Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.

Let $ ABC$ be a triangle, and let the angle bisectors of its angles $ CAB$ and $ ABC$ meet the sides $ BC$ and $ CA$ at the points $ D$ and $ F$, respectively. The lines $ AD$ and $ BF$ meet the line through the point $ C$ parallel to $ AB$ at the points $ E$ and $ G$ respectively, and we have $ FG \equal{} DE$. Prove that $ CA \equal{} CB$. [i]Original formulation:[/i] Let $ ABC$ be a triangle and $ L$ the line through $ C$ parallel to the side $ AB.$ Let the internal bisector of the angle at $ A$ meet the side $ BC$ at $ D$ and the line $ L$ at $ E$ and let the internal bisector of the angle at $ B$ meet the side $ AC$ at $ F$ and the line $ L$ at $ G.$ If $ GF \equal{} DE,$ prove that $ AC \equal{} BC.$

The $n$-factorial of a positive integer $x$ is the product of all positive integers less than or equal to $z$ that are congruent to $z$ modulo $n$. For example, for the number 16, its 2-factorial is $16 \times 14 \times 12 \times 10 \times 8 \times 6 \times 4 \times 2$, its 3-factorial is $16 \times 13 \times 10 \times 7 \times 4 \times 1$ and its 18-factorial is 16. A positive integer is called [i]olympic[/i] if it has $n$ digits, all different than zero, and if it is equal to the sum of the $n$-factorials of its digits. Find all positive olympic integers.

Let $A$ be a non-empty subset of the set of all positive integers $N^*$. If any sufficient big positive integer can be expressed as the sum of $2$ elements in $A$(The two integers do not have to be different), then we call that $A$ is a divalent radical. For $x \geq 1$, let $A(x)$ be the set of all elements in $A$ that do not exceed $x$, prove that there exist a divalent radical $A$ and a constant number $C$ so that for every $x \geq 1$, there is always $\left| A(x) \right| \leq C \sqrt{x}$.

Let $c$ be the least common multiple of positive integers $a$ and $b$, and $d$ be the greatest common divisor of $a$ and $b$. How many pairs of positive integers $(a,b)$ are there such that \[ \dfrac {1}{a} + \dfrac {1}{b} + \dfrac {1}{c} + \dfrac {1}{d} = 1? \] $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 2 $

Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that \[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\] for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that \[a_{k+2011}=a_{k+2011^{2011}}.\]

Ephramis taking his final exams. He has $7$ exams and his school holds finals over $3$ days. For a certain arrangement of finals, let $f$ be the maximum number of finals Ephram takes on any given day. Find the expected value of $f$ .

Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race in miles? [asy] size(10cm); filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray); draw((11,11)--(171,11),linetype("4 4")+white+linewidth(1.5)); draw((0,0)--(11,0)--(11,22)--(0,22)--cycle,linewidth(1.125)); draw((171,0)--(182,0)--(182,22)--(171,22)--cycle,linewidth(1.125)); draw((31,4.5)--(31,0)); draw((51,4.5)--(51,0)); draw((151,4.5)--(151,0)); label(scale(.9)*rotate(45)*"Water 1", (23,-13.5)); label(scale(.9)*rotate(45)*"Water 2", (43,-13.5)); label(scale(.9)*rotate(45)*"Water 7", (143,-13.5)); filldraw(circle((101,-13.5),.3)); filldraw(circle((97,-13.5),.3)); filldraw(circle((93,-13.5),.3)); filldraw(circle((89,-13.5),.3)); filldraw(circle((85,-13.5),.3)); label(scale(.9)*rotate(90)*"Start", (5.5,11)); label(scale(.9)*rotate(270)*"Finish", (176.5,11)); [/asy] $\textbf{(A) } 8\qquad\textbf{(B) } 16\qquad\textbf{(C) } 24\qquad\textbf{(D) } 48\qquad\textbf{(E) } 96$

Do there exist (a) four (b) five distinct positive integers such that the sum of any three of them is a prime number? (V Senderov)

Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the tangents to $\omega$ through $B,C$ meet each other at point $P$. Prove that the perpendicular bisector of $AB$ and the parallel to $AB$ through $P$ meet at line $AC$.

Find all pairs of positive integers $(m, n)$ such that $$m^n * n^m = m^m + n^n$$

A tetrahedron is inscribed in a sphere of radius $1$ such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.

In a tetrahedron, lengths of six edges are $2,3,3,4,5,5$. Find its largest volume.

Let $p$ be an odd prime number. Ana and Ben are playing a game with alternate moves as follows: in each move, the player which has the turn choose a number, which was not choosen before by any of the player, from the set $\{1,2,...,2p-3,2p-2\}$. This process continues until no number is left. After the end of the process, each player create the number by taking the product of the choosen numbers and then add 1. We say a player wins if the number that did create is divisible by $p$, while the number that did create the opponent it is not divisible by $p$, otherwise we say the game end in a draw. Ana start first move. Does it exist a strategy for any of the player to win the game? [i]Proposed by Dorlir Ahmeti, Kosovo[/i]

Through the vertices $A, B$ of the parallelogram $ABCD$ passes a circle that intersects for the second time diagonals $BD$ and $AC$ at points $X$ and $Y$, respectively. The circumsccribed circle of $\vartriangle ADX$ intersects diagonal $AC$ for the second time at the point $Z$. Prove that $AY = CZ$.

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. [list=i] [*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. [*] If no such pair exists, we write two times the number $0$. [/list] Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times. Proposed by [I]Serbia[/I].

Find all triplets $ (a,b,c) $ of nonzero complex numbers having the same absolute value and which verify the equality: $$ \frac{a}{b} +\frac{b}{c}+\frac{c}{a} =-1 $$

Two points $A$ and $B$ are given on the plane. An arbitrary circle passes through $B$ and intersects the straight line $AB$ for second time at a point $K$, different from $A$. A circle passing through $A$, $K$ and the center of the first circle intersects the first one for second time at point $M$. Find the locus of points $M$.

Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$. [i]Proposed by Max Jiang[/i]

Prove: There exists a positive integer $n$ with $2021$ prime divisors, satisfying $n|2^n+1$.

A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in $2007$. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in $2007$. A set of plates in which each possible sequence appears exactly once contains $N$ license plates. Find $\frac{N}{10}$.

For a positive integer $n$, denote by $\tau (n)$ the number of its positive divisors. For a positive integer $n$, if $\tau (m) < \tau (n)$ for all $m < n$, we call $n$ a good number. Prove that for any positive integer $k$, there are only finitely many good numbers not divisible by $k$.

Let $f:[0,\infty)\to\mathbb R$ be a continuous function s.t. $\lim_{x\to\infty}\frac {f(x)}x=0$. Let $(x_n)_n$ be a sequence of positive real numbers s.t. $\left(\frac{x_n}n\right)_n$ is bounded. Prove that $\lim_{n\to\infty}\frac{f(x_n)}n=0$. [i]Dorin Andrica, Eugen Paltanea[/i]

A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is: $ \textbf{(A)}\ \frac{1}{1} \qquad \textbf{(B)}\ \frac{1}{2} \qquad \textbf{(C)}\ \frac{2}{3} \qquad \textbf{(D)}\ \frac{2}{1} \qquad \textbf{(E)}\ \sqrt{\frac{5}{4}}$

Let $a, b, c$ be nonzero real numbers such that $a + b + c = 0$. Determine the maximum possible value of $\frac{a^2b^2c^2}{ (a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}$ .