It's known that there is always a prime between $n$ and $2n-7$ for all $n \ge 10$. Prove that, with the exception of $1$, $4$, and $6$, every natural number can be written as the sum of distinct primes.

Let $BD$ be a bisector of triangle $ABC$. Points $I_a$, $I_c$ are the incenters of triangles $ABD$, $CBD$ respectively. The line $I_aI_c$ meets $AC$ in point $Q$. Prove that $\angle DBQ = 90^\circ$.

Consider the parallelogram $ABCD$ such that $CD = 8$ and $BC = 14$. The diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$ and $AC = 16$. Consider a point $F$ on the segment $\overline{ED}$ with $F D =\frac{\sqrt{66}}{3}$. Compute $CF$.

Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$.

What is the smallest integer $n$ for which $\frac{10!}{n}$ is a perfect square?

There are two semi-infinite plane mirrors inclined physically at a non-zero angle with inner surfaces being reflective. [list] [*] Prove that all lines of incident/reflected rays are tangential to a particular circle for any given incident ray being incident on a reflective side. Assume that the incident ray lies on one of the normal planes to the mirrors.[/*] [*] Try to guess the radius of circle by the parameters you can observe. [/*] [/list]

Given a triangle $ABC$. Let $\Omega$ be the circumscribed circle of this triangle, and $\omega$ be the inscribed circle of this triangle. Let $\delta$ be a circle that touches the sides $AB$ and $AC$, and also touches the circle $\Omega$ internally at point $D$. The line $AD$ intersects the circle $\Omega$ at two points $P$ and $Q$ ($P$ lies between $A$ and $Q$). Let $O$ and $I$ be the centers of the circles $\Omega$ and $\omega$. Prove that $OD \parallel IQ$.

Find all functions $\,f: {\mathbb{N}}\rightarrow {\mathbb{N}}\,$ such that\[f(a)^{bf(b^2)}\le a^{f(b)^3}\hspace{0.2in}\text{for all}\,a,b\in \mathbb{N}. \]

Five points are on the surface of of a sphere of radius $1$. Let $a_{\text{min}}$ denote the smallest distance (measured along a straight line in space) between any two of these points. What is the maximum value for $a_{\text{min}}$, taken over all arrangements of the five points?

Let $a,b,c$ be positive real numbers. Prove the inequality \[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\] When does equality occur?

Find all positive integers $(r,s)$ such that there is a non-constant sequence $a_n$ os positive integers such that for all $n=1,2,\dots$ \[ a_{n+2}= \left(1+\frac{{a_2}^r}{{a_1}^s} \right ) \left(1+\frac{{a_3}^r}{{a_2}^s} \right ) \dots \left(1+\frac{{a_{n+1}}^r}{{a_n}^s} \right ).\] Proposed by Navid Safaei, Iran

Points $K$ and $L$ are on side $AB$ of triangle $ABC$ such that $KL=BC$ and $AK=LB$. Let $M$ be a midpoint of $AC$. Prove that $\angle KML = 90^{\circ}$

a) Consider the function $$f(x,y) = x^2 + xy + y^2$$ Prove that for the every point $(x,y)$ there exist such integers $(m,n)$, that $$f((x-m),(y-n)) \le 1/2$$ b) Let us denote with $g(x,y)$ the least possible value of the $f((x-m),(y-n))$ for all the integers $m,n$. The statement a) was equal to the fact $g(x,y) \le 1/2$. Prove that in fact, $$g(x,y) \le 1/3$$ Find all the points $(x,y)$, where $g(x,y)=1/3$. c) Consider function $$f_a(x,y) = x^2 + axy + y^2 \,\,\, (0 \le a \le 2)$$ Find any $c$ such that $g_a(x,y) \le c$. Try to obtain the closest estimation.

Which of the following series are convergent? $\textbf{(A)}~\sum_{n=1}^\infty\sqrt{\frac{2n^2+3}{5n^3+1}}$ $\textbf{(B)}~\sum_{n=1}^\infty\frac{(n+1)^n}{n^{n+3/2}}$ $\textbf{(C)}~\sum_{n=1}^\infty n^2x\left(1-x^2\right)^n$ $\textbf{(D)}~\text{None of the above}$

$22$ mathematicians are meeting together. Each mathematician has at least $3$ friends (friends are mutual). And each mathematician can pass his or her information to any mathematician through the transfer between friends. Is it possible to divide these $22$ mathematicians into $2$-person groups (that is, two people in each group, a total of $11$ groups), so that the mathematicians in each group are friends? [hide=original wording in Chinese]仃22位数学家一起开会.每位数学家都至少有3个朋友(朋友是相互的).而且每 位数学家都可以通过朋友之间的传递.将门已的资料传给任意一位数学家.问：是否一定可 以将这22位数学家两两分组(即每组两人,共11组),使得每组的数学家都是朋友？[/hide]

Find the number of subsets $ S$ of $ \{1,2, \dots 63\}$ the sum of whose elements is $ 2008$.

Consider an infinite sequence of positive integers $a_1, a_2, a_3, \dots$ such that $a_1 > 1$ and $(2^{a_n} - 1)a_{n+1}$ is a square for all positive integers $n$. Is it possible for two terms of such a sequence to be equal? [i]Proposed by Pavel Kozlov, Russia[/i]

Let $F$ be the set of all $n-tuples$ $(A_1,A_2,…,A_n)$ such that each $A_i$ is a subset of ${1,2,…,2019}$. Let $\mid{A}\mid$ denote the number of elements o the set $A$ . Find $\sum_{(A_1,…,A_n)\in{F}}^{}\mid{A_1\cup{A_2}\cup...\cup{A_n}}\mid$

Let $n$ be a positive integer. A mouse sits at each corner point of an $n\times n$ board, which is divided into unit squares as shown below for the example $n=5$. [asy] unitsize(5mm); defaultpen(linewidth(.5pt)); fontsize(25pt); for(int i=0; i<=5; ++i) { for(int j=0; j<=5; ++j) { draw((0,i)--(5,i)); draw((j,0)--(j,5)); }} dot((0,0)); dot((5,0)); dot((0,5)); dot((5,5)); [/asy] The mice then move according to a sequence of [i]steps[/i], in the following manner: (a) In each step, each of the four mice travels a distance of one unit in a horizontal or vertical direction. Each unit distance is called an [i]edge[/i] of the board, and we say that each mouse [i]uses[/i] an edge of the board. (b) An edge of the board may not be used twice in the same direction. (c) At most two mice may occupy the same point on the board at any time. The mice wish to collectively organize their movements so that each edge of the board will be used twice (not necessarily be the same mouse), and each mouse will finish up at its starting point. Determine, with proof, the values of $n$ for which the mice may achieve this goal.

Compute the sum of all the roots of $ (2x \plus{} 3)(x \minus{} 4) \plus{} (2x \plus{} 3)(x \minus{} 6) \equal{} 0$. $ \textbf{(A)}\ 7/2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 13$

A function $f$ satisfies the equation \[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\] for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.

Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.

A triangle $ABC$ has acute angles at $A$ and $B$. Isosceles triangles $ACD$ and $BCE$ with bases $AC$ and $BC$ are constructed externally to triangle $ABC$ such that $\angle ADC = \angle ABC$ and $\angle BEC = \angle BAC$. Let $S$ be the circumcenter of $\triangle ABC$. Prove that the length of the polygonal line $DSE$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.

Show that $\vert 12^m -5^n\vert \ge 7$ for all $m, n \in \mathbb{N}$.

Let $P_1(x) = x^2 + a_1x + b_1$ and $P_2(x) = x^2 + a_2x + b_2$ be two quadratic polynomials with integer coeffcients. Suppose $a_1 \ne a_2$ and there exist integers $m \ne n$ such that $P_1(m) = P_2(n), P_2(m) = P_1(n)$. Prove that $a_1 - a_2$ is even.