MathNet 2014

Prove that there exists a positive integer $n$ such that the last digits of $n^{3}$ are $\square 201320132013$.

Let $ABC$ be a triangle, $I$ its incenter, and $\omega$ a circle of center $I$. Points $A'$, $B'$, $C'$ are on $\omega$ such that rays $IA'$, $IB'$, $IC'$ starting from $I$ intersect perpendicularly sides $BC$, $CA$, $AB$, respectively. Prove that lines $AA'$, $BB'$, $CC'$ are concurrent.

Let $n \gt 3$ be an odd positive integer not divisible by $3$. Determine if it is possible to form an $n \times n$ array of numbers such that

a. The set of the numbers in each row is a permutation of $0, 1, \ldots, n-1$; the set of the numbers in each column is a permutation of $0, 1, \ldots, n-1$;

b. The board is totally non-symmetric: for $1 \leq i \lt j \leq n$ and $1 \leq i' \lt j' \leq n$, if $(i, j) \neq (i', j')$ then $(a_{i, j}, a_{j, i}) \neq (a_{i', j'}, a_{j', i'})$, where $a_{i, j}$ denotes the entry in the $i^{\text{th}}$ row and $j^{\text{th}}$ column.

Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.

Circles $\omega_{1}$ and $\omega_{2}$ meet at $P$ and $Q$. Segments $AC$ and $BD$ are chords of $\omega_{1}$ and $\omega_{2}$ respectively, such that segment $AB$ and ray $CD$ meet at $P$. Ray $BD$ and segment $AC$ meet at $X$. Point $Y$ lies on $\omega_{1}$ such that $PY \parallel BD$. Point $Z$ lies on $\omega_{2}$ such that $PZ \parallel AC$. Prove that points $Q, X, Y, Z$ are collinear.

Let $S$ be a set of positive real numbers with five elements such that for any distinct $a, b, c$ in $S$, the number $a b + b c + c a$ is rational. Prove that for any $a$ and $b$ in $S$, $\frac{a}{b}$ is a rational number.

Find all positive integers $n$ for which $1-5^{n}+5^{2 n+1}$ is a perfect square.

Let $ABC$ be a triangle, $D$ the midpoint of side $BC$ and $E$ the intersection point of the bisector of angle $\angle BAC$ with side $BC$. The perpendicular bisector of $AE$ intersects the bisectors of angles $\angle CBA$ and $\angle CDA$ at $M$ and $N$, respectively. The bisectors of angles $\angle CBA$ and $\angle CDA$ intersect at $P$. Prove that points $A, M, N, P$ are concyclic.

Let $\mathbb{R}$ be the set of real numbers. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$,
$$
f(xf(y) + x) = xy + f(x).
$$

Let $ABC$ be a triangle with $\angle B \leq \angle C$, $I$ its incenter and $D$ the intersection point of line $AI$ with side $BC$. Let $M$ and $N$ be points on sides $BA$ and $CA$, respectively, such that $BM = BD$ and $CN = CD$. The circumcircle of triangle $CMN$ intersects again line $BC$ at $P$. Prove that quadrilateral $DIM P$ is cyclic.

Let $0 \lt a_1 \lt a_2 \lt \dots \lt a_n$ be real numbers. Prove that

$$
\begin{aligned}
&(\frac{1}{1+a_1} + \frac{1}{1+a_2} + \dots + \frac{1}{1+a_n})^2 \le \frac{1}{a_1} + \frac{1}{a_2-a_1} + \frac{1}{a_3-a_2} \\
&+ \dots + \frac{1}{a_n-a_{n-1}}.
\end{aligned}
$$

There are $2015$ coins on a table. For $i=1,2, \ldots, 2015$ in succession, one must turn over exactly $i$ coins. Prove that it is always possible either to make all of the coins face up or to make all of the coins face down, but not both.

Two circles $\omega_{1}$ and $\omega_{2}$, with center $O_{1}$ and $O_{2}$ respectively, meet at points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_{1}$. Lines $X A$ and $Y A$ meet $\omega_{2}$ at $Z$ and $W$, respectively, such that $A$ lies between $X$ and $Z$ and between $Y$ and $W$. Let $M$ be the midpoint of $O_{1} O_{2}$, $S$ be the midpoint of $X A$ and $T$ be the midpoint of $W A$. Prove that $M S = M T$ if and only if $X, Y, Z$ and $W$ are concyclic.

Let $p \geq 2$ be a prime number and $\frac{a_{p}}{b_{p}} = 1 + \frac{1}{2} + \cdots + \frac{1}{p^{2}-1}$, where $a_{p}$ and $b_{p}$ are two relatively prime positive integers. Compute $\operatorname{gcd}\left(p, b_{p}\right)$.

Two players play the following game. They take turns writing a positive integer greater than one, but may not write a linear combination of previously written numbers with nonnegative integer coefficients. A player who cannot write a new number loses. Does either player have a winning strategy, and if so, which player? (Aleksandar Ilić / “Kvant”)

Let $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$ be nonzero real numbers. Prove that the polynomial
$$
P(X) = \prod_{k=0}^{4} \left( a_{k+1} X^{4} + a_{k+2} X^{3} + a_{k+3} X^{2} + a_{k+4} X + a_{k+5} \right),
$$
where $a_{5+i} = a_{i}$ for $i = 1, 2, 3, 4$, has a root with negative real part.

A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \geq 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n))=2$.

A perfect number is an integer that equals half the sum of its positive divisors. For example, because $2 \cdot 28 = 1 + 2 + 4 + 7 + 14 + 28$, $28$ is a perfect number.

a. A square-free integer is an integer not divisible by a square of any prime number. Find all square-free integers that are perfect numbers.

b. Prove that no perfect square is a perfect number.

Let $S=\{(a, b) \mid a, b=1,2,3,4$, and $a \neq b\}$, and consider all nonzero polynomials $p(X, Y)$ with integer coefficients such that $p(a, b)=0$ for every element $(a, b)$ in $S$.
(a) What is the minimal degree of such polynomial $p(X, Y)$?
(b) Determine all such polynomials $p(X, Y)$ with minimal degree.

The 2013 numbers
$$
\frac{1}{1 \times 2}, \frac{1}{2 \times 3}, \frac{1}{3 \times 4}, \ldots, \frac{1}{2013 \times 2014}
$$
are arranged randomly on a circle.

a. Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{4000}$.

b. Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{10000}$.