The four congruent circles below touch one another and each has radius 1. [asy] unitsize(30); fill(box((-1,-1), (1, 1)), gray); filldraw(circle((1, 1), 1), white); filldraw(circle((1, -1), 1), white); filldraw(circle((-1, 1), 1), white); filldraw(circle((-1, -1), 1), white); [/asy] What is the area of the shaded region?

Find all the three-digit numbers such that it equals to the arithmetic mean of the six numbers obtained by rearranging its digits.

A triangle with side lengths $16$, $18$, and $21$ has a circle with radius $6$ centered at each vertex. Find $n$ so that the total area inside the three circles but outside of the triangle is $n\pi$. [img]https://4.bp.blogspot.com/-dpCi7Gai3ZE/XoEaKo3C5wI/AAAAAAAALl8/KAuCVDT9R5MiIA_uTfRyoQmohEVw9cuVACK4BGAYYCw/s200/2018%2Bpc%2Bhs2.png[/img]

Suppose that a function $f:\mathbb R\to\mathbb R$ satisfies the following conditions: (i) $\left|f(a)-f(b)\right|\le|a-b|$ for all $a,b\in\mathbb R$; (ii) $f(f(f(0)))=0$. Prove that $f(0)=0$.

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

Seated in a circle are $11$ wizards. A different positive integer not exceeding $1000$ is pasted onto the forehead of each. A wizard can see the numbers of the other $10$, but not his own. Simultaneously, each wizard puts up either his left hand or his right hand. Then each declares the number on his forehead at the same time. Is there a strategy on which the wizards can agree beforehand, which allows each of them to make the correct declaration?

What is the radius of the largest circle centered at $(2, 2)$ that is completely bounded within the parabola $y = x^2 - 4x + 5$?

Let $ABC$ be an equilateral triangle with unit side length and circumcircle $\Gamma$. Let $D_1, D_2$ be the points on $\Gamma$ such that $BD_i = 3CD_i$. Let $E_1, E_2$ be the points on $\Gamma$ such that $CE_i = 3AE_i$. Let $F_1, F_2$ be the points on $\Gamma$ such that $AF_i = 3BF_i$. Then points $D_1, D_2, E_1, E_2, F_1, F_2$ are the vertices of a convex hexagon. What is the area of this hexagon?

The numbers $1, 2, 3, \ldots$ are arranged in a spiral in the vertices of an infinite square grid (see figure). Then in the centre of each square the sum of the numbers in its vertices is placed. Prove that for each positive integer n the centres of the squares contain infinitely many multiples of $n$.

Find all pairs of positive numbers $(x,y)$ which satisfy the system of equations $$\begin{cases} x^2 +y^2 = a^2 +b^2 \\ x^3 +y^3 = a^3 +b^3 \end{cases}$$ where $a$ and $b$ are given positive numbers.

In each square of the table below, we must write a different integer from $1$ to $17$, such that the sum of the numbers in each of the eight columns is the same, and the sum of the numbers in the top row is twice the sum of the numbers in the bottom row. Which number from $1$ to $17$ can be omitted? [img]https://wiki-images.artofproblemsolving.com//2/2b/Zrzut_ekranu_2023-05-22_o_10.28.33.png[/img]

What’s the greatest pyramid volume one can form using edges of length $2, 3, 3, 4, 5, 5$, respectively?

In the equilateral triangle $[ABC], D$ is the midpoint of $[AC], E$ and the orthogonal projection of $D$ over $[CB]$ and $F$ is the midpoint of $[DE]$. Prove that $[FB]$ and $[AE]$ are perpendicular. [img]https://1.bp.blogspot.com/-TjSyQotGIOM/X4XMolaXHvI/AAAAAAAAMng/cVsHfl-lrXAFE5LMdosE6vqK1Tf-8WOQgCLcBGAsYHQ/s0/2006%2Bportugal%2Bp2.png[/img]

Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$

Triangle $ABC$ is given. Circle $ \omega $ passes through $B$, touch $AC$ in $D$ and intersect sides $AB$ and $BC$ at $P$ and $Q$ respectively. Line $PQ$ intersect $BD$ and $AC$ at $M$ and $N$ respectively. Prove that $ \omega $, circumcircle of $DMN$ and circle, touching $PQ$ in $M$ and passes through B, intersects in one point.

Let $n!=ab^2$ where $a$ is free from squares. Prove, that for every $\epsilon>0$ for every big enough $n$ it is true, that $$2^{(1-\epsilon)n}<a<2^{(1+\epsilon)n}$$ [i]M. Ivanov[/i]

Given integer $n \geq 3$. There are $n$ dots marked $1$ to $n$ clockwise on a big circle. And between every two neighboring dots, there is a light. At first, every light were dark. A and B are playing a game, A pick up $n$ pairs from $\{ (i,j)|1 \leq i < j \leq n \}$ and for every pairs $(i,j)$. B starts from the point marked $i$ and choose to walk clockwise or counterclockwise to the point marked $j$. And B invert the status of all passing lights (bright $\leftrightarrow$ dark) A hopes the number of dark light can be as much as possible while B hopes the number of bright light can be as much as possible. Suppose A, B are both smart, how many lights are bright in the end? [i]Proposed by BlessingOfHeaven[/i] [img]https://pbs.twimg.com/profile_images/1014932415201120256/u9KAaMZ4_400x400.jpg[/img]

Let $(F_{n})_{n\geq1}$ the Fibonacci sequence. Find all $n \in \mathbb{N}$ such that for every $k=0,1,...,F_{n}$ \[ {F_{n}\choose k} \equiv (-1)^{k} \ (mod \ F_{n}+1) \]

In Vila Par, all the truth coins weigh an even quantity of grams and the false coins weigh an odd quantity of grams. The eletronic device only gives the parity of the weight of a set of coins. If there are $2020$ truth coins and $2$ false coins, detemine the least $k$, such that, there exists a strategy that allows to identify the two false coins using the eletronic device, at most, $k$ times.

Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor \equal{} 4x \plus{} 3$.

To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains $60$ pills, then the supply of medicine would last approximately $\text{(A)}\ 1\text{ month} \qquad \text{(B)}\ 4\text{ months} \qquad \text{(C)}\ 6\text{ months} \qquad \text{(D)}\ 8\text{ months} \qquad \text{(E)}\ 1\text{ year}$

Let $a, b$ be two co-prime positive integers. A number is called [i]good [/i] if it can be written in the form $ax + by$ for non-negative integers $x, y$. Define the function $f : Z\to Z $as $f(n) = n - n_a - n_b$, where $s_t$ represents the remainder of $s$ upon division by $t$. Show that an integer $n$ is [i]good [/i]if and only if the infinite sequence $n, f(n), f(f(n)), ...$ contains only non-negative integers.

Let $S$ be a set of $2022$ lines in the plane, no two parallel, no three concurrent. $S$ divides the plane into finite regions and infinite regions. Is it possible for all the finite regions to have integer area? [i]Proposed by Tony Wang[/i]

Let $\alpha = \cos^{-1} \left( \frac35 \right)$ and $\beta = \sin^{-1} \left( \frac35 \right) $. $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m}$$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.

Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b2,\ldots, b_n$ on the board. Prove that \[\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.\]