Let $ABC$ be a triangle and let $A' \in BC$, $B' \in CA$ and $C' \in AB$ be three collinear points. a) Prove that each pair of circles of diameters $AA'$, $BB'$ and $CC'$ has the same radical axis; b) Prove that the circumcenter of the triangle formed by the intersections of the lines $AA' , BB'$ and $CC'$ lies on the common radical axis found above.

Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that \[ \frac{1}{\sqrt{c}}\equal{}\frac{1}{\sqrt{a}}\plus{}\frac{1}{\sqrt{b}}\]

Consider the acute-angled triangle $ABC$, with orthocentre $H$ and circumcentre $O$. $D$ is the intersection point of lines $AH$ and $BC$ and $E$ lies on $\overline{AH}$ such that $AE=DH$. Suppose $EO$ and $BC$ meet at $F$. Prove that $BD=CF$. [i] (Călin Pop & Vlad Robu) [/i]

Let $S$ be a planar region. A $\emph{domino-tiling}$ of $S$ is a partition of $S$ into $1\times2$ rectangles. (For example, a $2\times3$ rectangle has exactly $3$ domino-tilings, as shown below.) [asy] import graph; size(7cm); pen dps = linewidth(0.7); defaultpen(dps); draw((0,0)--(3,0)--(3,2)--(0,2)--cycle, linewidth(2)); draw((4,0)--(4,2)--(7,2)--(7,0)--cycle, linewidth(2)); draw((8,0)--(8,2)--(11,2)--(11,0)--cycle, linewidth(2)); draw((1,0)--(1,2)); draw((2,1)--(3,1)); draw((0,1)--(2,1), linewidth(2)); draw((2,0)--(2,2), linewidth(2)); draw((4,1)--(7,1)); draw((5,0)--(5,2), linewidth(2)); draw((6,0)--(6,2), linewidth(2)); draw((8,1)--(9,1)); draw((10,0)--(10,2)); draw((9,0)--(9,2), linewidth(2)); draw((9,1)--(11,1), linewidth(2)); [/asy] The rectangles in the partition of $S$ are called $\emph{dominoes}$. (a) For any given positive integer $n$, find a region $S_n$ with area at most $2n$ that has exactly $n$ domino-tilings. (b) Find a region $T$ with area less than $50000$ that has exactly $100002013$ domino-tilings.

Rectangle $ ABCD$ has $ AB\equal{}8$ and $ BC\equal{}6$. Point $ M$ is the midpoint of diagonal $ \overline{AC}$, and E is on $ \overline{AB}$ with $ \overline{ME}\perp\overline{AC}$. What is the area of $ \triangle AME$? $ \textbf{(A)}\ \frac{65}{8} \qquad \textbf{(B)}\ \frac{25}{3} \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ \frac{75}{8} \qquad \textbf{(E)}\ \frac{85}{8}$

Let $f : (0,+\infty) \to \mathbb R$ be a function having the property that $f(x) = f\left(\frac{1}{x}\right)$ for all $x > 0.$ Prove that there exists a function $u : [1,+\infty) \to \mathbb R$ satisfying $u\left(\frac{x+\frac 1x }{2} \right) = f(x)$ for all $x > 0.$

Given that $a$ is a real solution to the polynomial equation $$nx^n-x^{n-1}-x^{n-2}-\cdots-x-1=0$$ where $n$ is a positive integer, show that $a=1$ or $-1<a<0$.

Consider a table with $n$ rows and $2n$ columns. we put some blocks in some of the cells. After putting blocks in the table we put a robot on a cell and it starts moving in one of the directions right, left, down or up. It can change the direction only when it reaches a block or border. Find the smallest number $m$ such that we can put $m$ blocks on the table and choose a starting point for the robot so it can visit all of the unblocked cells. (the robot can't enter the blocked cells.) Proposed by Seyed Mohammad Seyedjavadi and Alireza Tavakoli

The numbers $1- \sqrt{2}$, $\sqrt{2}$ and $1+\sqrt{2}$ are written on a blackboard. Every minute, if $x, y, z$ are the numbers written, then they are erased and the numbers, $x^2 + xy + y^2$, $y^2 + yz + z^2$ and $z^2 + zx + x^2$ are written. Determine whether it is possible for all written numbers to be rational numbers after a finite number of minutes.

6) In the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_1 = 400 g$, $M_2 = 200 g$, and $M_4 = 500 g$. What is the value of $M_3$ for the system to be in static equilibrium? A) 300 g B) 400 g C) 500 g D) 600 g E) 700 g

Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible? $ \textbf{(A)}\ 1930\qquad\textbf{(B)}\ 1931\qquad\textbf{(C)}\ 1932\qquad\textbf{(D)}\ 1933\qquad\textbf{(E)}\ 1934$

Let $f$ be a representation of the set $M = \{1, 2,..., 1988\}$ into $M$. For any natural $n$, let $x_1 = f(1)$, $x_{n+1} = f(x_n)$. Find out if there exists $m$ such that $x_{2m} = x_m$.

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

The incircle of $\vartriangle ABC$ is tangent to sides $\overline{BC}, \overline{AC}$, and $\overline{AB}$ at $D, E$, and $F$, respectively. Point $G$ is the intersection of lines $AC$ and $DF$ as shown. The sides of $\vartriangle ABC$ have lengths $AB = 73, BC = 123$, and $AC = 120$. Find the length $EG$. [img]https://cdn.artofproblemsolving.com/attachments/d/a/aede28071a1a6b94bbe3ad8e1e104822b89439.png[/img]

Solve in real numbers the equation: $ \frac{1}{x}\plus{}\frac{1}{x\plus{}2}\minus{}\frac{1}{x\plus{}4}\minus{}\frac{1}{x\plus{}6}\minus{}\frac{1}{x\plus{}8}\minus{}\frac{1}{x\plus{}10}\plus{}\frac{1}{x\plus{}12}\plus{}\frac{1}{x\plus{}14}\equal{}0.$

Find all functions $f:\mathbb{Z}\to \mathbb{Z}_{>0}$ for which \[f(x+f(y))^2+f(y+f(x))^2=f(f(x)+f(y))^2+1\] holds for any $x,y\in \mathbb{Z}$.

Let $X_1,X_2,..$ be independent random variables with the same distribution, and let $S_n=X_1+X_2+...+X_n, n=1,2,...$. For what real numbers $c$ is the following statement true: $$P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}$$

Let $n\ge  2$ be a positive integer and $p $ a prime such that $n|p-1$ and $p | n^3-1$. Show $ 4p-3$ is a square.

Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2. When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$.

On the sides $AC$ and $AB$ of the triangle $ABC$, the points $D$ and $E$ were chosen such that $\angle ABD =\angle CBD$ and $3 \angle ACE = 2\angle BCE$. Let $H$ be the point of intersection of $BD$ and $CE$, and $CD = DE = CH$. Find the angles of triangle $ABC$.

Let $n$ be a positive integer. Prove that the following two statements are equivalent. [list][*] $n$ is not divisible by $4$ [*] There exist $a, b \in \mathbb{Z}$ such that $a^{2}+b^{2}+1$ is divisible by $n$. [/list]

Let $N \geqslant 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating. [i]Proposed by Sutanay Bhattacharya and Anant Mudgal[/i]

Let $1\le k\le n$ be integers. At most how many $k$-element subsets can we select from $\{1,2,\dots,n\}$ such that for any two selected subsets, one of the subsets consists of the $k$ smallest elements of their union?

Let A$BC$ be a triangle with $AB = 5, AC = 4, BC = 6$. The internal angle bisector of $C$ intersects the side $AB$ at $D$. Points $M$ and $N$ are taken on sides $BC$ and $AC$, respectively, such that $DM\parallel AC$ and $DN \parallel BC$. If $(MN)^2 =\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers then what is the sum of the digits of $|p - q|$?

Problem 3 (MAR CP 1992) : From the digits $1,2,...,9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows : $123456789$, $123456798$, ..., $987654321$. What is the $100000th$ number ?