Consider a sequence of positive integers $a_1, a_2, a_3, . . .$ such that for $k \geq 2$ we have $a_{k+1} =\frac{a_k + a_{k-1}}{2015^i},$ where $2015^i$ is the maximal power of $2015$ that divides $a_k + a_{k-1}.$ Prove that if this sequence is periodic then its period is divisible by $3.$

Find all functions $u:R\rightarrow{R}$ for which there exists a strictly monotonic function $f:R\rightarrow{R}$ such that $f(x+y)=f(x)u(y)+f(y)$ for all $x,y\in{\mathbb{R}}$

What is the largest even integer that cannot be written as the sum of two odd composite numbers?

A plane rectangular grid is given and a “rational point” is defined as a point $(x, y)$ where $x$ and $y$ are both rational numbers. Let $A,B,A',B'$ be four distinct rational points. Let $P$ be a point such that $\frac{A'B'}{AB}=\frac{B'P}{BP} = \frac{PA'}{PA}.$ In other words, the triangles $ABP, A'B'P$ are directly or oppositely similar. Prove that $P$ is in general a rational point and find the exceptional positions of $A'$ and $B'$ relative to $A$ and $B$ such that there exists a $P$ that is not a rational point.

Given a prime $p$ in the form $4m+1$ ($m\in\mathbb{Z}$). Prove that the number $216p^3$ can't be represented in the form $x^2+y^2+z^9$, $x,y,z\in\mathbb{Z}$

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?

Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$. Find $\angle BCA$.

There are $N$ cities in a country. Any two of them are connected either by a road or by an airway. A tourist wants to visit every city exactly once and return to the city at which he started the trip. Prove that he can choose a starting city and make a path, changing means of transportation at most once.

A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

Given a nonconstant function $f : \mathbb{R}^+ \longrightarrow\mathbb{R}$ such that $f(xy) = f(x)f(y)$ for any $x, y > 0$, find functions $c, s : \mathbb{R}^+ \longrightarrow \mathbb{R}$ that satisfy $c\left(\frac{x}{y}\right) = c(x)c(y)-s(x)s(y)$ for all $x, y > 0$ and $c(x)+s(x) = f(x)$ for all $x > 0$.

Let $ABC$ be a triangle and $k < 1$ a positive real number. Let $A_1$, $B_1$, $C_1$ be points on the sides $BC$, $AC$, $AB$ such that $$\frac{A_1B}{BC} = \frac{B_1C}{AC} = \frac{C_1A}{AB} = k.$$ [b](a)[/b] Compute, in terms of $k$, the ratio between the areas of the triangles $A_1B_1C_1$ and $ABC$. [b](b)[/b] Generally, for each $n \ge 1$, the triangle $A_{n+1}B_{n+1}C_{n+1}$ is built such that $A_{n+1}$, $B_{n+1}$, $C_{n+1}$ are points on the sides $B_nC_n$, $A_nC_n$ e $A_nB_n$ satisfying $$\frac{A_{n+1}B_n}{B_nC_n} = \frac{B_{n+1}C_n}{A_nC_n} = \frac{C_{n+1}A_n}{A_nB_n} = k.$$ Compute the values of $k$ such that the sum of the areas of every triangle $A_nB_nC_n$, for $n = 1, 2, 3, \dots$ is equal to $\dfrac{1}{3}$ of the area of $ABC$.

Find all sequences of positive integers $(a_n)_{n\ge 1}$ which satisfy $$a_{n+2}(a_{n+1}-1)=a_n(a_{n+1}+1)$$ for all $n\in \mathbb{Z}_{\ge 1}$. [i](Bogdan Blaga)[/i]

Each face of a cube is painted either red or blue, each with probability $ 1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? $ \textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac{5}{16} \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac{7}{16} \qquad \textbf{(E)}\ \frac12$

Let $f(x)$ be the polynomial with integer coefficients ($f(x)$ is not constant) such that \[(x^3+4x^2+4x+3)f(x)=(x^3-2x^2+2x-1)f(x+1)\] Prove that for each positive integer $n\geq8$, $f(n)$ has at least five distinct prime divisors.

Let p(x) be the polynomial $x^3 + 14x^2 - 2x + 1$. Let $p^n(x)$ denote $p(p^(n-1)(x))$. Show that there is an integer N such that $p^N(x) - x$ is divisible by 101 for all integers x.

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be unit vectors in $R^3$. Prove that $$\sqrt{1-\overrightarrow{a}\cdot\overrightarrow{b}}\le \sqrt{1-\overrightarrow{a}\cdot\overrightarrow{c}}+\sqrt{1-\overrightarrow{c}\cdot\overrightarrow{b}}$$ (A.Mirotin)

For every integer $n \ge 2$ let $B_n$ denote the set of all binary $n$-nuples of zeroes and ones, and split $B_n$ into equivalence classes by letting two $n$-nuples be equivalent if one is obtained from the another by a cyclic permutation.(for example 110, 011 and 101 are equivalent). Determine the integers $n \ge 2$ for which $B_n$ splits into an odd number of equivalence classes.

Let $ T$ be a finite set of positive integers, satisfying the following conditions: 1. For any two elements of $ T$, their greatest common divisor and their least common multiple are also elements of $ T$. 2. For any element $ x$ of $ T$, there exists an element $ x'$ of $ T$ such that $ x$ and $ x'$ are relatively prime, and their least common multiple is the largest number in $ T$. For each such set $ T$, denote by $ s(T)$ its number of elements. It is known that $ s(T) < 1990$; find the largest value $ s(T)$ may take.

An ant is on the Cartesian plane. In a single move, the ant selects a positive integer $k$, then either travels [list] [*] $k$ units vertically (up or down) and $2k$ units horizontally (left or right); or [*] $k$ units horizontally (left or right) and $2k$ units vertically (up or down). [/list] Thus, for any $k$, the ant can choose to go to one of eight possible points. \\ Prove that, for any integers $a$ and $b$, the ant can travel from $(0, 0)$ to $(a, b)$ using at most $3$ moves.

The function $F (x)$ is defined on $R$ and has a second derivative for each value of the variable. Prove that there is a point $x_0$ such that the product $ F(x_0) F''(x_0)$ is non-negative. PS. In my [url=http://www.1543.su/olympiads/soros/20002001/1/1soros00.htm]source[/url], it is not clear if it means $ F(x_0) F''(x_0)$ or $ F(x_0) F'(x_0)$.

Show that if n is an arbitrary natural number and $\sqrt n \leq K \leq \frac{n}{2}$, then there exist n distinct integers, $k_j$ ( j = 1, ..., n ) such that $\bigg | \sum_ {j = 1} ^ ne ^ {ik_jt} \bigg | \geq K$ is satisfied on a subset of the interval $(- \pi, \pi)$ with Lebesgue measure at least $\frac{cn}{K^2}$ , where c is a suitable absolute constant.

Find all positive integers $n\geq 2$, for which there exist positive integers $a_1, a_2, ..., a_n$ such that both sets $$\{a_1, a_2, ..., a_n\}\;\;\;and\;\;\;\{a_1+a_2, a_2+a_3, ..., a_n+a_1\}$$ contain $n$ consecutive integers.

If positive real numbers $x,y,z$ satisfies $x+y+z=3,$ prove that $$\sum_{\text{cyc}} y^2z^2<3+\sum_{\text{cyc}} yz.$$ [i]Proposed by Li4 and Untro368.[/i]