For each positive integer $n$ find the largest subset $M(n)$ of real numbers possessing the property: \[n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i}\quad \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n)\] When does the inequality become an equality ?

Alice the architect and Bob the builder play a game. First, Alice chooses two points $P$ and $Q$ in the plane and a subset $\mathcal{S}$ of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair $A,\,B$ of cities, they are connected with a road along the line segment $AB$ if and only if the following condition holds: [center]For every city $C$ distinct from $A$ and $B$, there exists $R\in\mathcal{S}$ such[/center] [center]that $\triangle PQR$ is directly similar to either $\triangle ABC$ or $\triangle BAC$.[/center] Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy. [i]Note:[/i] $\triangle UVW$ is [i]directly similar[/i] to $\triangle XYZ$ if there exists a sequence of rotations, translations, and dilations sending $U$ to $X$, $V$ to $Y$, and $W$ to $Z$.

Consider a parabola. The [i]parabolic length[/i] of a segment is the length of the projection of this segment on a straight line perpendicular to the axis of symmetry of the parabola. In the parabola, two chords $AB$ and $CD$ are drawn, intersecting at the point $N{}$. Prove that the product of the parabolic lengths of the segments $AN$ and $BN$ is equal to the product of the parabolic lengths of the segments $CN$ and $DN$. [i]Proposed by M. Panov[/i]

For every integer $m\ge 1$, let $\mathbb{Z}/m\mathbb{Z}$ denote the set of integers modulo $m$. Let $p$ be a fixed prime and let $a\ge 2$ and $e\ge 1$ be fixed integers. Given a function $f\colon \mathbb{Z}/a\mathbb{Z}\to \mathbb{Z}/p^e\mathbb{Z}$ and an integer $k\ge 0$, the $k$[i]th finite difference[/i], denoted $\Delta^k f$, is the function from $\mathbb{Z}/a\mathbb{Z}$ to $\mathbb{Z}/p^e\mathbb{Z}$ defined recursively by \begin{align*} \Delta^0 f(n)&=f(n)\\ \Delta^k f(n)&=\Delta^{k-1}f(n+1)-\Delta^{k-1}f(n) & \text{for } k=1,2,\dots. \end{align*} Determine the number of functions $f$ such that there exists some $k\ge 1$ for which $\Delta^kf=f$. [i]Holden Mui[/i]

What is the minimum number of cells that can be colored black in white square $ 300 \times 300 $ so that no three black cells formed a corner, and after painting any white cell this condition violated?

Two rays start from a common point and have an angle of $60$ degrees. Circle $C$ is drawn with radius $42$ such that it is tangent to the two rays. Find the radius of the circle that has radius smaller than circle $C$ and is also tangent to $C$ and the two rays.

Toner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\%$ to Toner Drum. Let $N$ be the number of "representative'' sets of the $2015$ voters that could have been polled to correctly predict the winner of the election (i.e. more people in the set voted for Drum than Hilton). Compute the remainder when $N$ is divided by $2017$. [i] Proposed by Ashwin Sah [/i]

Determine all 2nd degree polynomials with integer coefficients of the form $P(X)=aX^{2}+bX+c$, that satisfy: $P(a)=b$, $P(b)=a$, with $a\neq b$.

Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, ..., 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a + b - c | > 10$. Determine the largest possible number of elements of $M$.

Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\le y\le x^3,$ then $(f(x))^2\le f(y) \le (f(x))^3.$

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Consider the sequence \[ 1, \minus{} 2,3, \minus{} 4,5, \minus{} 6,\ldots,\] whose $ n$th term is $ ( \minus{} 1)^{n \plus{} 1}\cdot n$. What is the average of the first $ 200$ terms of the sequence? $ \textbf{(A)}\minus{}\!1\qquad \textbf{(B)}\minus{}\!0.5\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 0.5\qquad \textbf{(E)}\ 1$

If $ 9^{x \plus{} 2} \equal{} 240 \plus{} 9^x$, then the value of $ x$ is: $ \textbf{(A)}\ 0.1 \qquad \textbf{(B)}\ 0.2\qquad \textbf{(C)}\ 0.3\qquad \textbf{(D)}\ 0.4\qquad \textbf{(E)}\ 0.5$

In the triangle $ABC$ we have $|AB|<|AC|$. The bisectors of the angles at $B$ and $C$ meet $AC$ and $AB$ at $D$ and $E$ respectively. $BD$ and $CE$ intersect at the incenter $I$ of $\triangle ABC$. Prove that $\angle BAC=60^\circ$ if and only if $|IE|=|ID|$

For which $n$ can we find positive integers $a_1,a_2,\dots,a_n$ such that \[ a_1^2+a_2^2+\cdots+a_n^2 \] is a square?

John and Mary each have a white $8 \times 8$ square divided into $1 \times 1$ cells. They have painted an equal number of cells on their respective squares in blue. Prove that one can cut up each of the two squares into $2 \times 1 $ dominoes so that it is possible to reassemble John's dominoes into a new square and Mary's dominoes into another square with the same pattern of blue cells. (A Shapovalov)

Prove that if $a, b, c, d$ are integers such that $d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}$ then $d$ is a perfect square.

Vanya was asked to write on the board an expression equal to $10$, using only the numbers $1$, the signs $+$ and $-$ and brackets (you cannot make up the numbers $11$, $111$, etc., as well as $(-1)$). He knows that the bully Anton will then correct all the $+$ signs to $-$ and vice versa. Help Vanya compose the required expression, which will remain equal to $10$ even after Anton's actions.

Line $PQ$ is tangent to the incircle of triangle $ABC$ in such a way that the points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively. On the sides $AB$ and $AC$ are selected points $M$ and $N$, respectively, so that $AM = BP$ and $AN = CQ$. Prove that all lines constructed in this manner $MN$ pass through one point

Find all strictly increasing sequences of positive integers $a_1, a_2, \ldots$ with $a_1=1$, satisfying $$3(a_1+a_2+\ldots+a_n)=a_{n+1}+\ldots+a_{2n}$$ for all positive integers $n$.

Consider a horizontal strip of $N+2$ squares in which the first and the last square are black and the remaining $N$ squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color tis neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let $w(N)$ be the expected number of white squares remaining. Find \[ \lim_{N\to\infty}\frac{w(N)}{N}.\]

Let $ A $ be a finite set of points in space having the property that for any of its points $ P, Q $ there is an isometry of space that transforms the set $ A $ into the set $ A $ and the point $ P $ into the point $ Q $. Prove that there is a sphere passing through all points of the set $ A $.

If $x^2+\frac{1}{x^2}=7,$ find all possible values of $x^5+\frac{1}{x^5}.$

Find all prime numbers $ p$ with $ 5^p\plus{}4p^4$ is the square of an integer.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.