A red coin is placed in a cell of $2n \times 2n$ board. Every move it can either move like a bishop and change its color (red to blue, blue to red), or move like a knight and not change its color. After some time the coin has visited every cell exactly twice. Prove that the number of cells in which the coin was both red and blue is even. [i]M. Zorka[/i]

Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm, we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as [i]value [/i] of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .

Prove that for any $\delta$ greater than 1 and any positive number $\epsilon$, there is an $n$ such that $\left \vert \frac{\sigma (n)}{n} -\delta \right \vert < \epsilon$.

Three parallel lines $d_a, d_b, d_c$ pass through the vertex of triangle $ABC$. The reflections of $d_a, d_b, d_c$ in $BC, CA, AB$ respectively form triangle $XYZ$. Find the locus of incenters of such triangles. (C.Pohoata)

Prove that if $ a_{1},a_{2},\ldots,a_{n}$, $ b_{1},b_{2},\ldots,b_{n}$ are arbitrary taken real numbers and $ c_{1},c_{2},\ldots,c_{n}$ are positive real numbers, than $ \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}a_{j}}{c_{i} \plus{} c_{j}}\right)\left(\sum_{i,j \equal{} 1}^{n}\frac {b_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)\ge \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)^{2}$.

In parallelogram $ABCD,$ $E$ is a point on $\overline{AD}$ such that $\overline{CE} \perp \overline{AD},$ $F$ is a point on $\overline{CD}$ such that $\overline{AF} \perp \overline{CD},$ and $G$ is a point on $\overline{BC}$ such that $\overline{AG} \perp \overline{BC}.$ Let $H$ be a point on $\overline{GF}$ such that $\overline{AH} \perp \overline{GF},$ and let $J$ be the intersection of lines $EF$ and $BC.$ Given that $AH=8, AE=6,$ and $EF=4,$ compute $CJ.$

Let $P(x)$ be a polynomial taking integer values at integer inputs. Are there infinitely many natural numbers that are not representable in the form $P(k)-2^n$ where $n{}$ and $k{}$ are non-negative integers? [i]Proposed by F. Petrov[/i]

For a convex hexagon $AHSIMC$ whose side lengths are all $1$, let $Z$ and $z$ be the maximum and minimum values, respectively, of the three diagonals $AI$, $HM$, and $SC$. If $\sqrt{x}\le Z \le \sqrt{y} $ and $\sqrt{q}\le z \le \sqrt{r} $ , find the product $qrxy$, if $q$,$ r$, $x$, and $y$ are all integers.

Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the $\emph{last}$ forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be?

Let $\mathbb{Q}$ be the set of rational numbers, $\mathbb{Z}$ be the set of integers. On the coordinate plane, given positive integer $m$, define $$A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}.$$ For segment $MN$, define $f_m(MN)$ as the number of points on segment $MN$ belonging to set $A_m$. Find the smallest real number $\lambda$, such that for any line $l$ on the coordinate plane, there exists a constant $\beta (l)$ related to $l$, satisfying: for any two points $M,N$ on $l$, $$f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l)$$

Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?

Determine all triples of positive integers $(A,B,C)$ for which some function $f \colon \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ satisfies \[ f^{f(y)} (y + f(2x)) + f^{f(y)} (2y) = (Ax+By)^{C} \] for all nonnegative integers $x$ and $y$, where $f^k$ as usual denotes $f$ composed $k$ times. Proposed by [i]Benny Wang[/i]

George has $150$ cups of flour and $200$ eggs. He can make a cupcake with $3$ cups of flour and $2$ eggs, or he can make an omelet with $4$ eggs. What is the maximum number of treats (both omelets and cupcakes) he canmake?

Let $ABC$ be an acute triangle with circumcenter $O$, and let $D$, $E$, and $F$ be the feet of altitudes from $A$, $B$, and $C$ to sides $BC$, $CA$, and $AB$, respectively. Denote by $P$ the intersection of the tangents to the circumcircle of $ABC$ at $B$ and $C$. The line through $P$ perpendicular to $EF$ meets $AD$ at $Q$, and let $R$ be the foot of the perpendicular from $A$ to $EF$. Prove that $DR$ and $OQ$ are parallel.

A function $f$ satisfies $f(\cos x) = \cos (17x)$ for every real $x$. Show that $f(\sin x) =\sin (17x)$ for every $x \in \mathbb{R}.$

In the figure below, the three small circles are congruent and tangent to each other. The large circle is tangent to the three small circles. [asy] import graph; unitsize(20); real r = sqrt(3) / 2; filldraw(Circle((0, 0), 1 + r), gray); filldraw(Circle(dir(90), r), white); filldraw(Circle(dir(210), r), white); filldraw(Circle(dir(330), r), white); [/asy] The area of the large circle is 1. What is the area of the shaded region?

A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y)$ if for any x, y, z real numbers $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$

Compute the sum of all positive integers $N$ for which there exists a unique ordered triple of non-negative integers $(a,b,c)$ such that $2a+3b+5c=200$ and $a+b+c=N$. [i]Proposed by Kyle Lee[/i]

Determine the set of all real numbers $\alpha$ with the following property: For each positive $c$ there exists a rational number $\frac mn~(m\in\mathbb Z,n\in\mathbb N)$ different than $\alpha$ such that $$\left|\alpha-\frac mn\right|<\frac cn.$$

If the conditions \[\begin{array}{rcl} f(2x+1)+g(3-x) &=& x \\ f((3x+5)/(x+1))+2g((2x+1)/(x+1)) &=& x/(x+1) \end{array}\] hold for all real numbers $x\neq 1$, what is $f(2013)$? $ \textbf{(A)}\ 1007 \qquad\textbf{(B)}\ \dfrac {4021}{3} \qquad\textbf{(C)}\ \dfrac {6037}7 \qquad\textbf{(D)}\ \dfrac {4029}{5} \qquad\textbf{(E)}\ \text{None of above} $

A sphere of radius $1$ is internally tangent to all four faces of a regular tetrahedron. Find the tetrahedron's volume.

Let $(p_1,p_2,..., p_n)$ be a random permutation of the set $\{1,2,...,n)$. If $n$ is odd, prove that the product $(p_1-1)\cdot (p_2-2)\cdot ...\cdot (p_n-n)$ is an even number. @below fixed.

Ten points are spaced equally around a circle. How many different chords can be formed by joining any 2 of these points? (A chord is a straight line joining two points on the circumference of a circle.) $\textbf{(A)}\ 9 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 55$

The number of distinct solutions of the equation $\big|x-|2x+1|\big| = 3$ is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }4$

Prove that if $ a \geq 0 $ and $ b \geq 0 $, then $$ \sqrt{a^2 + b^2} \geq a + b - (2 - \sqrt{2}) \sqrt{ab}.$$