The diagram below shows a $12$ by $20$ rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/9/e/ed6be5110d923965c64887a2ca8e858c977700.png[/img]

Find all integer numbers $n\ge 4$ which satisfy the following condition: from every $n$ different $3$-element subsets of $n$-element set it is possible to choose $2$ subsets, which have exactly one element in common.

Given two points $M$ and $K$ on the circumference with radius $r_1$ and centre $O_1$. The circumference with radius $r_2$ and centre $O_2$ is inscribed in $\angle MO_1K$ . Find the area of quadrangle $MO_1KO_2$ .

Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions $\star$ each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are [b]not[/b] relatively prime $\star$ all other pairs of the five integers [b]are[/b] relatively prime.

[b]8.[/b] Show that on any tetrahedron there can be found three acute bihedral angles such that the faces including these angles count among them all faces of tetrahedron. [b](G. 10)[/b]

Let $S_i$ be the sum of the first $i$ terms of the arithmetic sequence $a_1,a_2,a_3\ldots $. Show that the value of the expression \[\frac{S_i}{i}(j-k) + \frac{S_j}{j}(k-i) +\ \frac{S_k}{k}(i-j)\] does not depend on the numbers $i,j,k$ nor on the choice of the arithmetic sequence $a_1,a_2,a_3,\ldots$.

Let $ABC$ be an acute-angled scalene triangle and $\Gamma$ be its circumcircle. $K$ is the foot of the internal bisector of $\angle BAC$ on $BC$. Let $M$ be the midpoint of the arc $BC$ containing $A$. $MK$ intersect $\Gamma$ again at $A'$. $T$ is the intersection of the tangents at $A$ and $A'$. $R$ is the intersection of the perpendicular to $AK$ at $A$ and perpendicular to $A'K$ at $A'$. Show that $T, R$ and $K$ are collinear.

Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum \[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\] can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

Create $2024$ by $2024$ grid of integers numbered from $1$ to $2024^2$ such that the integer in row $n$ and column $k$ is $2024(n-1)+k$. Let the sum of the numbers in the grid be $S$. Arthur picks 2024 numbers such that their sum is $\dfrac{S}{2024}$. Prove that in every case, using the remaining numbers, Bob can create 2023 sets of 2024 numbers with equal numbers. For clarification, All numbers can only be used once

Find all sequence of consecutive positive numbers which the sum of them is equal with $2019$.

Let $ABC$ be a triangle and $A',B',C'$ be the midpoints of $BC,CA,AB$ respectively. Let $P$ and $P'$ be points in plane such that $PA=P'A',PB=P'B',PC=P'C'$. Prove that all $PP'$ pass through a fixed point.

Every point of the rectangle $R=[0,4] \times [0,40]$ is coloured using one of four colours. Show that there exist four points in $R$ with the same colour that form a rectangle having integer side lengths.

What is the sum of the mean, median, and mode of the numbers, $2,3,0,3,1,4,0,3$? $ \textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 7.5\qquad\textbf{(D)}\ 8.5\qquad\textbf{(E)}\ 9 $

Chris has a list of $5$ distinct numbers and every minute he independently and uniformly at random swaps a pair of them. Find the probability that after $4$ minutes the order of the list is the same as the original list.

Find the number of $6$-tuples $(A_1,A_2,...,A_6)$ of subsets of $M = \{1,..., n\}$ (not necessarily different) such that each element of $M$ belongs to zero, three, or six of the subsets $A_1,...,A_6$.

Given a board consists of $n \times n$ unit squares ($n \ge 3$). Each unit square is colored black and white, resembling a chessboard. In each step, TOMI can choose any $2 \times 2$ square and change the color of every unit square chosen with the other color (white becomes black and black becomes white). Find every $n$ such that after a finite number of moves, every unit square on the board has a same color.

Let $n$ be some positive integer. Free university accepts $n^2$ freshmen, where no two students know each other initially. It's known that students can only get to know eachother on parties, which are organized by the university's administration. The administration's goal is to ensure that there does not exist a group of $n$ students where none of them know each other. Organizing a party with $m$ members incurs a cost of $m^2 - m$. Determine the minimal cost for the administration to fulfill their goal. [i]Proposed by Luka Macharashvili, Georgia[/i]

[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins? [b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h$ if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc} & & a & b & c \\ + & & & d & e \\ \hline & f & a & g & c \\ x & b & b & h & \\ \hline f & f & e & g & c \\ \end{tabular}$ [b]p4.[/b] Is it possible to find a rectangle of perimeter $10$ m and cut it in rectangles (as many as you want) so that the sum of the perimeters is $500$ m? [b]p5.[/b] The picture shows a maze with chambers (shown as circles) and passageways (shown as segments). A cat located in chamber $C$ tries to catch a mouse that was originally in the chamber $M$. The cat makes the first move, moving from chamber $C$ to one of the neighboring chambers. Then the mouse moves, then the cat, and so forth. At each step, the cat and the mouse can move to any neighboring chamber or not move at all. The cat catches the mouse by moving into the chamber currently occupied by the mouse. Can the cat get the mouse? [img]https://cdn.artofproblemsolving.com/attachments/9/9/25f61e1499ff1cfeea591cb436d33eb2cdd682.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]

We color one of the numbers $1,...,8$ with white or black according to the following rules: i) number $4$ gets colored white and one at lest of the following numbers gets colored black ii) if two numbers $a,b$ are colored in a different color and $a+b\le 8$, then number $a+b$ gets colored black. iii) if two numbers $a,b$ are colored in a different color and $a\cdot b\le 8$, then number $a\cdot b$ gets colored white. If by those rules, all numbers get colored, find the color of each number.

Denote by $S_{n}$ the set of all permutations of the set $\{1,2,\dots, n\}$. Let $\sigma \in S_{n}$ be a permutation. We define the $\textit{displacement}$ of $\sigma$ to be the number $d(\sigma)=\sum_{i=1}^{n} \vert \sigma(i)-i \vert$. We saw that $\sigma$ is $\textit{maximally}$ $\textit{displacing}$ if $d(\sigma)$ is the largest possible, i.e. if $d(\sigma) \geq d({\pi})$, for all $\pi \in S_{n}$. $\text{a)}$ Suppose $\sigma$ is a maximally displacing permutation of $\{1,2, \dots, 2024\}$. Prove that $\sigma(i)\neq i$, for all $i \in \{1,2, \dots, 2024.\}$ $\text{b)}$ Does the statement of part a) hold for permutations of $\{1,2, \dots, 2025\}$?

Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(f(x - y)) = f(x)f(y) + f(x) - f(y) - xy.\]

Five numbers are chosen from $\{1, 2, \ldots, n\}$. Determine the largest $n$ such that we can always pick some of the 5 chosen numbers so that they can be made into two groups whose numbers have the same sum (a group may contain only one number).

For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.

A Pokemon fan walks into a store. An employee tells them that there are $2$ Pikachus, $3$ Eevees, $4$ Snorlaxes, and $5$ Bulbasaurs remaining inside the gacha machine. Given that this fan cannot see what is inside the Poké Balls before opening them, find the least number of Poké Balls they must buy in order to be sure to get one Pikachu and one Snorlax.