Let $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$. What is the value of $mn$?

Each vertex of an $m\times n$ grid is colored blue, green or red in such a way that all the boundary vertices are red. We say that a unit square of the grid is properly colored if: $(i)$ all the three colors occur at the vertices of the square, and $(ii)$ one side of the square has the endpoints of the same color. Show that the number of properly colored squares is even.

Points $A, B, C, D, E$ and F are given on a circle in such a way that the three chords $AB, CD$ and $EF$ intersect in one point. Express angle $\angle EFA$ in terms of angles $\angle ABC$ and $\angle CDE$ (find all possibilities).

Let $N$ be a positive integer. In each of the $N^2$ unit squares of an $N\times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N\times N$ board into $K$ regions. For each $N$, determine the smallest and the largest possible values of $K$. [asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((1,0)--(1,3), dotted); draw((2,0)--(2,3), dotted); draw((0,1)--(3,1), dotted); draw((0,2)--(3,2), dotted); draw((1,0)--(0,1)--(2,3)--(3,2)--(2,1)--(0,3)); draw((1,1)--(2,0)--(3,1)); label("$1$",(0.35,2)); label("$2$",(1,2.65)); label("$3$",(2,2)); label("$4$",(2.65,2.65)); label("$5$",(0.35,0.35)); label("$6$",(1.3,1.3)); label("$7$",(2.65,0.35)); label("Example with $N=3$, $K=7$",(0,-0.3)--(3,-0.3),S); [/asy]

Let $ABC$ be an acute-angled triangle. The line through $C$ perpendicular to $AC$ meets the external angle bisector of $\angle ABC$ at $D$. Let $H$ be the foot of the perpendicular from $D$ onto $BC$. The point $K$ is chosen on $AB$ so that $KH \parallel AC$. Let $M$ be the midpoint of $AK$. Prove that $MC = MB + BH$. Giorgi Arabidze, Georgia,

In a dancing hall, there are $n$ girls standing in one row and $n$ boys in the other row across them (so that all $2n$ dancers form a $2 \times n$ board). Each dancer gives her / his left hand to a neighboring person standing to the left, across, or diagonally to the left. The analogous rule applies for right hands. No dancer gives both hands to the same person. In how many ways can the dancers do this?

We say that a natural number $n$ is [i]charrua[/i] if it satisfy simultaneously the following conditions: - Every digit of $n$ is greater than 1. - Every time that four digits of $n$ are multiplied, it is obtained a divisor of $n$ Show that every natural number $k$ there exists a [i]charrua[/i] number with more than $k$ digits.

Given a positive real $C \geq 1$ and a sequence $a_1, a_2, a_3, \cdots$ satisfying for any positive integer $n,$ $a_n \geq 0$ and for any real $x \geq 1$, $$\left|x\lg x-\sum_{k=1}^{[x]}\left[\frac{x}{k}\right]a_k \right| \leq Cx,$$ where $[x]$ is defined as the largest integer that does not exceed $x$. Prove that for any real $y \geq 1$, $$\sum_{k=1}^{[y]}a_k < 3Cy.$$

For an integer $30 \le k \le 70$, let $M$ be the maximum possible value of \[ \frac{A}{\gcd(A,B)} \quad \text{where } A = \dbinom{100}{k} \text{ and } B = \dbinom{100}{k+3}. \] Find the remainder when $M$ is divided by $1000$. [i]Based on a proposal by Michael Tang[/i]

The Fibonacci sequence is determined by conditions $ F_0 \equal{} 0, F1 \equal{} 1$, and $ F_k\equal{}F_{k\minus{}1}\plus{}F_{k\minus{}2}$ for all $ k \ge 2$. Let $ n$ be a positive integer and let $ P(x) \equal{} a_mx^m \plus{}. . .\plus{} a_1x\plus{} a_0$ be a polynomial that satisfies the following two conditions: (1) $ P(F_n) \equal{} F_{n}^{2}$ ; (2) $ P(F_k) \equal{} P(F_{k\minus{}1}) \plus{} P(F_{k\minus{}2}$ for all $ k \ge 2$. Find the sum of the coefficients of P.

Circle $\omega$ is tangent to sides $AB, AC$ of triangle $ABC$. A circle $\Omega$ touches the side $AC$ and line $AB$ (produced beyond $B$), and touches $\omega$ at a point $L$ on side $BC$. Line $AL$ meets $\omega, \Omega$ again at $K, M$. It turned out that $KB \parallel CM$. Prove that $\triangle LCM$ is isosceles.

Circle $\omega_1$ and $\omega_2$ have centers $(0,6)$ and $(20,0)$, respectively. Both circles have radius $30$, and intersect at two points $X$ and $Y$. The line through $X$ and $Y$ can be written in the form $y = mx+b$. Compute $100m+b$. [i]Proposed by Evan Chen[/i]

Let $\triangle ABC$ be a triangle such that $AB = AC$, and let its circumcircle be $\Gamma$. Let $\omega$ be a circle which is tangent to $AB$ and $AC$ at $B$ and $C$. Point $P$ belongs to $\omega$, and lines $PB$ and $PC$ intersect $\Gamma$ again at $Q$ and $R$. $X$ and $Y$ are points on lines $BR$ and $CQ$ such that $AX = XB$ and $AY = YC$. Show that as $P$ varies on $\omega$, the circumcircle of $\triangle AXY$ passes through a fixed point other than $A$.

Compute the three-digit number that satisfies the following properties: $\bullet$ The hundreds digit and ones digit are the same, but the tens digit is different. $\bullet$ The number is divisible by $9$. $\bullet$ When the number is divided by $5$, the remainder is $1$.

Let $\Gamma_1$ and $\Gamma_2$ be two circles with different radii, with $\Gamma_1$ the smaller one. The two circles meet at distinct points $A$ and $B$. $C$ and $D$ are two points on the circles $\Gamma_1$ and $\Gamma_2$, respectively, and such that $A$ is the midpoint of $CD$. $CB$ is extended to meet $\Gamma_2$ at $F$, while $DB$ is extended to meet $\Gamma_1$ at $E$. The perpendicular bisector of $CD$ and the perpendicular bisector of $EF$ meet at $P$. (a) Prove that $\angle{EPF} = 2\angle{CAE}$. (b) Prove that $AP^2 = CA^2 + PE^2$.

Find all integers $a$, $b$, $c$ and $d$ such that $$a^2+5b^2-2c^2-2cd-3d^2=0$$

Let $K$ be the intersection of the diagonals of a convex quadrilateral $ABCD$. Let $L\in [AD]$, $M \in [AC]$, $N \in [BC]$ such that $KL\parallel AB$, $LM\parallel DC$, $MN\parallel AB$. Show that \[\dfrac{Area(KLMN)}{Area(ABCD)} < \dfrac {8}{27}.\]

Nine balls numbered $1,2,\cdots,9$ are put on nine poines that divide the circle into nine equal parts. The sum of absolute values of the difference between the number of two adjacent balls is $S$. Find the probablity of $S$ takes its minumum value. Note: If one way of putting balls can be the same as another one by rotating or specular-reflecting, then they are considered the same way.

Find all continous functions $f:[0,1]\to[0,2]$ with the property that $\left(\int_{\frac1{n+1}}^{\frac1n}xf(x)dx\right)^2=\int_{\frac1{n+1}}^{\frac1n}x^2f(x)dx, \forall n\in\mathbb N^*$. [i]Gabriel Marsanu, Andrei Nedelcu[/i]

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 $n$.