On competition which has $16$ teams, it is played $55$ games. Prove that among them exists $3$ teams such that they have not played any matches between themselves.

When Cirno walks into her perfect math class today, she sees a polynomial $P(x)=1$ (of degree 0) on the blackboard. As her teacher explains, for her pop quiz today, she will have to perform one of the two actions every minute: (a) Add a monomial to $P(x)$ so that the degree of $P$ increases by 1 and $P$ remains monic; (b) Replace the current polynomial $P(x)$ by $P(x+1)$. For example, if the current polynomial is $x^2+2x+3$, then she will change it to $(x+1)^2+2(x+1)+3=x^2+4x+6$. Her score for the pop quiz is the sum of coefficients of the polynomial at the end of 9 minutes. Given that Cirno (miraculously) doesn't make any mistakes in performing the actions, what is the maximum score that she can get? [i]Proposed by Yannick Yao[/i]

For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.

Find all positive real numbers $x$, that verify $x+\left[\frac{x}{3}\right]=\left[\frac{2x}{3}\right]+\left[\frac{3x}{5}\right]$.

Let $ b$ be an even positive integer for which there exists a natural number n such that $ n>1$ and $ \frac{b^n\minus{}1}{b\minus{}1}$ is a perfect square. Prove that $ b$ is divisible by 8.

Misha and Sahsa play a game on a $100\times 100$ chessboard. First, Sasha places $50$ kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins): At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture. Is there a winning strategy available for Sasha?

In the plane are given a circle with center $ O$ and radius $ r$ and a point $ A$ outside the circle. For any point $ M$ on the circle, let $ N$ be the diametrically opposite point. Find the locus of the circumcenter of triangle $ AMN$ when $ M$ describes the circle.

All four angles of quadrilateral are greater than $60^o$. Prove that we can choose three sides to make a triangle.

In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of [b]a.)[/b] all vertices $A$ of such triangles; [b]b.)[/b] all vertices $B$ of such triangles; [b]c.)[/b] all vertices $C$ of such triangles.

A analityc function $f:\mathbb{C}\to\mathbb{C}$ is call interesting if $f(z)$ is real along the parabola $Re (z) = (Im (z))^2$. a) Find an example of a non constant interesting function. b) Show that every interesting function $f$ satisfy that $f'(-3/4) = 0.$

Two rectangles of unit area overlap to form a convex octagon. Show that the area of the octagon is at least $\dfrac {1} {2}$. [i]Kvant Magazine [/i]

Determine all possible values of the expression$$x-\left [\frac{x}{2}\right ]-\left [\frac{x}{3}\right ]-\left [\frac{x} {6}\right ]$$by varying $x$ in the real numbers. Clarification: The brackets indicate the integer part of the number they enclose.

Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.

For an integer $n \ge 1$ we consider sequences of $2n$ numbers, each equal to $0, -1$ or $1$. The [i]sum product value[/i] of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together. For example, if we take $n = 2$ and the sequence $0,1, 1, -1$, then we find the products $0\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1$. Adding these six results gives the sum product value of this sequence: $0+0+0+1+(-1)+(-1) = -1$. The sum product value of this sequence is therefore smaller than the sum product value of the sequence $0, 0, 0, 0$, which equals $0$. Determine for each integer $n \ge 1$ the smallest sum product value that such a sequence of $2n$ numbers could have. [i]Attention: you are required to prove that a smaller sum product value is impossible.[/i]

In a soccer league with $2020$ teams every two team have played exactly once and no game have lead to a draw. The participating teams are ordered first by their points (3 points for a win, 1 point for a draw, 0 points for a loss) then by their goal difference (goals scored minus goals against) in a normal soccer table. Is it possible for the goal difference in such table to be strictly increasing from the top to the bottom? [i]Proposed by Abolfazl Asadi[/i]

A vertical line divides the triangle with vertices $(0,0)$, $(1,1)$, and $(9,1)$ in the $xy\text{-plane}$ into two regions of equal area. The equation of the line is $x=$ $\textbf {(A) } 2.5 \qquad \textbf {(B) } 3.0 \qquad \textbf {(C) } 3.5 \qquad \textbf {(D) } 4.0\qquad \textbf {(E) } 4.5$

If $ABCDEF$ is a convex cyclic hexagon, then its diagonals $AD$, $BE$, $CF$ are concurrent if and only if $\frac{AB}{BC}\cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$. [i]Alternative version.[/i] Let $ABCDEF$ be a hexagon inscribed in a circle. Then, the lines $AD$, $BE$, $CF$ are concurrent if and only if $AB\cdot CD\cdot EF=BC\cdot DE\cdot FA$.

Let $G, H$ be the centroid and orthocentre of $\vartriangle ABC$ which has an obtuse angle at $\angle B$. Let $\omega$ be the circle with diameter $AG$. $\omega$ intersects $\odot(ABC)$ again at $L \ne A$. The tangent to $\omega$ at $L$ intersects $\odot(ABC)$ at $K \ne L$. Given that $AG = GH$, prove $\angle HKG = 90^o$ . [i]Sam Bealing, United Kingdom[/i]

The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle. [img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]

$ABC$ is an acute triangle with $AB> AC$. $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for $ \Gamma_C$. Let $D$ be the intersection $\Gamma_B$ and $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle $ABC$. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.

Show that, if $a,b,...,n$ are distinct natural numbers, none divisible by any primes greater than $3$, then $$\frac{1}{a}+\frac{1}{b}+...+ \frac{1}{n}< 3$$

Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$ \[x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).\] Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$

Let $f(x)$ be a function, where if $q$ is an integer, then $f(\tfrac{1}{q})=q,$ and if $m$ and $n$ are real numbers, $f(m\cdot n)$ $ =$ $ f(m)\cdot f(n).$ If $f(\sqrt{2})$ can be written as $\tfrac{\sqrt{a}}{b}$ where $a$ is not divisible by the square of any prime and $b$ is a positive integer, then what is $a+b$?

A polynomial $ f(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}c$ is such that $ b<0$ and $ ab\equal{}9c$. Prove that the polynomial $ f$ has three different real roots.

Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$ [i]Proposed by Krit Boonsiriseth[/i]