Charlotte writes the integers $1,2,3,\ldots,2025$ on the board. Charlotte has two operations available: the GCD operation and the LCM operation. [list] [*]The GCD operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{gcd}(a, b)$. [*]The LCM operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{lcm}(a, b)$. [/list] An integer $N$ is called a [i]winning number[/i] if there exists a sequence of operations such that, at the end, the only integer left on the board is $N$. Find all winning integers among $\{1,2,3,\ldots,2025\}$ and, for each of them, determine the minimum number of GCD operations Charlotte must use. [b]Note:[/b] The number $\operatorname{gcd}(a, b)$ denotes the [i]greatest common divisor[/i] of $a$ and $b$, while the number $\operatorname{lcm}(a, b)$ denotes the [i]least common multiple[/i] of $a$ and $b$.

Solve the equation $\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x $

Let $N$ be a positive integer. It is given set of weights which satisfies following conditions: $i)$ Every weight from set has some weight from $1,2,...,N$; $ii)$ For every $i\in {1,2,...,N}$ in given set there exists weight $i$; $iii)$ Sum of all weights from given set is even positive integer. Prove that set can be partitioned into two disjoint sets which have equal weight

Consider the set $ M = \{1,2, \ldots ,2008\}$. Paint every number in the set $ M$ with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets: $ S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}$; $ S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}$. Prove that $ 2|S_1| > |S_2|$ (where $ |X|$ denotes the number of elements in a set $ X$).

A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $ 2/\pi $, as shown. What is the perimeter of this region? [asy] size(90); defaultpen(linewidth(0.7)); filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle,gray(0.5)); filldraw(arc((1,0),1,180,0, CCW)--cycle,gray(0.7)); filldraw(arc((0,1),1,90,270)--cycle,gray(0.7)); filldraw(arc((1,2),1,0,180)--cycle,gray(0.7)); filldraw(arc((2,1),1,270,90, CCW)--cycle,gray(0.7));[/asy] $ \textbf{(A) }\frac {4}\pi\qquad\textbf{(B) }2\qquad\textbf{(C) }\frac {8}\pi\qquad\textbf{(D) }4\qquad\textbf{(E) }\frac{16}{\pi} $

Find all polynomials $p(x)$ with real coefficients that have the following property: there exists a polynomial $q(x)$ with real coefficients such that $$p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)$$ for all positive integers $n$.

Let the sequence $\{a_n\}_{n\ge 1}$ be defined by $$a_n=\tan(n\theta)$$ where $\tan\theta =2$. Show that for all $n$, $a_n$ is a rational number which can be written with an odd denominator.

The positive real numbers $a,b, c,d$ satisfy the equality $ \frac {1}{a+1} + \frac {1}{b+1} + \frac {1}{c+1} + \frac{ 1}{d+1} = 3 $ . Prove the inequality $\sqrt [3]{abc} + \sqrt [3]{bcd} + \sqrt [3]{cda} + \sqrt [3]{dab} \le \frac {4}{3} $.

Let triangle $ABC$ be a given acute scalene triangle with altitudes $BE$ and $CF$. Let $D$ be the point where the incircle of $\,\triangle ABC$ touches side $BC$. The circumcircle of $\triangle BDE$ meets line $AB$ again at point $K$, the circumcircle of $\triangle CDF$ meets line $AC$ again at point $L$. The circumcircle of $\triangle BDE$ and $\triangle CDF$ meet line $KL$ again at $X$ and $Y$, respectively. Prove that the incenter of $\triangle DXY$ lies on the incircle of $\,\triangle ABC$. [i]Proposed by Luu Dong, Vietnam[/i]

The difference in the areas of two similar triangles is $18$ square feet, and the ratio of the larger area to the smaller is the square of an integer. The area of the smaller triange, in square feet, is an integer, and one of its sides is $3$ feet. The corresponding side of the larger triangle, in feet, is: $\textbf{(A)}\ 12\quad \textbf{(B)}\ 9\qquad \textbf{(C)}\ 6\sqrt{2}\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 3\sqrt{2}$

The integers $ c_{m,n}$ with $ m \geq 0, \geq 0$ are defined by \[ c_{m,0} \equal{} 1 \quad \forall m \geq 0, c_{0,n} \equal{} 1 \quad \forall n \geq 0,\] and \[ c_{m,n} \equal{} c_{m\minus{}1,n} \minus{} n \cdot c_{m\minus{}1,n\minus{}1} \quad \forall m > 0, n > 0.\] Prove that \[ c_{m,n} \equal{} c_{n,m} \quad \forall m > 0, n > 0.\]

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 $x(y^2-p)+y(x^2-p)=5p$