The functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$. What is the least period of the function $\cos(\sin(x))$? $\textbf{(A)}\ \frac{\pi}{2}\qquad\textbf{(B)}\ \pi\qquad\textbf{(C)}\ 2\pi\qquad\textbf{(D)}\ 4\pi\qquad\textbf{(E)}$ It's not periodic.

Some figures stand in certain cells of a chess board. It is known that a figure stands on each row, and that different rows have a different number of figures. Prove that it is possible to mark $8$ figures so that on each row and column stands exactly one marked figure.

Let $G$ be a simple graph with $n$ vertices and $m$ edges. Two vertices are called [i]neighbours[/i] if there is an edge between them. It turns out the $G$ does not contain any cycles of length from 3 to $2k$ (inclusive), where $k\geq2$ is a given positive integer. a) Prove that it is possible to pick a non-empty set $S$ of vertices of $G$ such that every vertex in $S$ has at least $\left\lceil \frac mn \right\rceil$ neighbours that are in $S$. ($\lceil x\rceil$ denotes the smallest integer larger than or equal to $x$.) b) Suppose a set $S$ as described in (a) is chosen. Let $H$ be the graph consisting of the vertices in $S$ and the edges between those vertices only. Let $v$ be a vertex of $H$. Prove that at least $\left\lceil \left(\frac mn -1\right)^k \right\rceil$ vertices of $H$ can be reached by starting at $v$ and travelling across the edges of $H$ for at most $k$ steps. (Note that $v$ itself satisfies this condition, since it can be reached by starting at $v$ and travelling along the edges of $H$ for 0 steps.)

Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?

[color=darkred]Let $f\colon [0,\infty)\to\mathbb{R}$ be a continuous function such that $\int_0^nf(x)f(n-x)\ \text{d}x=\int_0^nf^2(x)\ \text{d}x$ , for any natural number $n\ge 1$ . Prove that $f$ is a periodic function.[/color]

Divide the upper right quadrant of the plane into square cells with side length $1$. In this quadrant, $n^2$ cells are colored, show that there’re at least $n^2+n$ cells (possibly including the colored ones) that at least one of its neighbors are colored.

For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ and let $\{x\} = x - \lfloor x\rfloor$. If $a, b, c$ are distinct real numbers, prove that \[\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}\] is an integer if and only if $\{a\} + \{b\} + \{c\}$ is an integer.

Let the AP of the form $4$, $9$, $\ldots$ be $\mathbf{A}$, and the AP of the form $16$, $25$, $\ldots$ be $\mathbf{B}$. Find the number of integers from $1$ to $2024$ inclusive, that appear in only one of the AP's. For clarification, the AP's $\mathbf{A}$ and $\mathbf{B}$ start from 4 and 16 respectively. [hide=Answer]584[/hide]

Let $0<c<1$ and $n$ a positive integer. Alice and Bob are playing a game. Bob writes $n$ integers on the board, not all equal. On a player's turn, they erase two numbers from the board and write their arithmetic mean instead. Alice starts and performs at most $cn$ moves. After her, Bob makes moves until there are only two numbers left on the board. Alice wins if these two numbers are different, and otherwise, Bob wins. For which values of $c$ does Alice win for all large enough $n$?

An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.

Let $w(x)$ be largest odd divisor of $x$. Let $a,b$ be natural numbers such that $(a,b)=1$ and \\ $a+w(b+1)$ and $b+w(a+1)$ are powers of two. Prove that $a+1$ and $b+1$ are powers of two.

Given real numbers $x$ and $y$. Let $s_{1}=x+y, s_{2}=x^2+y^2, s_{3}=x^3+y^3, s_{4}=x^4+y^4$ and $t=xy$. [b]a)[/b] Prove, that number $t$ is rational, if $s_{2}, s_{3}$ and $s_{4}$ are rational numbers. [b]b)[/b] Prove, that number $s_{1}$ is rational, if $s_{2}, s_{3}$ and $s_{4}$ are rational numbers. [b]c)[/b] Can number $s_{1}$ be irrational, if $s_{2}$ and $s_{3}$ are rational numbers?

Find all solutions for positive integers $(x,y,k,m)$ such that \[ 20x^k+24y^m = 2024\] with $k, m > 1$

Let $ABCD$ be a rectangle. We consider the points $E\in CA,F\in AB,G\in BC$ such that $DC\perp CA,EF\perp AB$ and $EG\perp BC$. Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$.

Circle $\omega$ inscribed in triangle $ABC$ touches sides $BC$, $CA$, $AB$ at points $A_1$, $B_1$ and $C_1$ respectively. On the extension of segment $AA_1$, point $A$ is taken as point D such that $AD= AC_1$. Lines $DB_1$ and $DC_1$ intersect a second time circle $\omega$ at points $B_2$ and $C_2$. Prove that $B_2C_2$ is the diameter of circle of $\omega$.

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]