Triangle $ABC$ is inscribed in the circle $\Gamma$. Let $\Gamma_a$ denote the circle internally tangent to $\Gamma$ and also tangent to sides $AB$ and $AC$. Let $A'$ denote the point of tangency of $\Gamma$ and $\Gamma_a$. Define $B'$ and $C'$ similarly. Prove that $AA'$, $BB'$ and $CC'$ are concurrent.

Given a prime number $p$. It is known that for each integer $a$ such that $1<a<p/2$ there exist integer $b$ such that $p/2<b<p$ and $p|ab-1$. Find all such $p$.

A point in the cartesian plane with integer coordinates is called a lattice point. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold: (i) The endpoints of each selected segment are lattice points; (ii) Each selected segment is parallel to a coordinate axis or to one of the lines $y = \pm x$, (iii) Each selected segment contains exactly five lattice points, all of which are selected, (iv) Every two selected segments have at most one common point. A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and segments is a position. Prove or disprove that there exists an initial position such that the game can have infinitely many moves.

How many functions $f:\{1,2,3, \ldots, 7\} \rightarrow \{1,2,3, \ldots, 7\}$ are there such that the set $\mathcal{F} = \{f(i) : i\in\{1,\ldots, 7\}\}$ has cardinality four, while the set $\mathcal{G} = \{f(f(f(i))) : i\in\{1,\ldots, 7\}\}$ consists of a single element? [i]Proposed by Sam Delatore[/i]

Given $n$ positive real numbers $a_1, a_2, \ldots , a_n$ such that $a_1a_2 \cdots a_n = 1$, prove that \[(1 + a_1)(1 + a_2) \cdots (1 + a_n) \geq 2^n.\]

Points $K,L,M$ on the sides $AB,BC,CA$ respectively of a triangle $ABC$ satisfy $\frac{AK}{KB} = \frac{BL}{LC} = \frac{CM}{MA}$. Show that the triangles $ABC$ and $KLM$ have a common orthocenter if and only if $\triangle ABC$ is equilateral.

Bolek draw a trapezoid $ABCD$ trapezoid ($AB // CD$) on the board, with its midsegment line $EF$ in it. Point intersection of his diagonal $AC, BD$ denote by $P,$ and his rectangular projection on line $AB$ denote by $Q$. Lolek, wanting to tease Bolek, blotted from the board everything except segments $EF$ and $PQ$. When Bolek saw it, wanted to complete the drawing and draw the original trapezoid, but did not know how to do it. Can you help Bolek?

A wooden block (mass $M$) is hung from a peg by a massless rope. A speeding bullet (with mass $m$ and initial speed $v_\text{0}$) collides with the block at time $t = \text{0}$ and embeds in it. Let $S$ be the system consisting of the block and bullet. Which quantities are conserved between $t = -\text{10 s}$ and $ t = \text{+10 s}$? [asy] // Code by riben draw(circle((0,0),0.3),linewidth(2)); filldraw(circle((0,0),0.3),gray); draw((0,-0.8)--(0,-15.5),linewidth(2)); draw((5,-15.5)--(-5,-15.5)--(-5,-20.5)--(5,-20.5)--cycle,linewidth(2)); filldraw((5,-15.5)--(-5,-15.5)--(-5,-20.5)--(5,-20.5)--cycle,gray); draw((-15,-18)--(-16,-17)--(-18,-17)--(-18,-19)--(-16,-19)--cycle,linewidth(2)); filldraw((-15,-18)--(-16,-17)--(-18,-17)--(-18,-19)--(-16,-19)--cycle,gray); [/asy] (A) The total linear momentum of $S$. (B) The horizontal component of the linear momentum of $S$. (C) The mechanical energy of $S$. (D) The angular momentum of $S$ as measured about a perpendicular axis through the peg. (E) None of the above are conserved.

Given $\triangle ABC$ inscribed in $(O)$ an let $I$ and $I_a$ be it's incenter and $A$-excenter ,respectively. Tangent lines to $(O)$ at $C,B$ intersect the angle bisector of $A$ at $M,N$ ,respectively. Second tangent lines through $M,N$ intersect $(O)$ at $X,Y$. Prove that $XYII_a$ is cyclic.

Beatrix is going to place six rooks on a $6\times6$ chessboard where both the rows and columns are labelled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The [i]value[/i] of a square is the sum of its row number and column number. The [i]score[/i] of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let angle bisectors of angles $\angle BAC$ and $\angle ABC$ of triangle $ABC$ intersect sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let points $F$ and $G$ be foots of perpendiculars from point $C$ on lines $AD$ and $BE$, respectively. Prove that $FG \mid \mid AB$

Determine the minimum value of $$(r-1)^2 + \left(\frac{s}{r}-1 \right)^2 + \left(\frac{t}{s}-1 \right)^{2} + \left( \frac{4}{t} -1 \right)^2$$ for all real numbers $1\leq r \leq s \leq t \leq 4.$

How many pairs of integers $(a, b)$ satisfy $1 \le a < 1001^3$, $1 \le b < 1001^2$, and $1001^3 \mid a^3 + ab$?

$10$ points are drawn on each of two parallel lines. What is the largest number of acute triangles of positive area that can be formed using three of these $20$ points as vertices?

How many six-letter words formed from the letters of AMC do not contain the substring AMC? (For example, AMAMMC has this property, but AAMCCC does not.)

Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.

A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?

Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$. [i]Proposed by Anthony Yang and Andy Xu[/i]

Find all the pairs of positive numbers such that the last digit of their sum is 3, their difference is a primer number and their product is a perfect square.

How many members are there of the set $\{-79,-76,-73,\dots,98,101\}?$

How many functions $f: \left\{1,2,3\right\} \to \left\{1,2,3 \right\}$ satisfy $f(f(x))=f(f(f(x)))$ for every $ x $?

An "n-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\cdots,k,\cdots,n$, $n\geq 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong the $n$ pairs of sides numbered $k$ and $k+2$ until they meet. (A figure is shown for the case $n=5$) [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=704&sid=8da93909c5939e037aa99c429b2d157a[/img] Let $S$ be the degree-sum of the interior angles at the $n$ points of the star; then $S$ equals: $\text{(A)} \ 180 \qquad \text{(B)} \ 360 \qquad \text{(C)} \ 180(n+2) \qquad \text{(D)} \ 180(n-2) \qquad \text{(E)} \ 180(n-4)$

In one side of a hall there are $2N$ rooms numbered from 1 to $2N$. In each room $i$ between 1 and $N$ there are $p_i$ beds. Is needed to move every one of this beds to the roms from $N+ 1$ to $2N$, in such a way that for every $j$ between $N+1$ and $2N$ the room $j$ will have $p_j$ beds. Supose that each bed can be move once and the price of moving a bed from room $i$ to room $j$ is $(i-j)^2$. Find a way to move every bed such that the total cost is minimize. Note: The numbers $p_i$ are given and satisfy that $p_1 + p_2 + \cdots + p_N = p_{N+1} + p_{N+2} + \cdots+ p_{2N}.$

Let $C$ be a circle centered at point $O$, and let $P$ be a point in the interior of $C$. Let $Q$ be a point on the circumference of $C$ such that $PQ \perp OP$, and let $D$ be the circle with diameter $PQ$. Consider a circle tangent to $C$ whose circumference passes through point $P$. Let the curve $\Gamma$ be the locus of the centers of all such circles. If the area enclosed by $\Gamma$ is $1/100$ the area of $C$, then what is the ratio of the area of $C$ to the area of $D$?

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]