Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$

On a table lie $289$ coins that form a square array $17 \times 17$. All coins are facing with the crown up. In one move, it is possible to reverse any five coins lying in a row: vertical, horizontal or diagonal. Is it possible that after a number of such moves, all the coins to be arranged with tails up?

Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively. (a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$ (b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]

Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than $11$. P.s. the current year in the problem is $2010$.

Alan draws a convex $2020$-gon $\mathcal{A}=A_1A_2\dotsm A_{2020}$ with vertices in clockwise order and chooses $2020$ angles $\theta_1, \theta_2, \dotsc, \theta_{2020}\in (0, \pi)$ in radians with sum $1010\pi$. He then constructs isosceles triangles $\triangle A_iB_iA_{i+1}$ on the exterior of $\mathcal{A}$ with $B_iA_i=B_iA_{i+1}$ and $\angle A_iB_iA_{i+1}=\theta_i$. (Here, $A_{2021}=A_1$.) Finally, he erases $\mathcal{A}$ and the point $B_1$. He then tells Jason the angles $\theta_1, \theta_2, \dotsc, \theta_{2020}$ he chose. Show that Jason can determine where $B_1$ was from the remaining $2019$ points, i.e. show that $B_1$ is uniquely determined by the information Jason has. [i]Proposed by Andrew Gu.[/i]

A uniform horizontal circular plate of weight $ Q $ kG is supported at points $ A $, $ B $, $ C $ lying on the circumference of the plate, with $ AC = BC $ and $ ACB = 2\alpha $. What weight $ x $ kG must be placed on the plate at the other end $ D $ of the diameter drawn from point $ C $ so that the pressure of the plate on the support at $ C $G is equal to zero?

Let $ABC$ be an acute triangle with $| AB | > | AC |$. Let $D$ be the foot of the altitude from $A$ to $BC$, let $K$ be the intersection of $AD$ with the internal bisector of angle $B$, Let $M$ be the foot of the perpendicular from $B$ to $CK$ (it could be in the extension of segment $CK$) and$ N$ the intersection of $BM$ and $AK$ (it could be in the extension of the segments). Let $T$ be the intersection of$ AC$ with the line that passes through $N$ and parallel to $DM$. Prove that $BM$ is the internal bisector of the angle $\angle TBC$

A Pokemon Go player starts at $(0,0)$ and carries a pedometer that records the number of steps taken. He then takes steps with length $1$ unit in the north, south, east, or west direction, such that each move after the first is perpendicular to the move before it. Somehow, the player eventually returns to $(0, 0)$, but he had visited no point (except $(0, 0)$) twice. Let $n$ be the number on the pedometer when the player returns to $(0, 0)$. Of the numbers from $1$ to $2019$ inclusive, how many can be the value of $n$?

Find the number of nondegenerate triangles whose vertices lie in the set of points $(s,t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, $s$ and $t$ are integers.

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]

Let $a_1, a_2, \cdots, a_{2022}$ be nonnegative real numbers such that $a_1+a_2+\cdots +a_{2022}=1$. Find the maximum number of ordered pairs $(i, j)$, $1\leq i,j\leq 2022$, satisfying $$a_i^2+a_j\ge \frac 1{2021}.$$

1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer. 2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$.

Evaluate the sum $$\sum_{k=0}^{r} {r \choose k}{{12-r} \choose {6-k}} $$

In how many ways can each of the integers $1$ through $11$ be assigned one of the letters $L, M,$ and $T$ such that consecutive multiples of $n,$ for any positive integer $n,$ are not assigned the same letter?

For positive integers $a$ and $k$, define the sequence $a_1,a_2,\ldots$ by \[a_1=a,\qquad\text{and}\qquad a_{n+1}=a_n+k\cdot\varrho(a_n)\qquad\text{for } n=1,2,\ldots\] where $\varrho(m)$ denotes the product of the decimal digits of $m$ (for example, $\varrho(413)=12$ and $\varrho(308)=0$). Prove that there are positive integers $a$ and $k$ for which the sequence $a_1,a_2,\ldots$ contains exactly $2009$ different numbers.

Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.

Let $a,b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that \[\dfrac{1}{a^3+2b^2+2b+4}+\dfrac{1}{b^3+2c^2+2c+4}+\dfrac{1}{c^3+2a^2+2a+4}\leq \dfrac13.\]

Grant and Stephen are playing Square-Tac-Toe. In this game, players alternate placing $X$'s and $O$'s on a $3 \times 3$ board, and the first person to complete a $2 \times 2$ square with their respective symbols wins the game. If all tiles are filled and no such square exists, the game is a tie. Grant moves first. Given that Stephen plays randomly and Grant plays optimally (knowing that Stephen is playing randomly), the probability that Grant wins is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ ([i]Note: Grant playing "optimally" means he is maximizing his win probability[/i])

The set consists of equal three-cell corners ( $L$ -triminoes), the middle cells of which are marked with paint. A rectangular board has been covered with these triminoes in a single layer so that all triminoes were entirely on the board. Then the triminoes were removed leaving the paint marks where the marked cells were. Is it always possible to know the location of the triminoes on the board using only those paint marks? Alexandr Gribalko

All vertices of triangle $ABC$ lie inside square $K$. Prove that if all of them are reflected symmetrically with respect to the point of intersection of the medians of triangle $ABC$, then at least one of the resulting three points will be inside $K$.

Given an integer $N \geq 4$, determine the largest value the sum $$\sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right)$$ may achieve, where $k, n_1, \ldots, n_k$ run through the integers subject to $k \geq 3$, $n_1 \geq \ldots\geq n_k\geq 1$ and $n_1 + \ldots + n_k = N$.

Show that for every natural number $n$ the product \[\left( 4-\frac{2}{1}\right) \left( 4-\frac{2}{2}\right) \left( 4-\frac{2}{3}\right) \cdots \left( 4-\frac{2}{n}\right)\] is an integer.

Find all integers $x$ and $y$ satisfying the inequality \[x^4-12x^2+x^2y^2+30\leq 0.\]

Let $A,B,C,D,E,F$ be points in space such that the quadrilaterals $ABDE,BCEF, CDFA$ are parallelograms. Prove that the six midpoints of the sides $AB,BC,CD,DE,EF,FA$ are coplanar