For every positive integer $n$, define $S_n$ to be the sum \[ S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \, \pi}{2010} \right)^n . \] As $n$ approaches infinity, what value does $S_n$ approach?

Find all functions $ f,g:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify the relations $$ \left\{\begin{matrix} f(g(x)+g(y))=f(g(x))+y \\ g(f(x)+f(y))=g(f(x))+y\end{matrix}\right. , $$ for all $ x,y\in\mathbb{Q} . $

Let $ABC$ be a triangle and $\Gamma$ be the circle with diameter $[AB]$. The bisectors of $\angle BAC$ and $\angle ABC$ cut the circle $\Gamma$ again at $D$ and $E$, respectively. The incicrcle of the triangle $ABC$ cuts the lines $BC$ and $AC$ in $F$ and $G$, respectively. Show that the points $D, E, F$ and $G$ lie on the same line.

Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.

Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.

If $X=1+x+x^2+x^3+\cdots$ and $Y=1+y+y^2+y^3+\cdots$, what is $1+xy+x^2y^2+x^3y^3+\cdots$ in terms of $X$ and $Y$ only?

How many rectangles are there in the diagram below such that the sum of the numbers within the rectangle is a multiple of 7? [asy] int n; n=0; for (int i=0; i<=7;++i) { draw((i,0)--(i,7)); draw((0,i)--(7,i)); for (int a=0; a<=7;++a) { if ((a != 7)&&(i != 7)) { n=n+1; label((string) n,(a,i),(2,2)); } } } [/asy]

Show that for every integer $n \geq 6$, there exists a convex hexagon which can be dissected into exactly $n$ congruent triangles.

Given is an isosceles triangle $ABC$ with $CA=CB$ and angle bisector $BD$, $D \in AC$. The line through the center $O$ of $(ABC)$, perpendicular to $BD$, meets $BC$ at $E$. The line through $E$, parallel to $BD$, meets $AC$ at $F$. Prove that $CE=DF$.

When four gallons are added to a tank that is one-third full, the tank is then one-half full. The capacity of the tank in gallons is $\text{(A)}\ 8 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 48$

There are a board with $2n \cdot 2n \ (= 4n^2)$ squares and $4n^2-1$ cards numbered with different natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange two cards on two adjacent squares of a column (or a row) in a finite number of movements?

When the IMO is over and students want to relax, they all do the same thing: download movies from the internet. There is a positive number of rooms with internet routers at the hotel, and each student wants to download a positive number of bits. The load of a room is defined as the total number of bits to be downloaded from that room. Nobody likes slow internet, and in particular each student has a displeasure equal to the product of her number of bits and the load of her room. The misery of the group is defined as the sum of the students’ displeasures. Right after the contest, students gather in the hotel lobby to decide who goes to which room. After much discussion they reach a balanced configuration: one for which no student can decrease her displeasure by unilaterally moving to another room. The misery of the group is computed to be $M_1$, and right when they seemed satisfied, Gugu arrived with a serendipitous smile and proposed another configuration that achieved misery $M_2$. What is the maximum value of $M_1/M_2$ taken over all inputs to this problem? [i]Proposed by Victor Reis (proglote), Brazil.[/i]

Suppose $x,y,z$, and $w$ are positive reals such that \[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \] \[ xz + yw = 30. \] Find the largest possible value of $(xy + wz)^2$. [i]Author: Alex Zhu[/i]

Let $O$ be the center of the regular triangle $ABC$. Find the set of all points $M$ such that any line containing the point $M$ intersects one of the segments $AB, OC$.

Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of $$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$ where $x_{i+60}=x_i$.

Given $\triangle ABC$, let $R$ be its circumradius and $q$ be the perimeter of its excentral triangle. Prove that $q\le 6\sqrt{3} R$. Typesetter's Note: the excentral triangle has vertices which are the excenters of the original triangle.

[b]p1.[/b] Show that the $n \times n$ determinant $$\begin{vmatrix} 1+x & 1 & 1 & . & . & . & 1 \\ 1 & 1+x & 1 & . & . & . & 1 \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 1 & 1 & . & . & . & . & 1+x \\ \end{vmatrix}$$ has the value zero when $x = -n$ [b]p2.[/b] Let $c > a \ge b$ be the lengths of the sides of an obtuse triangle. Prove that $c^n = a^n + b^n$ for no positive integer $n$. [b]p3.[/b] Suppose that $p_1 = p_2^2+ p_3^2 + p_4^2$ , where $p_1$, $p_2$, $p_3$, and $p_4$ are primes. Prove that at least one of $p_2$, $p_3$, $p_4$ is equal to $3$. [b]p4.[/b] Suppose $X$ and $Y$ are points on tJhe boundary of the triangular region $ABC$ such that the segment $XY$ divides the region into two parts of equal area. If $XY$ is the shortest such segment and $AB = 5$, $BC = 4$, $AC = 3$ calculate the length of $XY$. Hint: Of all triangles having the same area and same vertex angle the one with the shortest base is isosceles. Clearly justify all claims. [b]p5.[/b] Find all solutions of the following system of simultaneous equations $$x + y + z = 7\,\, , \,\, x^2 + y^2 + z^2 = 31\,\,, \,\,x^3 + y^3 + z^3 = 154$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x_0=2024^{2024}$ and $x_{n+1}=|x_n-\pi|$ for $n \ge 0$. Show that there exists a value of $n$ such that $x_{n+2}=x_n$.

Calculate $ \left\lfloor \frac{(a^2+b^2+c^2)(a+b+c)}{a^3+b^3+c^3} \right\rfloor , $ where $ a,b,c $ are the lengths of the side of a triangle. [i]Costel Anghel[/i]

If $a,b,c$ are non-negative real numbers such that $a+b+c=3m,(m\ge1)$ then prove that $$(a^a+b^a+c^a)(a^b+b^b+c^b)(a^c+b^c+c^c)\ge27m^{3m}$$ [i]Proposed by Dorin Mărghidanu[/i]

Let $q>3$ be a rational number, such that $q^2-4$ is a perfect square of a rational number. The sequence $a_0, a_1, \ldots$ is defined by $a_0=2, a_1=q, a_{i+1}=qa_i-a_{i-1}$ for all $i \geq 1$. Is it true that there exist a positive integer $n$ and nonzero integers $b_0, b_1, \ldots, b_n$ with sum zero, such that if $\sum_{i=0}^{n} a_ib_i=\frac{A} {B}$ for $(A, B)=1$, then $A$ is squarefree?

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

A sequence is defined as follows $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}=6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum^{28}_{k=1} a_k$ is divided by 1000.

Prove that, given a positive integrer $n$, there exists a positive integrer $k_n$ with the following property: Given any $k_n$ points in the space, $4$ by $4$ non-coplanar, and associated integrer numbers between $1$ and $n$ to each sharp edge that meets $2$ of this points, there's necessairly a triangle determined by $3$ of them, whose sharp edges have associated the same number.

The bisectors of trapezoid's angles form a quadrilateral with perpendicular diagonals. Prove that this trapezoid is isosceles.