On the sides $AB, BC, CD$ and $DA$ of the parallelogram $ABCD$ marked the points $M, N, K$ and $F$. respectively. Is it possible to determine, using only compass, whether the area of ​​the quadrilateral $MNKF$ is equal to half the area of ​​the parallelogram $ABCD$?

Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?

In the figures, the vertices are marked with a circle. The segments that join vertices are called paths. Non-negative integers are distributed to the vertices and, to the paths, the differences between the numbers at their ends. [img]https://cdn.artofproblemsolving.com/attachments/d/6/e6fce93719a5b35dbf34d58652b01a8631de57.gif[/img] We will say that a distribution of numbers is [i]graceful [/i] if all the numbers from $1$ to $n$ appear in the paths, where $n$ is the number of paths. The following is an example of graceful distribution: [img]https://cdn.artofproblemsolving.com/attachments/1/1/a8c2b4fde673ca902b655804c4f5321f9666e9.gif[/img] Give -if possible- a graceful distribution for the following figures. If you can't do it, show why.

Two players play the following game: alternatively they write numbers $1$ or $0$ in the vertices of an $n$-gon. First player starts the game and wins if after any of his moves there exists a triangle, whose vertices are three consecutive vertices of the $n$-gon, such that the sum of numbers in it's vertices is divisible by $3$. Second player wins if he prevents this. Determine which player has a winning strategy if: a) $n=2019$ b) $n=2020$ c) $n=2021$

The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ in points $A_1, B_1$ and $C_1$ respectively. $C_1D$ is a diameter of the incircle. Finally, let $E$ be the intersection of the lines $B_1C_1$ and $A_1D$. Prove that the segments $CE$ and $CB_1$ have equal length.

On a straight track are several runners, each running at a different constant speed. They start at one end of the track at the same time. When a runner reaches any end of the track, he immediately turns around and runs back with the same speed (then he reaches the other end and turns back again, and so on). Some time after the start, all runners meet at the same point. Prove that this will happen again.

Let $(a_n)_{n=1}^{\infty}$ be a sequence with $a_n \in \{0,1\}$ for every $n$. Let $F:(-1,1) \to \mathbb{R}$ be defined by \[F(x)=\sum_{n=1}^{\infty} a_nx^n\] and assume that $F\left(\frac{1}{2}\right)$ is rational. Show that $F$ is the quotient of two polynomials with integer coefficients.

Prove that there are no integers $a,b,c,d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$, and equals $2$ at $x=62$.

For positive integers $n$, let $f(n)$ denote the number of integers $1 \leq a \leq 130$ for which there exists some integer $b$ such that $a^b-n$ is divisible by $131$, and let $g(n)$ denote the sum of all such $a$. Find the remainder when$$\sum_{n = 1}^{130} [f(n) \cdot g(n)]$$is divided by $131$. [i]Proposed by [b]ApraTrip[/b][/i]

For any $n\in \mathbb{N}^*$, let $H_n=\left\{\frac{k}{n!}\ |\ k\in \mathbb{Z}\right\}$. a) Prove that $H_n$ is a subgroup of the group $(Q,+)$ and that $Q=\bigcup_{n\in \mathbb{N}^*} H_n$; b) Prove that if $G_1,G_2,\ldots, G_m$ are subgroups of the group $(Q,+)$ and $G_i\neq Q,\ (\forall) 1\le i\le m$, then $G_1\cup G_2\cup \ldots \cup G_m\neq Q$ [i]Marian Andronache & Ion Savu[/i]

There are $ n ( \ge 4 ) $ people and some people shaked hands each other. Two people can shake hands at most 1 time. For arbitrary four people $ A, B, C, D$ such that $ (A,B), (B,C), (C,D) $ shaked hands, then one of $ (A,C), (A,D), (B,D) $ shaked hand each other. Prove the following statements. (a) Prove that $ n $ people can be divided into two groups, $ X, Y ( \ne \emptyset )$ , such that for all $ (x,y) $ where $ x \in X $ and $ y \in Y $, $ x $ and $ y $ shaked hands or $ x $ and $ y $ didn't shake hands. (b) There exist two people $ A , B $ such that the set of people who are not $ A $ and $ B $ that shaked hands with $ A $ is same wiith the set of people who are not $ A $ and $ B $ that shaked hands with $ B $.

Find all natural numbers that have a multiple consisting only of the digit 9.

Positive integers $p, q, r$ satisfy $gcd(a,b,c) = 1$. Prove that there exists an integer $a$ such that $gcd(p,q+ar) = 1$.

Let $n$ and $k$ be positive integers. An $n$-tuple $(a_1, a_2,\ldots , a_n)$ is called a permutation if every number from the set $\{1, 2, . . . , n\}$ occurs in it exactly once. For a permutation $(p_1, p_2, . . . , p_n)$, we define its $k$-mutation to be the $n$-tuple $$(p_1 + p_{1+k}, p_2 + p_{2+k}, . . . , p_n + p_{n+k}),$$ where indices are taken modulo $n$. Find all pairs $(n, k)$ such that every two distinct permutations have distinct $k$-mutations. [i]Remark[/i]: For example, when $(n, k) = (4, 2)$, the $2$-mutation of $(1, 2, 4, 3)$ is $(1 + 4, 2 + 3, 4 + 1, 3 + 2) = (5, 5, 5, 5)$. [i]Proposed by Borna Šimić[/i]

Let real numbers $a,b$ such that $a\ge b\ge 0$. Prove that \[ \sqrt{a^2+b^2}+\sqrt[3]{a^3+b^3}+\sqrt[4]{a^4+b^4} \le 3a+b .\]

Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$