MathNet 2011

Let positive integers $d,d'$ with $d'\gt d$ be divisors of a positive integer $n$. Prove $d'\gt d+d^2/n$.

Let $f = a X^{2} + b X + c \in \mathbb{Z}[X]$ be a polynomial such that for every positive integer $n$, $f(n)$ is a perfect square. Prove that $f = g^{2}$ for some polynomial $g \in \mathbb{Z}[X]$.

Let $x$, $y$ be distinct positive integers. Prove that the number
$$
\frac{(x+y)^2}{x^3 + x y^2 - x^2 y - y^3}
$$
is not an integer.

There are 999 academicians. Each scientific topic interests exactly three academicians, and each pair of academicians shares exactly one topic. Prove that 250 topics can be selected so that no academician is interested in more than one selected topic. (A. Magazinov)

Let $ABC$ be a triangle with $AB \neq AC$. Its incircle has center $I$ and touches the side $BC$ at point $D$. Line $AI$ intersects the circumcircle $\mathcal{C}$ of triangle $ABC$ at $M$ and $DM$ intersects again $\mathcal{C}$ at $P$. Prove that $\widehat{API}=90^{\circ}$.

Let $O$ be the center of rectangle $ABCD$, and let $K$ lie on its circumcircle. Line $CK$ intersects segment $AD$ at $M$, with $AM:MD=2$. Prove that the centroid of triangle $OKD$ lies on the circumcircle of triangle $COD$. (V. Shmarov)

Consider a triangle $ABC$. Let $A_{1}$ be the symmetric point of $A$ with respect to the line $BC$, $B_{1}$ the symmetric point of $B$ with respect to the line $CA$, and $C_{1}$ the symmetric point of $C$ with respect to the line $AB$. Determine the possible set of angles of triangle $ABC$ for which $A_{1}B_{1}C_{1}$ is equilateral.

Let $ABC$ be an acute-angled triangle. A circle passing through its vertex $B$ and its circumcenter $O$ intersects the sides $BC$ and $BA$ at points $P \neq B$ and $Q \neq B$ respectively. Prove that the orthocenter of triangle $POQ$ lies on line $AC$. (T. Emelyanova, L. Emelyanov)

Some cells of a $100 \times 100$ desk contain a token. We call a cell beautiful if the total number of tokens in its neighbors is even (two cells are neighbors if they share a common side). Can it appear that there exists exactly one beautiful cell? (K. Knop)

An acute triangle $ABC$ is inscribed in circle $\omega$. Tangents at $B,C$ meet the tangent at $A$ at $K,L$ respectively. The line through $K$ parallel to $AB$ meets the line through $L$ parallel to $AC$ at $P$. Prove $BP=CP$. (P. Kozhevnikov)

2011 storehouses are connected by roads so that every storehouse is reachable from every other. Storehouse $i$ contains $x_i$ kg of cement. One trip may transport any amount along a direct road. The target is $y_i$ kg at storehouse $i$, with $\sum x_i=\sum y_i$. Find the least $N$ guaranteeing completion in $N$ trips for every admissible distribution and road system.

Points $A$, $B$, $C$, $D$ lie on a line in this order. Draw parallel lines $a$ and $b$ through $A$ and $B$, respectively, and parallel lines $c$ and $d$ through $C$ and $D$, respectively, such that their points of intersection are vertices of a square. Prove that the side length of this square does not depend on the length of segment $B C$.

Find all integers $n$, $n \geq 2$, such that the numbers $1!$, $2!$, $\ldots$, $(n-1)!$ give distinct remainders when divided by $n$.

A pile contains more than $n^2$ stones. Pete and Basil alternately take stones, Pete first. A move may take $k$ stones if $k=1$, or $k$ is prime and less than $n$, or $n$ divides $k$. Whoever takes the last stone wins. Prove that Pete has a winning strategy. (S. Berlov)

2011 real numbers are written on a blackboard. It appears that for each three written numbers $a$, $b$, $c$, the sum $a+b+c$ is also written. Find the least possible number of zeroes among the written numbers.

Let $n$ be a positive integer.
Find all real numbers $x_{1}, x_{2}, \ldots, x_{n}$ such that
$$
\prod_{k=1}^{n}\left(x_{k}^{2}+(k+2) x_{k}+k^{2}+k+1\right)=\left(\frac{3}{4}\right)^{n}(n!)^{2}
$$

In a triangle $A B C$ let $O$ be the circumcenter, $H$ the orthocenter, and $M$ the midpoint of the segment $A H$. The perpendicular at $M$ onto $O M$ intersects lines $A B$ and $A C$ at $P$ and $Q$, respectively. Prove that $M P = M Q$.

Find all polynomials $P$ with real coefficients such that for all $x, y, z \in \mathbb{R}$,
$$
P(x) + P(y) + P(z) + P(x + y + z) = P(x + y) + P(y + z) + P(z + x).
$$

Let $a_{1} + a_{2} + \ldots + a_{n} = 0$ and $|a_{1}| + |a_{2}| + \ldots + |a_{n}| = 1$.
Prove that
$$
\left|a_{1} + 2 a_{2} + \ldots + n a_{n}\right| \leq \frac{n-1}{2}
$$

The shape of a military base is an equilateral triangle of side $10$ kilometers. Security constraints make cellular phone communication possible only within $2.5$ kilometers. Each of $17$ soldiers patrols the base randomly and tries to contact all others. Prove that at each moment at least two soldiers can communicate.