The incircle of a scalene triangle $ABC$ touches the sides $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Triangles $APE$ and $AQF$ are constructed outside the triangle so that \[AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.\]Let $M$ be the midpoint of $BC$. Find $\angle QMP$ in terms of the angles of the triangle $ABC$. [i]Iran, Shayan Talaei[/i]

Determine all triples $(x,y,z)$ of real numbers that satisfy all of the following three equations: $$\begin{cases} \lfloor x \rfloor + \{y\} =z \\ \lfloor y \rfloor + \{z\} =x \\ \lfloor z \rfloor + \{x\} =y \end{cases}$$

For $x$ real, the inequality $1\le |x-2|\le 7$ is equivalent to $\textbf{(A) }x\le 1\text{ or }x\ge 3\qquad\textbf{(B) }1\le x\le 3\qquad\textbf{(C) }-5\le x\le 9\qquad$ $\textbf{(D) }-5\le x\le 1\text{ or }3\le x\le 9\qquad \textbf{(E) }-6\le x\le 1\text{ or }3\le x\le 10$

[b]p1.[/b] What is $7\%$ of one half of $11\%$ of $20000$ ? [b]p2.[/b] Three circles centered at $A, B$, and $C$ are tangent to each other. Given that $AB = 8$, $AC = 10$, and $BC = 12$, find the radius of circle $ A$. [b]p3. [/b]How many positive integer values of $x$ less than $2012$ are there such that there exists an integer $y$ for which $\frac{1}{x} +\frac{2}{2y+1} =\frac{1}{y}$ ? [b]p4. [/b]The positive difference between $ 8$ and twice $x$ is equal to $11$ more than $x$. What are all possible values of $x$? [b]p5.[/b] A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line? [b]p6.[/b] A polygon is composed of only angles of degrees $138$ and $150$, with at least one angle of each degree. How many sides does the polygon have? [b]p7.[/b] $M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \ne 0$ and $L \ne 0$. Given that the sum $MATH +LMT$, where each letter represents a digit, equals $2012$, what is the average of all possible values of the three-digit integer $LMT$? [b]p8. [/b]A square with side length $\sqrt{10}$ and two squares with side length $\sqrt{7}$ share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger square? [b]p9.[/b] Consider the sequence $2012, 12012, 20120, 20121, ...$. This sequence is the increasing sequence of all integers that contain “$2012$”. What is the $30$th term in this sequence? [b]p10.[/b] What is the coefficient of the $x^5$ term in the simplified expansion of $(x +\sqrt{x} +\sqrt[3]{x})^{10}$ ? PS. You had better use hide for answers.

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 $$\frac{a}{b}=\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}$$