There are $2024$ rectangles $1 \times n$ for $n=1, 2, \ldots, 2024$. Is it possible to make a square using some of them, such that the side length of the square is greater than $1$?

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.

Knights, who always tell truth, and liars, who always lie, live on an island. They have been distributed into two teams $A$ and $B$ for a game of tennis, and team $A$ had more members than team $B$. Two players from different teams started the game, whenever a player loses the game, he leaves it forever and he is replaces by a member of his team (that has never played before). The team, all of whose members left the game, loses. After the tournament, every member of team $A$ was asked: "Is it true that you have lost to a liar in some game?", and every member of team $B$ was asked: "Is it true that you have won at least two games, in which your opponent was a knight?". It turns out that every single answer was positive. Which team won?

The positive integers $1, 2, \ldots, 1000$ are written in some order on one line. Show that we can find a block of consecutive numbers, whose sum is in the interval $(100000; 100500]$.

Let $ABC$ be an isosceles triangle with $BA=BC$. The points $D, E$ lie on the extensions of $AB, BC$ beyond $B$ such that $DE=AC$. The point $F$ lies on $AC$ is such that $\angle CFE=\angle DEF$. Show that $\angle ABC=2\angle DFE$.

There are 7 different numbers on the board, their sum is $10$. For each number on the board, Petya wrote the product of this number and the sum of the remaining 6 numbers in his notebook. It turns out that the notebook only has 4 distinct numbers in it. Determine one of the numbers that is written on the board.

There is a circle which is 1 meter in circumference and a point marked on it. Two cockroaches start running in the same direction from the marked point with different speeds. Whenever the fast one would catch up with the slow one, the slow one would instantly turn around and start running in tho other direction with the same speed. Whenever they would meet face-to-face, the fast one would instantly turn around and start running in tho other direction with the same speed. How far from the marked point could their 100th meeting be?

Let $ABC$ be an acute triangle and let $P, Q$ lie on the segment $BC$ such that $BP=PQ=CQ$. The feet of the perpendiculars from $P, Q$ to $AC, AB$ are $X, Y$. Show that the centroid of $ABC$ is equidistant from the lines $QX$ and $PY$.

An equilateral triangle $T$ with side $111$ is partitioned into small equilateral triangles with side $1$ using lines parallel to the sides of $T$. Every obtained point except the center of $T$ is marked. A set of marked points is called $\textit{linear}$ if the points lie on a line, parallel to a side of $T$ (among the drawn ones). In how many ways we can split the marked point into $111$ $\textit{linear}$ sets?

Does there exist a positive integer $n>10^{100}$, such that $n^2$ and $(n+1)^2$ satisfy the following property: every digit occurs equal number of times in the decimal representations of each number?

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the lateral sides of all such trapezoids share a common point.

There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.

The quadrilateral $ABCD$ has perpendicular diagonals that meet at $O$. The incenters of triangles $ABC, BCD, CDA, DAB$ form a quadrilateral with perimeter $P$. Show that the sum of the inradii of the triangles $AOB, BOC, COD, DOA$ is less than or equal to $\frac{P} {2}$.

Do there exist distinct reals $x, y, z$, such that $\frac{1}{x^2+x+1}+\frac{1}{y^2+y+1}+\frac{1}{z^2+z+1}=4$?

Are there $10$ consecutive positive integers, such that if we consider the digits that appear in the decimal representations of those numbers as a multiset, every digit appears the same number of times in this multiset?

Let $ABCD$ be a quadrilateral such that $\angle A=\angle C=90^{\circ}$. If $A, D$ and the midpoints of $BA, BC$ are concyclic, show that the midpoints of $AD, DC$ and $B, C$ are concyclic.

Find all triplets $(a, b, c)$ of positive integers, such that $a+bc, b+ac, c+ab$ are primes and all divide $(a^2+1)(b^2+1)(c^2+1)$.

There are $2024$ people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were $0,1, \ldots, 2023$. Every knight answered truthfully, while every liar changed the real answer by exactly $1$. What is the minimal number of liars?

Let $x_1<x_2< \ldots <x_{2024}$ be positive integers and let $p_i=\prod_{k=1}^{i}(x_k-\frac{1}{x_k})$ for $i=1,2, \ldots, 2024$. What is the maximal number of positive integers among the $p_i$?

Let $XY$ be a segment, which is a diameter of a semi-circle. Let $Z$ be a point on $XY$ and 9 rays from $Z$ are drawn that divide $\angle XZY=180^{\circ}$ into $10$ equal angles. These rays meet the semi-circle at $A_1, A_2, \ldots, A_9$ in this order in the direction from $X$ to $Y$. Prove that the sum of the areas of triangles $ZA_2A_3$ and $ZA_7A_8$ equals the area of the quadrilateral $A_2A_3A_7A_8$.

The equation $$t^4+at^3+bt^2=(a+b)(2t-1)$$has $4$ positive real roots $t_1<t_2<t_3<t_4$. Show that $t_1t_4>t_2t_3$.

Teacher has 100 weights with masses $1$ g, $2$ g, $\dots$, $100$ g. He wants to give 30 weights to Petya and 30 weights to Vasya so that no 11 Petya's weights have the same total mass as some 12 Vasya's weights, and no 11 Vasya's weights have the same total mass as some 12 Petya's weights. Can the teacher do that?

Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.

3 segments $AA_1$, $BB_1$, $CC_1$ in space share a common midpoint $M$. Turns out, the sphere circumscribed about the tetrahedron $MA_1B_1C_1$ is tangent to plane $ABC$ at point $D$. Point $O$ is the circumcenter of triangle $ABC$. Prove that $MO = MD$.

Let $n>100$ be a positive integer and originally the number $1$ is written on the blackboard. Petya and Vasya play the following game: every minute Petya represents the number of the board as a sum of two distinct positive fractions with coprime nominator and denominator and Vasya chooses which one to delete. Show that Petya can play in such a manner, that after $n$ moves, the denominator of the fraction left on the board is at most $2^n+50$, no matter how Vasya acts.