Let $f:[0,1]\to\mathbb R$ be a continuous function such that $f(0)=f(1)=0$. Prove that the set $$A:=\{h\in[0,1]:f(x+h)=f(x)\text{ for some }x\in[0,1]\}$$is Lebesgue measureable and has Lebesgue measure at least $\frac12$.

In the triangle $ABC$, the angle  $\alpha = \angle BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$

Let $ABC$ and $AMN$ be two similar triangles with the same orientation, such that $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral. [i]Valentin Vornicu[/i]

There are $2n$ points on a circle, $n$ are red and $n$ are blue. Janson found a red frog and a blue frog at a red point and a blue point on the circle respectively. Every minute, the red frog moves to the next red point in the clockwise direction and the blue frog moves to the next blue point in the anticlockwise direction. Prove that for any initial position of the two frogs, Janson can draw a line through the circle, such that the two frogs are always on opposite sides of the line.

The quadratic equation $x^2+mx+n=0$ has roots that are twice those of $x^2+px+m=0$, and none of $m$, $n$, and $p$ is zero. What is the value of $\frac{n}{p}$? $\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 4 \qquad \text{(D)} \ 8\qquad \text{(E)} \ 16$

Let $\triangle ABC$ be a triangle with a right angle $\angle ABC$. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AC}$, and let $F$ be the midpoint of $\overline{AB}$. Let $G$ be the midpoint of $\overline{EC}$. One of the angles of $\triangle DFG$ is a right angle. What is the least possible value of $\frac{BC}{AG}$?

The Fibonacci sequence is defined by \[ a_{n+1} = a_n + a_{n-1}, n \geq 1, a_0 = 0, a_1 = a_2 = 1. \] Find the greatest common divisor of the 1960-th and 1988-th terms of the Fibonacci sequence.

Given a polynomial with integer coefficients $$P(x) = x^{20} + a_{19}x^{19} +... + a_1x + a_0,$$ having $20$ different real roots. Determine the maximum number of roots such a polynomial $P$ can have in the interval $(99, 100)$.

Find all real solutions of the equation \[\sqrt[3]{x-1}+\sqrt[3]{3x-1}=\sqrt[3]{x+1}.\]

Let $P(x)$ be a nonzero polynomial of degree $n>1$ with nonnegative coefficients such that function $y=P(x)$ is odd. Is that possible thet for some pairwise distinct points $A_{1}, A_{2}, \dots A_{n}$ on the graph $G: y = P(x)$ the following conditions hold: tangent to $G$ at $A_{1}$ passes through $A_{2}$, tangent to $G$ at $A_{2}$ passes through $A_{3}$, $\dots$, tangent to $G$ at $A_{n}$ passes through $A_{1}$?

Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. Which of these arrangements gives the dog the greater area to roam, and by how many square feet? [asy]defaultpen(linewidth(0.7)); size(7cm); D((0,0)--(16,0)--(16,-16)--(0,-16)--cycle, black); D((16,-8)--(24,-8), black); label('Dog', (24, -8), SE); label('I', (8,-8), (0,0)); MP('8', (16,-4), W); MP('8', (16,-12),W); MP('8', (20,-8), N); label('Rope', (20,-8),S); D((0,-20)--(16,-20)--(16,-36)--(0,-36)--cycle, black); D((16,-24)--(24,-24), black); label("II", (8,-28), (0,0)); MP('4', (16,-22), W); MP('8', (20,-24), N); label("Dog",(24,-24),SE); label("Rope", (20,-24), S); dot((24,-24)^^(24,-8));[/asy] $ \textbf{(A)}\text{ I, by }8\pi\qquad\textbf{(B)}\text{ I, by }6\pi\qquad\textbf{(C)}\text{ II, by }4\pi\qquad\textbf{(D) }\text{II, by }8\pi\qquad\textbf{(E)}\text{ II, by }10\pi $

Find maximum value of $|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.

For some positive numbers $a$, $b$, $c$ and $d$, we know that $$ \frac{1}{a^3 + 1}+ \frac{1}{b^3 + 1}+ \frac{1}{c^3 + 1} + \frac{1}{d^3 + 1} = 2. $$ Prove that $$ \frac{1 - a}{a^2 - a + 1} + \frac{1-b}{b^2 - b + 1} + \frac{1-c}{c^2 - c + 1} +\frac{1-d}{d^2 - d + 1} \geq 0. $$

A moth starts at vertex $A$ of a certain cube and is trying to get to vertex $B$, which is opposite $A$, in five or fewer “steps,” where a step consists in traveling along an edge from one vertex to another. The moth will stop as soon as it reaches $B$. How many ways can the moth achieve its objective?

In triangle $ABC$, points $M_1, M_2$ and $M_3$ are midpoints of sides $AB$, $BC$ and $AC$, respectively, while points $H_1, H_2$ and $H_3$ are bases of altitudes drawn from $C$, $A$ and $B$, respectively. Prove that one can construct a triangle from segments $H_1M_2, H_2M_3$ and $H_3M_1$. [i](3 points)[/i]

A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.

Let $a,b,c,d \in \mathbb{R}^+$ and suppose that all roots of the equation \begin{align*} x^5-ax^4+bx^3-cx^2+dx=1 \end{align*} are real. Prove \begin{align*} \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \le \frac{3}{5} \end{align*}

For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.

Find all pairs of nonnegative integers $(m,n)$ such that \[(m+n-5)^2=9mn.\]

Let $S$ be a finite set of size $s\geq 1$ defined with a uniform probability $\mathbb{P}$( i.e. for any subset $X\subset S$ of size $x$, $\mathbb{P}(x)=\frac{x}{s}$). Suppose $A$ and $B$ are subsets of $S$. They are said to be independent iff $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(A\cap B)$. Which if these is sufficient for independence? (A)$|A\cup B|=|A|+|B|$ (B)$|A\cap B|=|A|+|B|$ (C)$|A\cup B|=|A|\cdot |B|$ (D)$|A\cap B|=|A|\cdot |B|$

Let be a natural number $ n\ge 2, $ and a matrix $ A\in\mathcal{M}_n\left( \mathbb{C} \right) $ whose determinant vanishes. Show that $$ \left( A^* \right)^2 =A^*\cdot\text{tr} A^*, $$ where $ A^* $ is the adjugate of $ A. $

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]

The incircle of a scalene triangle $ABC$ touches the sides $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Triangles $APE$ and $AQF$ are constructed outside the triangle so that \[AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.\]Let $M$ be the midpoint of $BC$. Find $\angle QMP$ in terms of the angles of the triangle $ABC$. [i]Iran, Shayan Talaei[/i]

[b]5.[/b] Denote by $c_n$ the $n$th positive integer which can be represented in the form $c_n = k^{l} (k,l = 2,3, \dots )$. Prove that $\sum_{n=1}^{\infty}\frac{1}{c_n-1}=1$ [b](N. 18)[/b]

Two fair six-sided dice are rolled. The probability that the positive difference between the two rolls is at least $4$ can be written in simplest form as $\frac{m}{n}$. Compute $m + n$.