IMO Problem Collection: Table of contents

Given a triangle \(ABC\), let \(J\) be the center of the excircle opposite to the vertex \(A\). This circle is tangent to lines \(AB\), \(AC\), and \(BC\) at \(K\), \(L\), and \(M\), respectively. The lines \(BM\) and \(JF\) meet at \(F\), and the lines \(KM\) and \(CJ\) meet at \(G\). Let \(S\) be the intersection of \(AF\) and \(BC\), and let \(T\) be the intersection of \(AG\) and \(BC\). Prove that \(M\) is the midpoint of \(BC\).

Let \(a_2\), \(a_3\), \(\dots\), \(a_n\) be positive real numbers that satisfy \(a_2\cdot a_3\cdots a_n=1\). Prove that \[\left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+1\right)^n > n^n.\]

The *liar’s guessing game* is a game played between two players \(A\) and \(B\). The rules of the game depend on two positive integers \(k\) and \(n\) which are known to both players.

At the start of the game the player \(A\) chooses integers \(x\) and \(N\) with \(1\leq x\leq N\). Player \(A\) keeps \(x\) secret, and truthfully tells \(N\) to the player \(B\). The player \(B\) now tries to obtain information about \(x\) by asking player \(A\) questions as follows: each question consists of \(B\) specifying an arbitrary set \(S\) of positive integers (possibly one specified in some previous question), and asking \(A\) whether \(x\) belongs to \(S\). Player \(B\) may ask as many questions as he wishes. After each question, player \(A\) must immediately answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is that, among any \(k+1\) consecutive answers, at least one answer must be truthful.

After \(B\) has asked as many questions as he wants, he must specify a set \(X\) of at most \(n\) positive integers. If \(x\in X\), then \(B\) wins; otherwise, he loses. Prove that:

(a) If \(n\geq 2^k\) then \(B\) has a winning strategy.

(b) There exists a positive integer \(k_0\) such that for every \(k\geq k_0\) there exists an integer \(n\geq 1.99^k\) for which \(B\) cannot guarantee a victory.

Find all functions \(f:\mathbb Z\to\mathbb Z\) such that, for all integers \(a\), \(b\), \(c\) with \(a+b+c=0\) the following equality holds: \[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]

Given a triangle \(ABC\), assume that \(\angle C=90^{\circ}\). Let \(D\) be the foot of the perpendicular from \(C\) to \(AB\), and let \(X\) be any point of the segment \(CD\). Let \(K\) and \(L\) be the points on the segments \(AX\) and \(BX\) such that \(BK=BC\) and \(AL=AC\), respectively. Let \(M\) be the intersection of \(AL\) and \(BK\).

Prove that \(MK=ML\).

Find all positive integers \(n\) for which there exist non-negative integers \(a_1\), \(a_2\), \(\dots\), \(a_n\) such that \[\frac1{2^{a_1}}+\frac1{2^{a_2}}+\cdots+\frac1{2^{a_n}}=\frac1{3^{a_1}}+\frac2{3^{a_2}}+\cdots+\frac{n}{3^{a_n}}=1.\]