$\odot A$, centered at point $A$, has radius $14$ and $\odot B$, centered at point $B$, has radius $15$. $AB = 13$. The circles intersect at points $C$ and $D$. Let $E$ be a point on $\odot A$, and $F$ be the point where line $EC$ intersects $\odot B$, again. Let the midpoints of $DE$ and $DF$ be $M$ and $N$, respectively. Lines $AM$ and $BN$ intersect at point $G$. If point $E$ is allowed to move freely on $\odot A$, what is the radius of the locus of $G$?

Prove that every positive integer can be represented in the form \[3^{u_1} \ldots 2^{v_1} + 3^{u_2} \ldots 2^{v_2} + \ldots + 3^{u_k} \ldots 2^{v_k}\] with integers $u_1, u_2, \ldots , u_k, v_1, \ldots, v_k$ such that $u_1 > u_2 >\ldots > u_k\ge 0$ and $0 \le v_1 < v_2 <\ldots < v_k$.

Given a polynomial of degree $2$, $p(x) = ax^2 +bx+c$ define the function $$S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.$$ Determine the real number$ r$such that, for any polynomial $p(x)$ of degree $2$ with real roots, holds $S(p) \ge ra^2$

Let $ABC$ be an acute triangle and let $M$, $N$, and $ P$ be on $CB$, $AC$, and $AB$, respectively, such that $AB = AN + PB$, $BC = BP + MC$, $CA = CM + AN$. Let $\ell$ be a line in a different half plane than $C$ with respect to to the line $AB$ such that if $X, Y$ are the projections of $A, B$ on $\ell$ respectively, $AX = AP$ and $BY = BP$. Prove that $NXYM$ is a cyclic quadrilateral.

Compute$$\int \limits_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{(1+\ln x)\cos x+x\sin x\ln x}{\cos^2 x + x^2 \ln^2 x}dx$$

Kostya marked the points $A(0, 1), B(1, 0), C(0, 0)$ in the coordinate plane. On the legs of the triangle ABC he marked the points with coordinates $(\frac{1}{2},0), (\frac{1}{3},0), \cdots, (\frac{1}{n+1},0)$ and $(0,\frac{1}{2}), (0,\frac{1}{3}), \cdots, (0,\frac{1}{n+1}).$ Then Kostya joined each pair of marked points with a segment. Sasha drew a $1 \times n$ rectangle and joined with a segment each pair of integer points on its border. As a result both the triangle and the rectangle are divided into polygons by the segments drawn. Who has the greater number of polygons: Sasha or Kostya? [i](M. Alekseyev )[/i]

Let \[ S = \sum_{i = 1}^{2012} i!. \] The tens and units digits of $S$ (in decimal notation) are $a$ and $b$, respectively. Compute $10a + b$. [i]Proposed by Lewis Chen[/i]

Prove that the number $$\sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}}$$ is rational for all natural $n$.

A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$. a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?

There is a board with the shape of an equilateral triangle with side $n$ divided into triangular cells with the shape of equilateral triangles with side $ 1$ (the figure below shows the board when $n = 4$). Each and every triangular cell is colored either red or blue. What is the least number of cells that can be colored blue without two red cells sharing one side? [img]https://cdn.artofproblemsolving.com/attachments/0/1/d1f034258966b319dc87297bdb311f134497b5.png[/img]

From time $t = 0$ to time $t = 1$ a population increased by $i\%$, and from time $t = 1$ to time $t = 2$ the population increased by $j\%$. Therefore, from time $t = 0$ to time $t = 2$ the population increased by $ \textbf{(A)}\ (i + j)\%\qquad\textbf{(B)}\ ij\%\qquad\textbf{(C)}\ (i+ij)\%\qquad\textbf{(D)}\ \left(i + j + \frac{ij}{100}\right)\%\qquad\textbf{(E)}\left( i + j + \frac{i + j}{100}\right)\% $

For function $ f: \mathbb{R} \to \mathbb{R}$ given that $ f(x^2 +x +3) +2 \cdot f(x^2 - 3x + 5) = 6x^2 - 10x +17$, calculate $ f(2009)$.

For a positive integer $n$, let $S(n) $be the set of complex numbers $z = x+iy$ ($x,y \in R$) with $ |z| = 1$ satisfying $(x+iy)^n+(x-iy)^n = 2x^n$ . (a) Determine $S(n)$ for $n = 2,3,4$. (b) Find an upper bound (depending on $n$) of the number of elements of $S(n)$ for $n > 5$.

Without using a calculator, determine which number is greater: $17^{24}$ or $31^{19}$

Solve in the real numbers the equation $ \sqrt[5]{2^x-2^{-1}} -\sqrt[5]{2^x+2^{-1}} =-1. $ [i]Mihai Neagu[/i]

Prove that for any triangle each exradius is less than four times the circumradius.

In a convex quadrilateral $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. If $MP$ and $NQ$ divide $ABCD$ in four quadrilaterals with the same area, prove that $ABCD$ is a parallelogram.

Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?

A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying $$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$ then the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$ (a) Prove that $M=20-\frac{1}{20}$ is not $strong$. (b) Prove that $M=20-\frac{1}{21}$ is $strong$.

Find the positive integer $k$ such that the roots of $x^3 - 15x^2 + kx -1105$ are three distinct collinear points in the complex plane.

(a) On a long pavement, a sequence of $999$ integers is written in chalk. The numbers need not be in increasing order and need not be distinct. Merlijn encircles $500$ of the numbers with red chalk. From left to right, the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$. Next, Jeroen encircles $500$ of the numbers with blue chalk. From left to right, the numbers circled in blue are precisely the numbers $500, 499, 498, ...,2,1$. Prove that the middle number in the sequence of $999$ numbers is circled both in red and in blue. (b) Merlijn and Jeroen cross the street and find another sequence of $999$ integers on the pavement. Again Merlijn circles $500$ of the numbers with red chalk. Again the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$ from left to right. Now Jeroen circles $500$ of the numbers, not necessarily the same as Merlijn, with green chalk. The numbers circled in green are also precisely the numbers $1, 2, 3, ...,499, 500$ from left to right. Prove: there is a number that is circled both in red and in green that is not the middle number of the sequence of $999$ numbers.

A 16 ×16 square sheet of paper is folded once in half horizontally and once in half vertically to make an 8 × 8 square. This square is again folded in half twice to make a 4 × 4 square. This square is folded in half twice to make a 2 × 2 square. This square is folded in half twice to make a 1 × 1 square. Finally, a scissor is used to make cuts through both diagonals of all the layers of the 1 × 1 square. How many pieces of paper result?

Suppose $P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$ where $a_0, a_1, \ldots, a_{n-1}$ are reals for $n\geq 1$ (monic $n$th-degree polynomial with real coefficients). If the inequality \[ 3(P(x)+P(y)) \geq P(x+y) \] holds for all reals $x,y$, determine the minimum possible value of $P(2024)$.

The trapeze $ABCD$ with bases $AB$ and $CD$ is inscribed in a circle $\Gamma$. Let $X$ be a variable point of the arc $\overarc{AB}$ that does not contain either $C$ or $D$. Let $Y$ be the point of intersection of $AB$ and $DX$, and let $Z$ be the point of the segment $CX$ such that $\frac{XZ}{XC}=\frac{AY}{AB}$. Prove that the measure of the angle $\angle AZX$ does not depend on the choice of $X$.

If the graph of $x^2+y^2=m$ is tangent to that of $x+y=\sqrt{2m}$, then: $\textbf{(A) }m\text{ must equal }\tfrac12\qquad \textbf{(B) }m\text{ must equal }\tfrac1{\sqrt2}\qquad$ $\textbf{(C) }m\text{ must equal }\sqrt2\qquad \textbf{(D) }m\text{ must equal }2\qquad$ $\textbf{(E) }m\text{ may be any nonnegative real number}$