Each of the cells of a table 2014 x 2014 is colored in white or black. It is known that each square 2 x 2 contains an even number of black cells and each cross (3 x 3 square without its corner cells) contains an odd number of black cells. Prove that the 4 corner cells of the table are in the same color.

Let define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that : $i)$$f(p)=1$ for all prime numbers $p$. $ii)$$f(xy)=xf(y)+yf(x)$ for all positive integers $x,y$ find the smallest $n \geq 2016$ such that $f(n)=n$

Let $a,b,c$ and $d$ be positive real numbers such that $a^2+b^2+c^2+d^2 \ge 4$. Prove that $$(a+b)^3+(c+d)^3+2(a^2+b^2+c^2+d^2) \ge 4(ab+bc+cd+da+ac+bd)$$

Let $ABC$ be an isosceles triangle with $AC = BC$. On the arc $CA$ of its circumcircle, which does not contain $ B$, there is a point $ P$. The projection of $C$ on the line $AP$ is denoted by $E$, the projection of $C$ on the line $BP$ is denoted by $F$. Prove that the lines $AE$ and $BF$ have equal lengths. (W. Janous, WRG Ursulincn, Innsbruck)

The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers?

Prove that in each polyhedron there exist two faces with the same number of edges.

In triangle $ABC$ with $AB = 7$, $AC = 8$, and $BC = 9$, the $A$-excircle is tangent to $BC$ at point $D$ and also tangent to lines $AB$ and $AC$ at points $ $ and $F$, respectively. Find $[DEF]$. (The $A$-excircle is the circle tangent to segment $BC$ and the extensions of rays $AB$ and $AC$. Also, $[XY Z]$ denotes the area of triangle $XY Z$.)

Find the biggest positive integer $n$ such that $n$ is $167$ times the amount of it's positive divisors.

Solve in positive integers: $\frac{1}{x^2}+\frac{y}{xz}+\frac{1}{z^2}=\frac{1}{2013}$ .

Do there exist 2021 points with integer coordinates on the plane such that the pairwise distances between them are pairwise distinct consecutive integers?

Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable. [i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]

In a circle with center $O$ is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.

At Mallard High School there are three intermural sports leagues: football, basketball, and baseball. There are 427 students participating in these sports: 128 play on football teams, 291 play on basketball teams, and 318 play on baseball teams. If exactly 36 students participate in all three of the sports, how many students participate in exactly two of the sports?

In triangle $ABC$, let $J$ be the center of the excircle tangent to side $BC$ at $A_{1}$ and to the extensions of the sides $AC$ and $AB$ at $B_{1}$ and $C_{1}$ respectively. Suppose that the lines $A_{1}B_{1}$ and $AB$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular from $C_{1}$ to line $DJ$. Determine the angles $\angle{BEA_{1}}$ and $\angle{AEB_{1}}$. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Circles with centres $O_1$ and $O_2$ intersect in two points, let one of which be $A$. The common tangent of these circles touches them respectively in points $P$ and $Q$. It is known that points $O_1, A$ and $Q$ are on a common straight line and points $O_2, A$ and $P$ are on a common straight line. Prove that the radii of the circles are equal.

$x,y,z$ are three real numbers inequal to zero satisfying $x+y+z=xyz$. Prove that $$ \sum (\frac{x^2-1}{x})^2 \geq 4$$ [i]Proposed by Amin Fathpour[/i]

A lock has $16$ keys arranged in a $4 \times 4$ array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too (see diagram). Show that no matter what the starting positions are, it is always possible to open this lock. (Only one key at a time can be switched.)

Ankit wants to create a pseudo-random number generator using modular arithmetic. To do so he starts with a seed $x_0$ and a function $f(x) = 2x + 25$ (mod $31$). To compute the $k$-th pseudo random number, he calls $g(k)$ dened as follows: $$g(k) = \begin{cases} x_0 \,\,\, \text{if} \,\,\, k = 0 \\ f(g(k- 1)) \,\,\, \text{if} \,\,\, k > 0 \end{cases}$$ If $x_0$ is $2017$, compute $\sum^{2017}_{j=0} g(j)$ (mod $31$).

Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders? $\textbf{(A) }\text{The second height is 10\% less than the first.}$ $\textbf{(B) }\text{The first height is 10\% more than the second.}$ $\textbf{(C) }\text{The second height is 21\% less than the first.}$ $\textbf{(D) }\text{The first height is 21\% more than the second.}$ $\textbf{(E) }\text{The second height is 80\% of the first.}$

Let $ABC$ be a non-isosceles triangle with altitudes $AD$, $BE$, $CF$ with orthocenter $H$. Suppose that $DF \cap HB = M$, $DE \cap HC = N$ and $T$ is the circumcenter of triangle $HBC$. Prove that $AT\perp MN$.

Let $k\ge 1$ be a positive integer. We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip. Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.

Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$.

Let $ABC$ be an acute triangle, $(O)$ be the circumcircle, and $AB<AC.$ Let $I$ be the midpoint of arc $BC$ (not containing $A$). $K$ lies on $AC,$ $K\ne C$ such that $IK=IC.$ $BK$ intersects $(O)$ at the second point $D,$ $D\ne B$ and intersects $AI$ at $E.$ $DI$ intersects $AC$ at $F.$ a) Prove that $EF=\frac{BC}{2}.$ b) $M$ lies on $DI$ such that $CM$ is parallel to $AD.$ $KM$ intersects $BC$ at $N.$ The circumcircle of triangle $BKN$ intersects $(O)$ at the second point $P.$ Prove that $PK$ passes through the midpoint of segment $AD.$

Which of the following numbers is a perfect square? $\textbf{(A) }4^45^56^6\qquad\textbf{(B) }4^45^66^5\qquad\textbf{(C) }4^55^46^6\qquad\textbf{(D) }4^65^46^5\qquad\textbf{(E) }4^65^56^4$

Suppose $p > 3$ is a prime number and \[S = \sum_{2 \le i < j < k \le p-1} ijk\] Prove that $S+1$ is divisible by $p$.