International Competitions 2011

Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 + a_2 + a_3 + a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i,j)$ with $1 \leq i < j \leq 4$ for which $a_i + a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal{S}$ are collinear. A windmill is a process that starts with a line $\ell$ going through a single point $P \in \mathcal{S}$. The line rotates clockwise about the pivot $P$ until the first time that the line meets some other point belonging to $\mathcal{S}$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $\mathcal{S}$. This process continues indefinitely.

Show that we can choose a point $P$ in $\mathcal{S}$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal{S}$ as a pivot infinitely many times.

Let $f: \mathbb{R} \to \mathbb{R}$ be a real-valued function defined on the set of real numbers that satisfies $$f(x + y) \leq yf(x) + f(f(x))$$ for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$.

Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \ldots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.

Determine the number of ways in which this can be done.

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m - n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$.

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a$, $\ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a$, $\ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$.

Initially, only the integer $44$ is written on a board. An integer $a$ on the board can be replaced with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac{1}{4}(a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b_2, \ldots, b_n$ on the board. Prove that $$\frac{b_1^2 + b_2^2 + \ldots + b_n^2}{n} \geq 2011.$$

Let $n \geq 3$ be an integer. John and Mary play the following game: First John labels the sides of a regular $n$-gon with the numbers $1, 2, \ldots, n$ in whatever order he wants, using each number exactly once. Then Mary divides this $n$-gon into triangles by drawing $n - 3$ diagonals which do not intersect each other inside the $n$-gon. All these diagonals are labeled with number $1$. Into each of the triangles the product of the numbers on its sides is written. Let $S$ be the sum of those $n - 2$ products.

Determine the value of $S$ if Mary wants the number $S$ to be as small as possible and John wants $S$ to be as large as possible and if they both make the best possible choices.

In a plane the circles $\mathcal{K}_1$ and $\mathcal{K}_2$ with centers $I_1$ and $I_2$, respectively, intersect in two points $A$ and $B$. Assume that $\angle I_1AI_2$ is obtuse. The tangent to $\mathcal{K}_1$ in $A$ intersects $\mathcal{K}_2$ again in $C$ and the tangent to $\mathcal{K}_2$ in $A$ intersects $\mathcal{K}_1$ again in $D$. Let $\mathcal{K}_3$ be the circumcircle of the triangle $BCD$. Let $E$ be the midpoint of that arc $CD$ of $\mathcal{K}_3$ that contains $B$. The lines $AC$ and $AD$ intersect $\mathcal{K}_3$ again in $K$ and $L$, respectively. Prove that the line $AE$ is perpendicular to $KL$.

Let $k$ and $m$, with $k > m$, be positive integers such that the number $km(k^2 - m^2)$ is divisible by $k^3 - m^3$. Prove that $(k - m)^3 > 3km$.

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that the equality $$y^{2} f(x) + x^{2} f(y) + xy = x y f(x + y) + x^{2} + y^{2}$$ holds for all $x, y \in \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers.

Let $a, b, c$ be positive real numbers such that $$\frac{a}{1 + a} + \frac{b}{1 + b} + \frac{c}{1 + c} = 2.$$ Prove that $$\frac{\sqrt{a} + \sqrt{b} + \sqrt{c}}{2} \geqslant \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}} + \frac{1}{\sqrt{c}}.$$

For an integer $n \geq 3$, let $\mathcal{M}$ be the set $\{(x, y) \mid x, y \in \mathbb{Z}, 1 \leq x \leq n, 1 \leq y \leq n\}$ of points in the plane. ($\mathbb{Z}$ is the set of integers.)

What is the maximum possible number of points in a subset $S \subseteq \mathcal{M}$ which does not contain three distinct points being the vertices of a right triangle?

Let $n\geq 3$ be an integer. At a MEMO-like competition, there are $3n$ participants, there are $n$ languages spoken, and each participant speaks exactly three different languages.

Prove that at least $\left\lceil\dfrac{2n}{9}\right\rceil$ of the spoken languages can be chosen in such a way that no participant speaks more than two of the chosen languages.

($\lceil x\rceil$ is the smallest integer which is greater than or equal to $x$.)

Let $ABCDE$ be a convex pentagon with all five sides equal in length. The diagonals $AD$ and $EC$ meet in $S$ with $\angle ASE=60^{\circ}$. Prove that $ABCDE$ has a pair of parallel sides.

Let $ABC$ be an acute triangle. Denote by $B_{0}$ and $C_{0}$ the feet of the altitudes from vertices $B$ and $C$, respectively. Let $X$ be a point inside the triangle $ABC$ such that the line $BX$ is tangent to the circumcircle of the triangle $AXC_{0}$ and the line $CX$ is tangent to the circumcircle of the triangle $AXB_{0}$. Show that the line $AX$ is perpendicular to $BC$.

Let $A$ and $B$ be disjoint nonempty sets with $A\cup B=\{1,2,3,\ldots,10\}$. Show that there exist elements $a\in A$ and $b\in B$ such that the number $a^{3}+ab^{2}+b^{3}$ is divisible by $11$.

We call a positive integer $n$ amazing if there exist positive integers $a,b,c$ such that the equality $$n=(b,c)(a,bc)+(c,a)(b,ca)+(a,b)(c,ab)$$ holds. Prove that there exist $2011$ consecutive positive integers which are amazing.

(By $(m,n)$ we denote the greatest common divisor of positive integers $m$ and $n$.)