A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$. Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows: A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.

Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic. Proposed by Navid Safaei

Find all integers $k$ such that all roots of the following polynomial are also integers: $$f(x)=x^3-(k-3)x^2-11x+(4k-8).$$

Prove that if $ E \subset \mathbb{R}$ is a bounded set of positive Lebesgue measure, then for every $ u < 1/2$, a point $ x\equal{}x(u)$ can be found so that \[ |(x\minus{}h,x\plus{}h) \cap E| \geq uh\] and \[ |(x\minus{}h,x\plus{}h) \cap (\mathbb{R} \setminus E)| \geq uh\] for all sufficiently small positive values of $ h$. [i]K. I. Koljada[/i]

Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.

In an urn there are n balls numbered $1, 2, \cdots, n$. They are drawn at random one by one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum? A term in a sequence is a local maximum if it is greater than all its neighbors.

On the legs $AC$ and $BC$ of an isosceles right-angled triangle with a right angle $C$, points $D$ and $E$ are taken, respectively, so that $CD = CE$. Perpendiculars on line $AE$ from points $C$ and $D$ intersect segment $AB$ at points $P$ and $Q$, respectively. Prove that $BP = PQ$.

2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

$A; B; C; D; E$ and $F$ are points in space such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, $CD$ is parallel to $FA$, but $AB \neq DE$. Prove that all six points lie in the same plane. [i](4 points)[/i]

Let $x_1,x_2,\dots,x_{2014}$ be integers among which no two are congurent modulo $2014$. Let $y_1,y_2,\dots,y_{2014}$ be integers among which no two are congurent modulo $2014$. Prove that one can rearrange $y_1,y_2,\dots,y_{2014}$ to $z_1,z_2,\dots,z_{2014}$, so that among \[x_1+z_1,x_2+z_2,\dots,x_{2014}+z_{2014}\] no two are congurent modulo $4028$.

Prove that there exists an unique sequence $ \left( c_n \right)_{n\ge 1} $ of real numbers from the interval $ (0,1) $ such that$$ \int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } , $$ for all natural numbers $ m, $ and calculate $ \lim_{k\to\infty } kc_k^k. $ [i]Radu Pop[/i]

Let $ABCD$ be rhombus $ABCD$ where the triangles $ABD$ and $BCD$ are equilateral. Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, such that $\angle MAN = 30^o$. Let $X$ be the intersection point of the diagonals $AC$ and $BD$. Prove that $\angle XMN = \angle\ DAM$ and $\angle XNM = \angle BAN$.

What is the largest positive integer which is equal to the sum of its digits? [i]Proposed by Evan Chen[/i]

On the grid plane all possible broken lines with the following properties are constructed: each of them starts at the point $(0, 0)$, has all its vertices at integer points, each linear segment goes either up or to the right along the grid lines. For each such broken line consider the corresponding [i]worm[/i], the subset of the plane consisting of all the cells that share at least one point with the broken line. Prove that the number of worms that can be divided into dominoes (rectangles $2\times 1$ and $1\times 2$) in exactly $n > 2$ different ways, is equal to the number of positive integers that are less than n and relatively prime to $n$. (Ilke Chanakchi, Ralf Schiffler)

Isosceles $\triangle$ $ABC$ has equal side lengths $AB$ and $BC$. In the figure below, segments are drawn parallel to $\overline{AC}$ so that the shaded portions of $\triangle$ $ABC$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of $h$ of $\triangle$ $ABC$? [asy] size(12cm); draw((5,10)--(5,6.7),dashed+gray+linewidth(.5)); draw((5,3)--(5,5.3),dashed+gray+linewidth(.5)); filldraw((1.5,3)--(8.5,3)--(10,0)--(0,0)--cycle,lightgray); draw((0,0)--(10,0)--(5,10)--cycle,linewidth(1.3)); dot((0,0)); dot((5,10)); dot((10,0)); label(scale(.8)*"$11$", (5,6.5),S); dot((17.5,0)); dot((27.5,0)); dot((22.5,10)); draw((22.5,1.3)--(22.5,0),dashed+gray+linewidth(.5)); draw((22.5,2.5)--(22.5,3.6),dashed+gray+linewidth(.5)); draw((17.5,0)--(27.5,0)--(22.5,10)--cycle,linewidth(1.3)); filldraw((19.3,3.6)--(25.7,3.6)--(22.5,10)--cycle,lightgray); label(scale(.8)*"$5$", (22.5,1.9)); draw((5,10)--(22.5,10),dashed+gray+linewidth(.5)); draw((10,0)--(17.5,0),dashed+gray+linewidth(.5)); draw((13.75,4.3)--(13.75,0),dashed+gray+linewidth(.5)); draw((13.75,5.7)--(13.75,10),dashed+gray+linewidth(.5)); label(scale(.8)*"$h$", (13.75,5)); label(scale(.7)*"$A$", (0,0), S); label(scale(.7)*"$C$", (10,0), S); label(scale(.7)*"$B$", (5,10), N); label(scale(.7)*"$A$", (17.5,0), S); label(scale(.7)*"$C$", (27.5,0), S); label(scale(.7)*"$B$", (22.5,10), N); [/asy] $\textbf{(A) } 14.6 \qquad \textbf{(B) } 14.8 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 15.2 \qquad \textbf{(E) } 15.4$

On $[1,2]$ if two functions $f(x)=x^2+px+q$ and $g(x)=x+\frac{1}{x^2}$ get their minumum value at the same point, then the maximum value of $f(x)$ on $[1,2]$ is $\text{(A)}4+\frac{11}{2}\sqrt[3]{2}+\sqrt[3]{4}\qquad\text{(B)}4-\frac{5}{2}\sqrt[3]{2}+\sqrt[3]{4}$ $\text{(C)}1-\frac{1}{2}\sqrt[3]{2}+\sqrt[3]{4}\qquad\text{(D)}$ none above

Prove that $2^{n - 1}$ divides $n!$ if and only if $n = 2^{k - 1}$ for some positive integer $k$.

Three equal circles of radius $r$ each pass through the centres of the other two. What is the area of intersection that is common to all the three circles?

Find all integers that can be written as $\frac{(a+b)(b+c)(c+a)}{abc}$, where $a,b,c$ are pairwise coprime positive integers.

Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.

Proce that number $$\underbrace{11...11}_{2n \,\, digits}-\underbrace{22 ... 22}_{n \,\, digits}$$ is, for every natural $n$, a perfect square.

Find the number of $4 \times 4$ array whose entries are from the set $\{ 0 , 1, 2, 3 \}$ and which are such that the sum of the numbers in each of the four rows and in each of the four columns is divisible by $4$.

Find all pairs $(m,n)$ of integers that satisfy the equation \[(m-n)^{2}=\frac{4mn}{m+n-1}.\]

Prove that there exists a set $ S$ of $ n \minus{} 2$ points inside a convex polygon $ P$ with $ n$ sides, such that any triangle determined by $3$ vertices of $ P$ contains exactly one point from $ S$ inside or on the boundaries.

Prove that if a vector space is the union of some of it's proper subspaces, then number of these subspaces can not be less than the number of elements of the field of that vector space.