and let $r$ be the number of ways of placing $p$ checkers on a $p \times p$ checkerboard so that not all checkers are in the same row (but they may all be in the same column). Show that $r$ is divisible by $p^{5}$. Here, we assume that all the checkers are identical. (Solution) Note that $r=\binom{p^{2}}{p}-p$. Hence, it suffices to show that

$$(p^2-1)(p^2-2)\cdots (p^2-(p-1))-(p-1)!\equiv 0(p^4)\backslash left(p^\{2\}-1\backslash right)\backslash left(p^\{2\}-2\backslash right)\; \backslash cdots\backslash left(p^\{2\}-(p-1)\backslash right)-(p-1)!\backslash equiv\; 0\; \backslash quad\backslash left(\backslash bmod\; p^\{4\}\backslash right)$$

Now, let

$$f(x):=(x-1)(x-2)\cdots (x-(p-1))=x^{p-1}+s_{p-2}{x}^{p-2}+\cdots +s_{1}x+s_0\mathrm{.}f(x):=(x-1)(x-2)\; \backslash cdots(x-(p-1))=x^\{p-1\}+s\_\{p-2\}\; x^\{p-2\}+\backslash cdots+s\_\{1\}\; x+s\_\{0\}\; .$$

Then the congruence equation (1) is same as $f\left(p^{2}\right)-s_{0} \equiv 0\left(\bmod p^{4}\right)$. Therefore, it suffices to show that $s_{1} p^{2} \equiv 0\left(\bmod p^{4}\right)$ or $s_{1} \equiv 0\left(\bmod p^{2}\right)$.

Since $a^{p-1} \equiv 1(\bmod p)$ for all $1 \leq a \leq p-1$, we can factor

$$x^{p-1}-1\equiv (x-1)(x-2)\cdots (x-(p-1))(p)x^\{p-1\}-1\; \backslash equiv(x-1)(x-2)\; \backslash cdots(x-(p-1))\; \backslash quad(\backslash bmod\; p)$$

Comparing the coefficients of the left hand side of (3) with those of the right hand side of (2), we obtain $p \mid s_{i}$ for all $1 \leq i \leq p-2$ and $s_{0} \equiv-1(\bmod p)$. On the other hand, plugging $p$ for $x$ in (2), we get

$$f(p)=(p-1)!=s_0=p^{p-1}+s_{p-2}{p}^{p-2}+\cdots +s_{1}p+s_{0}f(p)=(p-1)!=s\_\{0\}=p^\{p-1\}+s\_\{p-2\}\; p^\{p-2\}+\backslash cdots+s\_\{1\}\; p+s\_\{0\}$$

which implies

$$p^{p-1}+s_{p-2}{p}^{p-2}+\cdots +s_2{p}^2=-s_{1}pp^\{p-1\}+s\_\{p-2\}\; p^\{p-2\}+\backslash cdots+s\_\{2\}\; p^\{2\}=-s\_\{1\}\; p$$

Since $p \geq 5, p \mid s_{2}$ and hence $s_{1} \equiv 0\left(\bmod p^{2}\right)$ as desired.

Problem 4. Let $A, B$ be two distinct points on a given circle $O$ and let $P$ be the midpoint of the line segment $A B$. Let $O_{1}$ be the circle tangent to the line $A B$ at $P$ and tangent to the circle $O$. Let $\ell$ be the tangent line, different from the line $A B$, to $O_{1}$ passing through $A$. Let $C$ be the intersection point, different from $A$, of $\ell$ and $O$. Let $Q$ be the midpoint of the line segment $B C$ and $O_{2}$ be the circle tangent to the line $B C$ at $Q$ and tangent to the line segment $A C$. Prove that the circle $O_{2}$ is tangent to the circle $O$. (Solution) Let $S$ be the tangent point of the circles $O$ and $O_{1}$ and let $T$ be the intersection point, different from $S$, of the circle $O$ and the line $S P$. Let $X$ be the tangent point of $\ell$ to $O_{1}$ and let $M$ be the midpoint of the line segment $X P$. Since $\angle T B P=\angle A S P$, the triangle $T B P$ is similar to the triangle $A S P$. Therefore,

$$\frac{PT}{PB}=\frac{PA}{PS}\backslash frac\{P\; T\}\{P\; B\}=\backslash frac\{P\; A\}\{P\; S\}$$

Since the line $\ell$ is tangent to the circle $O_{1}$ at $X$, we have

$$\mathrm{\angle }SPX={90}^{\circ }-\mathrm{\angle }XSP={90}^{\circ }-\mathrm{\angle }APM=\mathrm{\angle }PAM\backslash angle\; S\; P\; X=90^\{\backslash circ\}-\backslash angle\; X\; S\; P=90^\{\backslash circ\}-\backslash angle\; A\; P\; M=\backslash angle\; P\; A\; M$$

which implies that the triangle $P A M$ is similar to the triangle $S P X$. Consequently,

$$\frac{XS}{XP}=\frac{MP}{MA}=\frac{XP}{2MA}\text{ and }\frac{XP}{PS}=\frac{MA}{AP}\backslash frac\{X\; S\}\{X\; P\}=\backslash frac\{M\; P\}\{M\; A\}=\backslash frac\{X\; P\}\{2\; M\; A\}\; \backslash quad\; \backslash text\; \{\; and\; \}\; \backslash quad\; \backslash frac\{X\; P\}\{P\; S\}=\backslash frac\{M\; A\}\{A\; P\}$$

From this and the above observation follows

$$\frac{XS}{XP}\cdot \frac{PT}{PB}=\frac{XP}{2MA}\cdot \frac{PA}{PS}=\frac{XP}{2MA}\cdot \frac{MA}{XP}=\frac{1}{2}\mathrm{.}\backslash frac\{X\; S\}\{X\; P\}\; \backslash cdot\; \backslash frac\{P\; T\}\{P\; B\}=\backslash frac\{X\; P\}\{2\; M\; A\}\; \backslash cdot\; \backslash frac\{P\; A\}\{P\; S\}=\backslash frac\{X\; P\}\{2\; M\; A\}\; \backslash cdot\; \backslash frac\{M\; A\}\{X\; P\}=\backslash frac\{1\}\{2\}\; .$$

Let $A^{\prime}$ be the intersection point of the circle $O$ and the perpendicular bisector of the chord $B C$ such that $A, A^{\prime}$ are on the same side of the line $B C$, and $N$ be the intersection point of the lines $A^{\prime} Q$ and $C T$. Since

$$\mathrm{\angle }NCQ=\mathrm{\angle }TCB=\mathrm{\angle }TCA=\mathrm{\angle }TBA=\mathrm{\angle }TBP\backslash angle\; N\; C\; Q=\backslash angle\; T\; C\; B=\backslash angle\; T\; C\; A=\backslash angle\; T\; B\; A=\backslash angle\; T\; B\; P$$

and

$$\mathrm{\angle }C{A}^{\mathrm{\prime }}Q=\frac{\mathrm{\angle }CAB}{2}=\frac{\mathrm{\angle }XAP}{2}=\mathrm{\angle }PAM=\mathrm{\angle }SPX,\backslash angle\; C\; A^\{\backslash prime\}\; Q=\backslash frac\{\backslash angle\; C\; A\; B\}\{2\}=\backslash frac\{\backslash angle\; X\; A\; P\}\{2\}=\backslash angle\; P\; A\; M=\backslash angle\; S\; P\; X,$$

the triangle $N C Q$ is similar to the triangle $T B P$ and the triangle $C A^{\prime} Q$ is similar to the triangle $S P X$. Therefore

$$\frac{QN}{QC}=\frac{PT}{PB}\text{ and }\frac{QC}{Q{A}^{\mathrm{\prime }}}=\frac{XS}{XP}\mathrm{.}\backslash frac\{Q\; N\}\{Q\; C\}=\backslash frac\{P\; T\}\{P\; B\}\; \backslash quad\; \backslash text\; \{\; and\; \}\; \backslash quad\; \backslash frac\{Q\; C\}\{Q\; A^\{\backslash prime\}\}=\backslash frac\{X\; S\}\{X\; P\}\; .$$

and hence $Q A^{\prime}=2 Q N$ by (1). This implies that $N$ is the midpoint of the line segment $Q A^{\prime}$. Let the circle $O_{2}$ touch the line segment $A C$ at $Y$. Since

$$\mathrm{\angle }ACN=\mathrm{\angle }ACT=\mathrm{\angle }BCT=\mathrm{\angle }QCN\backslash angle\; A\; C\; N=\backslash angle\; A\; C\; T=\backslash angle\; B\; C\; T=\backslash angle\; Q\; C\; N$$

and $|C Y|=|C Q|$, the triangles $Y C N$ and $Q C N$ are congruent and hence $N Y \perp A C$ and $N Y=N Q=N A^{\prime}$. Therefore, $N$ is the center of the circle $O_{2}$, which completes the proof.

Remark: Analytic solutions are possible: For example, one can prove for a triangle $A B C$ inscribed in a circle $O$ that $A B=k(2+2 t), A C=k(1+2 t), B C=k(1+4 t)$ for some positive numbers $k, t$ if and only if there exists a circle $O_{1}$ such that $O_{1}$ is tangent to the side $A B$ at its midpoint, the side $A C$ and the circle $O$.

One obtains $A B=k^{\prime}\left(1+4 t^{\prime}\right), A C=k^{\prime}\left(1+2 t^{\prime}\right), B C=k^{\prime}\left(2+2 t^{\prime}\right)$ by substituting $t=1 / 4 t^{\prime}$ and $k=2 k^{\prime} t^{\prime}$. So, there exists a circle $O_{2}$ such that $O_{2}$ is tangent to the side $B C$ at its midpoint, the side $A C$ and the circle $O$.

In the above, $t=\tan ^{2} \alpha$ and $k=\frac{4 R \tan \alpha}{\left(1+\tan ^{2} \alpha\right)\left(1+4 \tan ^{2} \alpha\right)}$, where $R$ is the radius of $O$ and $\angle A=2 \alpha$. Furthermore, $t^{\prime}=\tan ^{2} \gamma$ and $k^{\prime}=\frac{4 R \tan \gamma}{\left(1+\tan ^{2} \gamma\right)\left(1+4 \tan ^{2} \gamma\right)}$, where $\angle C=2 \gamma$. Observe that $\sqrt{t t^{\prime}}=\tan \alpha \cdot \tan \gamma=\frac{X S}{X P} \cdot \frac{P T}{P B}=\frac{1}{2}$, which implies $t t^{\prime}=\frac{1}{4}$. It is now routine easy to check that $k=2 k^{\prime} t^{\prime}$.

Problem 5. In a circus, there are $n$ clowns who dress and paint themselves up using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set of colours and no more than 20 clowns may use any one particular colour. Find the largest number $n$ of clowns so as to make the ringmaster's order possible. (Solution) Let $C$ be the set of $n$ clowns. Label the colours $1,2,3, \ldots, 12$. For each $i=1,2, \ldots, 12$, let $E_{i}$ denote the set of clowns who use colour $i$. For each subset $S$ of ${1,2, \ldots, 12}$, let $E_{S}$ be the set of clowns who use exactly those colours in $S$. Since $S \neq S^{\prime}$ implies $E_{S} \cap E_{S^{\prime}}=\emptyset$, we have

$$\sum _S\mid E_S\mid =\mathrm{\mid }C\mathrm{\mid }=n\backslash sum\_\{S\}\backslash left|E\_\{S\}\backslash right|=|C|=n$$

where $S$ runs over all subsets of ${1,2, \ldots, 12}$. Now for each $i$,

$$E_S\subseteq E_i\text{ if and only if }i\in SE\_\{S\}\; \backslash subseteq\; E\_\{i\}\; \backslash quad\; \backslash text\; \{\; if\; and\; only\; if\; \}\; \backslash quad\; i\; \backslash in\; S$$

and hence

$$\mid E_i\mid =\sum _{i\in S}\mid E_S\mid \backslash left|E\_\{i\}\backslash right|=\backslash sum\_\{i\; \backslash in\; S\}\backslash left|E\_\{S\}\backslash right|$$

By assumption, we know that $\left|E_{i}\right| \leq 20$ and that if $E_{S} \neq \emptyset$, then $|S| \geq 5$. From this we obtain

$$20\times 12\ge \sum _{i=1}^{12}\mid E_i\mid =\sum _{i=1}^{12}\left(\sum _{i\in S}\mid E_S\mid \right)\ge 5\sum _S\mid E_S\mid =5n20\; \backslash times\; 12\; \backslash geq\; \backslash sum\_\{i=1\}^\{12\}\backslash left|E\_\{i\}\backslash right|=\backslash sum\_\{i=1\}^\{12\}\backslash left(\backslash sum\_\{i\; \backslash in\; S\}\backslash left|E\_\{S\}\backslash right|\backslash right)\; \backslash geq\; 5\; \backslash sum\_\{S\}\backslash left|E\_\{S\}\backslash right|=5\; n$$

Therefore $n \leq 48$. Now, define a sequence $\left{c_{i}\right}_{i=1}^{52}$ of colours in the following way: $1234|5678| 9101112 \mid$ $4123|8567| 1291011 \mid$ $3412|7856| 1112910 \mid$ $2341|6785| 1011129 \mid 1234$

The first row lists $c_{1}, \ldots, c_{12}$ in order, the second row lists $c_{13}, \ldots, c_{24}$ in order, the third row lists $c_{25}, \ldots, c_{36}$ in order, and finally the last row lists $c_{37}, \ldots, c_{52}$ in order. For each $j, 1 \leq j \leq 48$, assign colours $c_{j}, c_{j+1}, c_{j+2}, c_{j+3}, c_{j+4}$ to the $j$-th clown. It is easy to check that this assignment satisfies all conditions given above. So, 48 is the largest for $n$.

Remark: The fact that $n \leq 48$ can be obtained in a much simpler observation that

$$5n\le 12\times 20=2405\; n\; \backslash leq\; 12\; \backslash times\; 20=240$$

There are many other ways of constructing 48 distinct sets consisting of 5 colours. For example, consider the sets

Each of the above 8 sets has 6 distinct subsets consisting of exactly 5 colours. It is easy to check that the 48 subsets obtained in this manner are all distinct.