Suppose $ A$, $ B$, and $ C$ are three numbers for which $ 1001C\minus{}2002A\equal{}4004$ and $ 1001B\plus{}3003A\equal{}5005$.The average of the three numbers $ A$, $ B$, and $ C$ is $ \text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ \text{not uniquely determined}$

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

Let $a, b, c, d$ be non-negative real numbers such that \[\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=3.\] Prove that \[3(ab+bc+ca+ad+bd+cd)+\frac{4}{a+b+c+d}\leqslant 5.\][i]Vasile Cîrtoaje and Leonard Giugiuc[/i]

Find the remainder when $2+4+\cdots+2014$ is divided by $1+3+\cdots+2013$. Justify your answer. [i]Proposed by Evan Chen[/i]

Miss Levans has 169 students in her history class and wants to seat them all in a $13\times13$ grid of desks. Each desk is placed at a different vertex of a 12 meter by 12 meter square grid of points she has marked on the floor. The distance between neighboring vertices is exactly 1 meter. Each student has at most three best friends in the class. Best-friendship is mutual: if Lisa is one of Shannon's best friends, then Shannon is also one of Lisa's best friends. Miss Levans knows that if any two best friends sit at points that are 3 meters or less from each other then they will be disruptive and nobody will learn any history. And that is bad. Prove that Miss Levans can indeed place all $169$ students in her class without any such disruptive pairs.

Let $P(x)$ be a polynomial with real coefficients such that $P(12)=20$ and \[ (x-1) \cdot P(16x)= (8x-1) \cdot P(8x) \] holds for all real numbers $x$. Compute the remainder when $P(2014)$ is divided by $1000$. [i]Proposed by Alex Gu[/i]

Find all non negative integers $(a, b)$ such that $$a + 2b - b^2 =\sqrt{2a + a^2 + |2a + 1 - 2b|}.$$

Let $ABC$ be a triangle with sides of length $a$, $b$ and $c$. Let the bisector of the angle $C$ cut $AB$ in $D$. Prove that the length of $CD$ is \[ \frac{2ab\cos \frac{C}{2}}{a+b}. \]

Let $\mathcal{P}$ be the set of vectors defined by \[\mathcal{P} = \left\{\begin{pmatrix} a \\ b \end{pmatrix} \, \middle\vert \, 0 \le a \le 2, 0 \le b \le 100, \, \text{and} \, a, b \in \mathbb{Z}\right\}.\] Find all $\mathbf{v} \in \mathcal{P}$ such that the set $\mathcal{P}\setminus\{\mathbf{v}\}$ obtained by omitting vector $\mathbf{v}$ from $\mathcal{P}$ can be partitioned into two sets of equal size and equal sum.

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.