Let $ABCD$ be a trapezoid which is not a parallelogram, such that $AD$ is parallel to $BC$. Let $O=BD\cap AC$ and $S$ be the second intersection of the circumcircles of triangles $AOB$ and $DOC$. Prove that the circumcircles of triangles $ASD$ and $BSC$ are tangent.

Given a $n\times n$ table, where $n$ is odd. There is either $1$ or $-1$ in its every field. A product of the numbers in the column is written under every column. A product of the numbers in the row is written to the right of every row. Prove that the sum of $2n$ products doesn't equal to $0$.

Compute the exact value of \[ \sum_{a=1}^{\infty}\sum_{b=1}^{\infty} \frac{\gcd(a, b)}{(a + b)^3}. \] If necessary, you may express your answer in terms of the Riemann zeta function, $Z(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$, for integers $s \geq 2$.

$a,b$ are odd numbers. Prove, that exists natural $k$ that $b^k-a^2$ or $a^k-b^2$ is divided by $2^{2018}$.

The sequence $ (x_n)$ is defined as; $ x_1\equal{}a$, $ x_2\equal{}b$ and for all positive integer $ n$, $ x_{n\plus{}2}\equal{}2008x_{n\plus{}1}\minus{}x_n$. Prove that there are some positive integers $ a,b$ such that $ 1\plus{}2006x_{n\plus{}1}x_n$ is a perfect square for all positive integer $ n$.

Let $x$ satisfy $x^4+x^3+x^2+x+1=0$. Compute the value of $(5x+x^2)(5x^2+x^4)(5x^3+x^6)(5x^4+x^8)$. [i]Proposed by Andrew Zhao[/i]

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

Point $X$ moves along side $AB$ of triangle $ABC$, and point $Y$ moves along its circumcircle in such a way that line $XY$ passes through the midpoint of arc $AB$. Find the locus of the circumcenters of triangles $IXY$ , where I is the incenter of $ ABC$.

Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying (1) for each $n_i$, its digits belong to the set $\{1,2\}$; (2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right. Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.

Find all $5$-tuples of different four-digit integers with the same initial digit such that the sum of the five numbers is divisible by four of them.

In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.

Given a triangle $ ABC $. Find the rectangle of smallest area containing the triangle.

Let $\theta_1,\theta_2,\cdots,\theta_n$ be $n$ real numbers such that $\sin \theta_1+\sin \theta_2+\cdots+\sin \theta_n=0$. Prove that \[|\sin \theta_1+2 \sin \theta_2+\cdots +n \sin \theta_n| \leq \left[ \frac{n^2}{4} \right]\]

A polynomial $ f(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}c$ is such that $ b<0$ and $ ab\equal{}9c$. Prove that the polynomial $ f$ has three different real roots.

A salesman and a customer altogether have $1999$ rubles in coins and bills of $1, 5, 10, 50, 100, 500 , 1000$ rubles. The customer has enough money to buy a Cat in the Bag which costs the integer number of rubles. Prove that the customer can buy the Cat and get the correct change.

Compute the remainder when $2^{3^5}+ 3^{5^2}+ 5^{2^3}$ is divided by $30$.

A triangle $ABC$ with circumcircle $\omega$ satisfies $|AB| > |AC|$. Points $X$ and $Y$ on $\omega$ are different from $A$, such that the line $AX$ passes through the midpoint of $BC$, $AY$ is perpendicular to $BC$, and $XY$ is parallel to $BC$. Find $\angle BAC$.

Let [i]n[/i] $\ge$ 3 be an integer and let ([i]$p_1$[/i], [i]$p_2$[/i], [i]$p_3$[/i], $\dots$, [i]$p_n$[/i]) be a permutation of {1, 2, 3, $\dots$ [i]n[/i]}. For this permutation we say that [i]$p_t$[/i] is a [i]turning point[/i] if 2$\le$ [i]t[/i] $\le$ [i]n[/i]-1 and ([i]$p_t$[/i] - [i]$p_{t-1}$[/i])([i]$p_t$[/i] - [i]$p_{t+1}$[/i]) > 0 For example, for [i]n[/i] = 8, the permutation (2, 4, 6, 7, 5, 1, 3, 8) has two turning points: [i]$p_4$[/i] = 7 and [i]$p_6$[/i] = 1. For fixed [i]n[/i], let [i]q[/i]([i]n)[/i] denote the number of permutations of {1, 2, 3, $\dots$ [i]n[/i]} with exactly one turning point. Find all [i]n[/i] $\ge$ 3 for which [i]q[/i]([i]n)[/i] is a perfect square.

Let $M$ be the midpoint of the side $BC$ of the $\triangle ABC$. The perpendicular at $A$ to $AM$ meets $(ABC)$ at $K$. The altitudes $BE,CF$ of the triangle $ABC$ meet $AK$ at $P, Q$. Show that the radical axis of the circumcircles of the triangles $PKE, QKF$ is perpendicular to $BC$.

Aladdin has a set of coins with weights $ 1, 2, \ldots, 20$ grams. He can ask Genie about any two coins from the set which one is heavier, but he should pay Genie some other coin from the set before. (So, with every question the set of coins becomes smaller.) Can Aladdin find two coins from the set with total weight at least $ 28$ grams?

Given the natural numbers $a$ and $b$, with $1 \le a <b$, prove that there exist natural numbers $n_1<n_2< ...<n_k$, with $k \le a$ such that $$\frac{a}{b}=\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}$$

Let $ f(n)$ be defined for $ n \in \mathbb{N}$ by $ f(1)\equal{}2$ and $ f(n\plus{}1)\equal{}f(n)^2\minus{}f(n)\plus{}1$ for $ n \ge 1$. Prove that for all $ n >1:$ $ 1\minus{}\frac{1}{2^{2^{n\minus{}1}}}<\frac{1}{f(1)}\plus{}\frac{1}{f(2)}\plus{}...\plus{}\frac{1}{f(n)}<1\minus{}\frac{1}{2^{2^n}}$

Let $C$ be a fixed unit circle in the cartesian plane. For any convex polygon $P$ , each of whose sides is tangent to $C$, let $N( P, h, k)$ be the number of points common to $P$ and the unit circle with center at $(h, k).$ Let $H(P)$ be the region of all points $(x, y)$ for which $N(P, x, y) \geq 1$ and $F(P)$ be the area of $H(P).$ Find the smallest number $u$ with $$ \frac{1}{F(P)} \int \int N(P,x,y)\;dx \;dy <u$$ for all polygons $P$, where the double integral is taken over $H(P).$

Real numbers $x,y,z$ satisfy $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0$$ and none of them lies in the open interval $(-1,1)$. Find the maximum value of $x+y+z$. [i]Proposed by Jaromír Šimša[/i]

Addison and Emerson are playing a card game with three rounds. Addison has the cards $1, 3$, and $5$, and Emerson has the cards $2, 4$, and $6$. In advance of the game, both designate each one of their cards to be played for either round one, two, or three. Cards cannot be played for multiple rounds. In each round, both show each other their designated card for that round, and the person with the higher-numbered card wins the round. The person who wins the most rounds wins the game. Let $m/n$ be the probability that Emerson wins, where $m$ and $n$ are relatively prime positive integers. Find $m +n$. [i]Proposed by Ada Tsui[/i]