Given a regular $ 2n$-gon, show that each of its sides and diagonals can be assigned in such a way that the sum of the obtained vectors equals zero.

Seller reduced price of one shirt for $20\%$,and they raised it by $10\%$. Does he needs to reduce or raise the price and how many, so that price of shirt will be reduced by $10\%$ from the original price

Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$. [i]Author: Kestutis Cesnavicius, Lithuania[/i]

Find all positive integers $n \ge 4$ for which there are $n$ points in general position on the plane such that an arbitrary triangle with vertices belonging to the convex hull of these $n$ points, containing exactly one of $n - 3$ points inside remained.

If $a > b > 0$, prove the inequality $$\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.$$

Let $n$ be a positive integer divisible by $39$. What is the smallest possible sum of digits that $n$ can have (in base $10$)?

A nonnegative integer $n$ is said to be $\textit{squarish}$ is it satisfies the following conditions: $\textbf{(i)}$ it contains no digits of $9$; $\textbf{(ii)}$ no matter how we increase exactly one of its digits with $1$, the outcome is a square. Find all squarish nonnegative integers. $\textit{(Vlad Robu)}$

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

$\triangle ABC$ has side lengths $13$, $14$, and $15$. Let the feet of the altitudes from $A$, $B$, and $C$ be $D$, $E$, and $F$, respectively. The circumcircle of $\triangle DEF$ intersects $AD$, $BE$, and $CF$ at $I$, $J$, and $K$ respectively. What is the area of $\triangle IJK$?

Find the number of ways $66$ identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

Let $A_1A_2A_3A_4A_5$ be a convex pentagon. Suppose rays $A_2A_3$ and $A_5A_4$ meet at the point $X_1$. Define $X_2$, $X_3$, $X_4$, $X_5$ similarly. Prove that $$\displaystyle\prod_{i=1}^{5} X_iA_{i+2} = \displaystyle\prod_{i=1}^{5} X_iA_{i+3}$$ where the indices are taken modulo 5.

The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.

Let the boxes in picture $1$ be marked as in picture $2$ below (from top to bottom in layers). In one move it is allowed to switch the empty box with another box adjacent to it (two boxes are adjacent if they share a common side). Can the arrangement of the numbers in picture $3$ be obtained after finitely many moves?

Prove that there are no integers $x$ and $y$ satisfying $x^{2}=y^{5}-4$.

Decide whether there are infinitely many primes $p$ having a multiple in the form $n^2 + n + 1$ for some natural number $n$

A cart rolls down a hill, traveling 5 inches the first second and accelerating so that each successive 1-second time interval, it travels 7 inches more than during the previous 1-second interval. The cart takes 30 seconds to reach the bottom of the hill. How far, in inches, does it travel? $\textbf{(A) }215 \qquad \textbf{(B) }360 \qquad \textbf{(C) }2992 \qquad \textbf{(D) }3195 \qquad \textbf{(E) }3242$

Let $ n$ and $ k$ be given natural numbers, and let $ A$ be a set such that \[ |A| \leq \frac{n(n+1)}{k+1}.\] For $ i=1,2,...,n+1$, let $ A_i$ be sets of size $ n$ such that \[ |A_i \cap A_j| \leq k \;(i \not=j)\ ,\] \[ A= \bigcup_{i=1}^{n+1} A_i.\] Determine the cardinality of $ A$. [i]K. Corradi[/i]

In the rectangle there is a broken line, the neighboring links of which are perpendicular and equal to the smaller side of the rectangle (see the figure). Find the ratio of the sides of the rectangle. [img]https://2.bp.blogspot.com/-QYj53KiPTJ8/XT_mVIw876I/AAAAAAAAKbE/gJ1roU4Bx-kfGVfJxYMAuLE0Ax0glRbegCK4BGAYYCw/s1600/oral%2Bmoscow%2B2016%2B8.9%2Bp2.png[/img]

In a country, mathematicians chose an $\alpha> 2$ and issued coins in denominations of 1 ruble, as well as $\alpha ^k$ rubles for each positive integer k. $\alpha$ was chosen so that the value of each coins, except the smallest, was irrational. Is it possible that any natural number of rubles can be formed with at most 6 of each denomination of coins?

2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations: [list] [*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip. [*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list] Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it. [i]Brian Hamrick.[/i]

The vertex $A$, center $O$ and Euler line $\ell$ of a triangle $ABC$ is given. It is known that $\ell$ intersects $AB,AC$ at two points equidistant from $A$. Restore the triangle.

For which natural numbers $n$ do there exist $n$ natural numbers $a_i\ (1\le i\le n)$ such that $\sum_{i=1}^n a_i^{-2}=1$?

For a natural number $x$ we define $f(x)$ to be the sum of all natural numbers less than $x$ and coprime with it. Let $m$ and $n$ be some natural numbers where $n$ is odd. Prove that there exist $x$, which is a multiple of $m$ and for which $f(x)$ is a perfect n-th power.

Let $\triangle ABC$ be a triangle such that $AB=6, BC=8,$ and $AC=10$. Let $M$ be the midpoint of $BC$. Circle $\omega$ passes through $A$ and is tangent to $BC$ at $M$. Suppose $\omega$ intersects segments $AB$ and $AC$ again at points $X$ and $Y$, respectively. If the area of $AXY$ can be expressed as $\frac{p}{q}$ where $p, q$ are relatively prime integers, compute $p+q$.