$ABCD$ is a rectangle with side lengths $AB = CD = 1$ and $BC = DA = 2$. Let $ M$ be the midpoint of $AD$. Point $P$ lies on the opposite side of line $MB$ to $A$, such that triangle $MBP$ is equilateral. Find the value of $\angle PCB$.

Ana draws a checkered board that has at least 20 rows and at least 24 columns. Then, Beto must completely cover that board, without holes or overlaps, using only pieces of the following two types: Each piece must cover exactly 4 or 3 squares of the board, as shown in the figure, without leaving the board. It is permitted to rotate the pieces and it is not necessary to use all types of pieces. Explain why, regardless of how many rows and how many columns Ana's board has, Beto can always complete his task.

Find the equation of the line tangent to the parabola $y = 1/3(x^2-2x+4)$ and a circle of unit radius centered at the origin. (List all solutions.)

For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?

Find all the pairs of positive integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+3n$, $B=2n^2+3mn+m^2$, $C=3n^2+mn+2m^2$ are consecutive in some order.

A ten-level $2$-tree is drawn in the plane: a vertex $A_1$ is marked, it is connected by segments with two vertices $B_1$ and $B_2$, each of $B_1$ and $B_2$ is connected by segments with two of the four vertices $C_1, C_2, C_3, C_4$ (each $C_i$ is connected with one $B_j$ exactly); and so on, up to $512$ vertices $J_1, \ldots, J_{512}$. Each of the vertices $J_1, \ldots, J_{512}$ is coloured blue or golden. Consider all permutations $f$ of the vertices of this tree, such that (i) if $X$ and $Y$ are connected with a segment, then so are $f(X)$ and $f(Y)$, and (ii) if $X$ is coloured, then $f(X)$ has the same colour. Find the maximum $M$ such that there are at least $M$ permutations with these properties, regardless of the colouring.

Let $a_1, a_2, \dots, a_n$ be positive real numbers, whose sum is $1$. Prove that \[ \frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+\dots+\frac{a_{n-1}^2}{a_{n-1}+a_n}+\frac{a_n^2}{a_n+a_1}\ge \frac{1}{2} \]

Find all real values of $x$ for which $$\frac{1}{\sqrt{x} + \sqrt{x - 2}} +\frac{1}{\sqrt{x+2} + \sqrt{x }} =\frac14.$$

There are 20 students in a high school class, and each student has exactly three close friends in the class. Five of the students have bought tickets to an upcoming concert. If any student sees that at least two of their close friends have bought tickets, then they will buy a ticket too. Is it possible that the entire class buys tickets to the concert? (Assume that friendship is mutual; if student $A$ is close friends with student $B$, then $B$ is close friends with $A$.)

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

Let $\mathbb{R}^+$ the set of positive real numbers. Determine all the functions $f, g: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that, for all positive real numbers $x, y$ we have that $$f(x + g(y)) = f(x + y) + g(y) \text{ and } g(x + f(y)) = g(x + y) + f(y)$$

Triangle $ABC$ has $\overline{AB} = 5, \overline{BC} = 4, \overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $\overline{AD} = \overline{AE} = \overline{BC}$. Let $CD$ and $BE$ intersect at $F$ and let $AF$ and $DE$ intersect at $G$. The length of $FG$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$ in simplified form. What is $a + b + c$?

There is a square table with side $n$. Is it possible to enter the numbers $0$, $1$ or $2$ into the cells of this table so that all sums of numbers in rows and columns are different and take values from $1$ to $2n$, if: a) $n = 7$ ? b) $n = 8$ ?

Find the smallest real $\lambda$, such that for any positive integers $n, a, b$, such that $n \nmid a+b$, there exists a positive integer $1 \leq k \leq n-1$, satisfying $$\{\frac{ak} {n}\}+\{\frac{bk} {n}\} \leq \lambda.$$

Let $ABC$ be a triangle. On the sides $BC$, $CA$, $AB$ of the triangle, construct outwardly three squares with centres $O_a$, $O_b$, $O_c$ respectively. Let $\omega$ be the circumcircle of $\vartriangle O_aO_bO_c$. Given that $A$ lies on $\omega$, prove that the centre of $\omega$ lies on the perimeter of $\vartriangle ABC$. [i]Sam Bealing, United Kingdom[/i]

Let A be a subset of natural numbers and let k , r be positive integers. Suppose that for any r different elements selected from A , their greatest common divisor has at most k different prime factors. Prove that A can be partitioned into B and C , where any element of B has at most k + 1 different prime divisors and $$\sum_{n\in C} \frac{1}{n} <\infty$$

Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

For a natural number $m>1$ we'll denote with $f(m)$ the sum of all natural numbers less than $m$, which are also coprime to $m$. Find all natural numbers $n$, such that there exist natural numbers $k$ and $\ell$ which satisfy $f(n^{k})=n^{\ell}$.

Let $O_1$ and $O_2$ be concentric circles with radii 4 and 6, respectively. A chord $AB$ is drawn in $O_1$ with length $2$. Extend $AB$ to intersect $O_2$ in points $C$ and $D$. Find $CD$.

Points $A$, $B$, and $C$ lie on a line, in that order, with $AB=8$ and $BC=2$. $B$ is rotated $20^\circ$ counter-clockwise about $A$ to a point $B'$, tracing out an arc $R_1$. $C$ is then rotated $20^\circ$ clockwise about $A$ to a point $C'$, tracing out an arc $R_2$. What is the area of the region bounded by arc $R_1$, segment $B'C$, arc $R_2$, and segment $C'B$? [i]Proposed by Thomas Lam[/i]

Let $ABC$ be a triangle, $AD$ be the altitude from $A$ on to $BC$. Draw perpendiculars $DD_1$ and $DD_2$ from $D$ on to $AB$ and $AC$ respectively and let $p(A)$ be the length of the segment $D_1D_2$. Similarly define $p(B)$ and $p(C)$. Prove that $\frac{p(A)p(B)p(C)}{s^3}\le \frac18$ , where s is the semi-perimeter of the triangle $ABC$.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Given a $n \times n$ square board. Two players by turn remove some side of unit square if this side is not a bound of $n \times n$ square board. The player lose if after his move $n \times n$ square board became broken into two parts. Who has a winning strategy?

Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.

Determine the largest $k\ge0$ such that the inequality \[\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2\] holds for every $n\ge2$ and any $n$-tuple $x_1,\ldots,x_n$ of non-negative numbers (given that $x_{n+1}=x_1$)