Find all functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$, $$f(x-f(y))=1-x-y.$$

Points $D$ and $E$ lie on the lateral sides $AB$ and $BC$ respectively of an isosceles triangle $ABC$ in such a way that $\angle BED = 3\angle BDE$. Let $D'$ be the reflection of $D$ about $AC$. Prove that the line $D'E$ passes through the incenter of $ABC$.

An unbiased coin is tossed $n$ times. What is the expected value of $|H-T|$, where $H$ is the number of heads and $T$ is the number of tails?

(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail? (b) The same question for regular pentagons. (A Kanel)

The European zoos with exactly $100$ types of species each are separated into two groups $\hat{A}$ and $\hat{B}$ in such a way that every pair of zoos $(A, B)$ $(A\in\hat{A}, B\in\hat{B})$ have some animal in common. Prove that we can colour the cages in $3$ colours (all animals of the same type live in the same cage) such that no zoo has cages of only one colour

Let $\{a_n\}_{n\ge 0}$ be a sequence of rational numbers given by $a_0 = a_1 = a_2 = a_3 = 1$ and for all $n \ge 4$ we have $a_{n-4}a_n = a_{n-3}a_{n-1} + a^2_{n-2}$. Prove that all the terms of the sequence are integers.

Let ABC be an acute isosceles triangle ($AB=BC$) inscribed in a circle with center $O$ . The line through the midpoint of the chord $AB$ and point $O$ intersects the line $AC$ at $L$ and the circle at the point $P$. Let the bisector of angle $BAC$ intersects the circle at point $K$. Lines $AB$ and $PK$ intersect at point $D$. Prove that the points $L,B,D$ and $P$ lie on the same circle.

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

Infinitely many math beasts stand in a line, all six feet apart, wearing masks, and with clean hands. Grogg starts at the front of the line, holding $n$ pieces of candy, $ n \ge 1,$ and everyone else has none. He passes his candy to the beasts behind him, one piece each to the next $n$ beasts in line. Then, Grogg leaves the line. The other beasts repeat this process: the beast in front, who has $k$ pieces of candy, passes one piece each to the next $k$ beasts in line, and then leaves the line. For some values of $n,$ another beast, besides Grogg, temporarily holds all the candy. For which values of $n$ does this occur?

Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$, \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$. [i]Pakawut Jiradilok[/i]

Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

What is the least positive integer $n$ such that $n!$ is a multiple of $2012^{2012}$?

Find all real numbers $q$, such that for all real $p \geq 0$, the equation $x^2-2px+q^2+q-2=0$ has at least one real root in $(-1;0)$.

Let $\gamma_1, \gamma_2, \gamma_3$ be three circles with radii $3, 4, 9,$ respectively, such that $\gamma_1$ and $\gamma_2$ are externally tangent at $C,$ and $\gamma_3$ is internally tangent to $\gamma_1$ and $\gamma_2$ at $A$ and $B,$ respectively. Suppose the tangents to $\gamma_3$ at $A$ and $B$ intersect at $X.$ The line through $X$ and $C$ intersect $\gamma_3$ at two points, $P$ and $Q.$ Compute the length of $PQ.$ [i]Proposed by Kyle Lee[/i]

Reduce the fraction \[\frac{2121212121210}{1121212121211}\] to its simplest form.

$ ABC $ and $ ADE $ are two triangles with $ \angle ABC=\angle ADE =90^{\circ } $ and such that $ AB=AD. $ The projection of $ B $ on $ AC $ is $ F, $ and the projection of $ D $ on $ AE $ is $ G. $ Prove that $ B,F,E $ are collinear if and only if $ D,G,C $ are collinear.

The bisector of the angle $A$ of a triangle $ABC$ intersects $BC$ in a point $D$ and intersects the circumcircle of the triangle $ABC$ in a point $E$. Let $K,L,M$ and $N$ be the midpoints of the segments $AB,BD,CD$ and $AC$, respectively. Let $P$ be the circumcenter of the triangle $EKL$, and $Q$ be the circumcenter of the triangle $EMN$. Prove that $\angle PEQ=\angle BAC$.

Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eight row and eight column) and places the other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest and the least number that appear in the second row from the top? [asy] add(grid(7,7)); label("$\dots$", (0.5,0.5)); label("$\dots$", (1.5,0.5)); label("$\dots$", (2.5,0.5)); label("$\dots$", (3.5,0.5)); label("$\dots$", (4.5,0.5)); label("$\dots$", (5.5,0.5)); label("$\dots$", (6.5,0.5)); label("$\dots$", (1.5,0.5)); label("$\dots$", (0.5,1.5)); label("$\dots$", (0.5,2.5)); label("$\dots$", (0.5,3.5)); label("$\dots$", (0.5,4.5)); label("$\dots$", (0.5,5.5)); label("$\dots$", (0.5,6.5)); label("$\dots$", (6.5,0.5)); label("$\dots$", (6.5,1.5)); label("$\dots$", (6.5,2.5)); label("$\dots$", (6.5,3.5)); label("$\dots$", (6.5,4.5)); label("$\dots$", (6.5,5.5)); label("$\dots$", (0.5,6.5)); label("$\dots$", (1.5,6.5)); label("$\dots$", (2.5,6.5)); label("$\dots$", (3.5,6.5)); label("$\dots$", (4.5,6.5)); label("$\dots$", (5.5,6.5)); label("$\dots$", (6.5,6.5)); label("$17$", (1.5,1.5)); label("$18$", (1.5,2.5)); label("$19$", (1.5,3.5)); label("$20$", (1.5,4.5)); label("$21$", (1.5,5.5)); label("$16$", (2.5,1.5)); label("$5$", (2.5,2.5)); label("$6$", (2.5,3.5)); label("$7$", (2.5,4.5)); label("$22$", (2.5,5.5)); label("$15$", (3.5,1.5)); label("$4$", (3.5,2.5)); label("$1$", (3.5,3.5)); label("$8$", (3.5,4.5)); label("$23$", (3.5,5.5)); label("$14$", (4.5,1.5)); label("$3$", (4.5,2.5)); label("$2$", (4.5,3.5)); label("$9$", (4.5,4.5)); label("$24$", (4.5,5.5)); label("$13$", (5.5,1.5)); label("$12$", (5.5,2.5)); label("$11$", (5.5,3.5)); label("$10$", (5.5,4.5)); label("$25$", (5.5,5.5)); [/asy] $\textbf{(A) }367 \qquad \textbf{(B) }368 \qquad \textbf{(C) }369 \qquad \textbf{(D) }379 \qquad \textbf{(E) }380$

Given a triangle $ \triangle{ABC} $ whose incenter is $ I $ and $ A $-excenter is $ J $. $ A' $ is point so that $ AA' $ is a diameter of $ \odot\left(\triangle{ABC}\right) $. Define $ H_{1}, H_{2} $ to be the orthocenters of $ \triangle{BIA'} $ and $ \triangle{CJA'} $. Show that $ H_{1}H_{2} \parallel BC $

Let $U=\{1,2,\ldots ,n\}$, where $n\geq 3$. A subset $S$ of $U$ is said to be [i]split[/i] by an arrangement of the elements of $U$ if an element not in $S$ occurs in the arrangement somewhere between two elements of $S$. For example, 13542 splits $\{1,2,3\}$ but not $\{3,4,5\}$. Prove that for any $n-2$ subsets of $U$, each containing at least 2 and at most $n-1$ elements, there is an arrangement of the elements of $U$ which splits all of them.

The Tower of Hanoi is a puzzle with $n$ disks of different sizes and $3$ vertical rods on it. All of the disks are initially placed on the leftmost rod, sorted by size such that the largest disk is on the bottom. On each turn, one may move the topmost disk of any nonempty rod onto any other rod, provided that it is smaller than the current topmost disk of that rod, if it exists. (For instance, if there were two disks on different rods, the smaller disk could move to either of the other two rods, but the larger disk could only move to the empty rod.) The puzzle is solved when all of the disks are moved to the rightmost rod. The specifications normally include an intelligent monk to move the disks, but instead there is a monkey making random moves (with each valid move having an equal probability of being selected). Given $64$ disks, what is the expected number of moves the monkey will have to make to solve the puzzle?

Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$ .

You are given $n\ge 4$ positive real numbers. It turned out that all $\frac{n(n-1)}{2}$ of their pairwise products form an arithmetic progression in some order. Show that all given numbers are equal. [i](Proposed by Anton Trygub)[/i]

$CL$ is bisector of $\angle C$ of $ABC$ and intersect circumcircle at $K$. $I$ - incenter of $ABC$. $IL=LK$. Prove, that $CI=IK$ [i]D. Shiryaev [/i]

Do there exist two real monic polynomials $P(x)$ and $Q(x)$ of degree 3,such that the roots of $P(Q(X))$ are nine pairwise distinct nonnegative integers that add up to $72$? (In a monic polynomial of degree 3, the coefficient of $x^{3}$ is $1$.)