Find the number of real solutions $(x,y)$ to the system of equations: $$\begin{cases} \sin(x^2-y) = 0 \\ |x|+|y|=2\pi \end{cases}$$

For what integer values of $n$ is $1+n+\frac{n^2}{2}+\cdots +\frac{n^n}{n!}$ an integer?

Let us have an infinite grid of unit squares. We write in every unit square a real number, such that the absolute value of the sum of the numbers from any $n*n$ square is less or equal than $1$. Prove that the absolute value of the sum of the numbers from any $m*n$ rectangular is less or equal than $4$.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Define the sequence $(a_n)$ as follows: $a_0=a_1=1$ and for $k\ge 2$, $a_k=a_{k-1}+a_{k-2}+1$. Determine how many integers between $1$ and $2004$ inclusive can be expressed as $a_m+a_n$ with $m$ and $n$ positive integers and $m\not= n$.

Let $a^b$ denote the number $ab$. The order of operations in the expression 7^7^7^7^7^7^7 must be determined by parentheses ($5$ pairs of parentheses are needed). Is it possible to put parentheses in two distinct ways so that the value of the expression be the same?

Let $p$ be an odd prime number and let $a_1,a_2,...,a_n \in \mathbb Q^+$ be rational numbers. Prove that \[\mathbb Q(\sqrt[p]{a_1}+\sqrt[p]{a_2}+...+\sqrt[p]{a_n})=\mathbb Q(\sqrt[p]{a_1},\sqrt[p]{a_2},...,\sqrt[p]{a_n}).\]

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Find all five-digit numbers that satisfy the following conditions: 1. The number is a palindrome. 2. The middle digit is twice the value of the first digit. 3. The number is a perfect square. [i]Proposed by Tamar Turashvili, Georgia [/i]

Let be a number $ x $ and three positive numbers $ a,b,c $ such that $ a^x+b^x=c^x. $ Prove that $ a^y,b^y,c^y $ are the lenghts of the sides of an obtuse triangle if and only if $ y<x<2y. $

On a circle noted $n$ points. It turned out that among the triangles with vertices in these points exactly half of the acute. Find all values $n$ in which this is possible.

Given two internally tangent circles; in the bigger one we inscribe an equilateral triangle. From each of the vertices of this triangle, we draw a tangent to the smaller circle. Prove that the length of one of these tangents equals the sum of the lengths of the two other tangents.

The points $A_1,B_1,C_1$ belong to $[BC],[CA],[AB]$ sides of the $ABC$ triangle respectively. The $[AA_1], [BB_1], [CC_1]$ segments split the $ABC$ onto $4$ smaller triangles and $3$ quadrangles. It is known, that the smaller triangles have the same area. Prove that the quadrangles have equal areas. What is the quadrangle area, it the small triangle has the unit area?

Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$. [i](Karl Czakler)[/i]

Let $E$ be the midpoint of side $[AB]$ of square $ABCD$. Let the circle through $B$ with center $A$ and segment $[EC]$ meet at $F$. What is $|EF|/|FC|$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ \dfrac{3}{2} \qquad\textbf{(C)}\ \sqrt{5}-1 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \sqrt{3} $

Given a circle, point $A$ on it and point $M$ inside it. We consider the chords $BC$ passing through $M$. Prove that the circles passing through the midpoints of the sides of all the triangles $ABC$ are tangent to a fixed circle.

Raina the frog is playing a game in a circular pond with six lilypads around its perimeter numbered clockwise from $1$ to $6$ (so that pad $1$ is adjacent to pad $6$). She starts at pad $1$, and when she is on pad i, she may jump to one of its two adjacent pads, or any pad labeled with $j$ for which $j - i$ is even. How many jump sequences enable Raina to hop to each pad exactly once?

Determine the sum of the two largest prime factors of the integer $89! + 90!.$

Let triangle $\triangle{ABC}$ be acute. Point $D$ is the foot of the altitude of $\triangle{ABC}$ from $A$ to $\overline{BC},$ and $E$ is the foot of the altitude of $\triangle{ABC}$ from $B$ to $\overline{AC}.$ Let $F$ denote the point of intersection between $\overline{BE}$ and $\overline{AD},$ and let $G$ denote the point of intersection between $\overline{CF}$ and $\overline{DE}.$ The areas of triangles $\triangle{EFG}, \triangle{CDG},$ and $\triangle{CEG}$ are $1,4,$ and $3,$ respectively. Find the area of $\triangle{ABC}.$

All sides and diagonals of a $25$-gon are drawn either red or white. Show that at least $500$ triangles, having all three sides are in same color and having all three vertices from the vertices of the $25$-gon, can be found.

There are $ 6n \plus{} 4$ mathematicians participating in a conference which includes $ 2n \plus{} 1$ meetings. Each meeting has one round table that suits for $ 4$ people and $ n$ round tables that each table suits for $ 6$ people. We have known that two arbitrary people sit next to or have opposite places doesn't exceed one time. 1. Determine whether or not there is the case $ n \equal{} 1$. 2. Determine whether or not there is the case $ n > 1$.

On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?

The feet of the altitudes drawn from vertices $A$ and $B$ of an acute triangle $ABC$ are $K$ and $L$, respectively. Prove that if $|BK| = |KL|$ then the triangle $ABC$ is isosceles.

Square $S_{1}$ is $1\times 1.$ For $i\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$ [asy] size(250); path p=rotate(45)*polygon(4); int i; for(i=0; i<5; i=i+1) { draw(shift(2-(1/2)^(i-1),0)*scale((1/2)^i)*p); } label("$S_1$", (0,-0.75)); label("$S_2$", (1,-0.75)); label("$S_3$", (3/2,-0.75)); label("$\cdots$", (7/4, -3/4)); label("$\cdots$", (2.25, 0));[/asy]

In decimal representation, we call an integer [i]$k$-carboxylic[/i] if and only if it can be represented as a sum of $k$ distinct integers, all of them greater than $9$, whose digits are the same. For instance, $2008$ is [i]$5$-carboxylic[/i] because $2008=1111+666+99+88+44$. Find, with an example, the smallest integer $k$ such that $8002$ is [i]$k$-carboxylic[/i].