Let $ABC$ be a triangle with incenter $I$ such that $AB=20$ and $AC=19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA=IP=IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Ankit Bisain[/i]

[b]i.)[/b] Let $ABC$ be a triangle with $AB = 12$ and $AC = 16.$ Suppose $M$ is the midpoint of side $BC$ and points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, and suppose that lines $EF$ and $AM$ intersect at $G.$ If $AE = 2 \cdot AF$ then find the ratio \[ \frac{EG}{GF} \] [b]ii.)[/b] Let $E$ be a point external to a circle and suppose that two chords $EAB$ and $EDC$ meet at angle of $40^{\circ}.$ If $AB = BC = CD$ find the size of angle $ACD.$

Given is an acute triangle $ABC$. $M$ is an arbitrary point at the side $AB$ and $N$ is midpoint of $AC$. The foots of the perpendiculars from $A$ to $MC$ and $MN$ are points $P$ and $Q$. Prove that center of the circumcircle of triangle $PQN$ lies on the fixed line for all points $M$ from the side $AB$.

Let $ b$ be an even positive integer for which there exists a natural number n such that $ n>1$ and $ \frac{b^n\minus{}1}{b\minus{}1}$ is a perfect square. Prove that $ b$ is divisible by 8.

For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$. What is the minimum possible value of $M(S)/m(S)$ ?

A right-angled triangle $ABC$ is given. Its catheus $AB$ is the base of a regular triangle $ADB$ lying in the exterior of $ABC$, and its hypotenuse $AC$ is the base of a regular triangle $AEC$ lying in the interior of $ABC$. Lines $DE$ and $AB$ meet at point $M$. The whole configuration except points $A$ and $B$ was erased. Restore the point $M$.

Find all pairs $(m, n)$ of positive integers $m$ and $n$ for which one has $$\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}$$

This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant $X$, let $t(X)$ be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$ For any two friends $X'$ and $X''$, we have $t(X') \neq t(X''),$ $2)$ For every contestant $X$, the set $\{ t(Y) : Y \text{ is a friend of } X \}$ consists of consecutive integers. The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?

Which of the following sets of $ x$-values satisfy the inequality $ 2x^2 \plus{} x < 6?$ $ \textbf{(A)}\ \minus{} 2 < x < \frac{3}{2} \qquad \textbf{(B)}\ x > \frac32 \text{ or }x < \minus{} 2 \qquad \textbf{(C)}\ x < \frac32 \qquad \textbf{(D)}\ \frac32 < x < 2 \qquad \textbf{(E)}\ x < \minus{} 2$

Goldfish are sold at $ 15$ cents each. The rectangular coordinate graph showing the cost of $ 1$ to $ 12$ goldfish is: $ \textbf{(A)}\ \text{a straight line segment} \qquad \\ \textbf{(B)}\ \text{a set of horizontal parallel line segments}\qquad \\ \textbf{(C)}\ \text{a set of vertical parallel line segments}\qquad \\ \textbf{(D)}\ \text{a finite set of distinct points}\qquad \textbf{(E)}\ \text{a straight line}$

Find all prime number $p$, such that there exist an infinite number of positive integer $n$ satisfying the following condition: $p|n^{ n+1}+(n+1)^n.$ (September 29, 2012, Hohhot)

Let $f:[0,1]\rightarrow [0,1]$ a continuous function in $0$ and in $1$, which has one-side limits in any point and $f(x-0)\le f(x)\le f(x+0),\ (\forall)x\in (0,1)$. Prove that: a)for the set $A=\{x\in [0,1]\ |\ f(x)\ge x\}$, we have $\sup A\in A$. b)there is $x_0\in [0,1]$ such that $f(x_0)=x_0$. [i]Mihai Piticari[/i]

Let $n,s,$ and $t$ be positive integers and $0<\lambda<1.$ A simple graph on $n$ vertices with at least $\lambda n^2$ edges is given. We say that $(x_1,\ldots,x_s,y_1,\ldots,y_t)$ is a [i]good intersection[/i] if letters $x_i$ and $y_j$ denote not necessarily distinct vertices and every $x_iy_j$ is an edge of the graph $(1\leq i\leq s,$ $1\leq j\leq t).$ Prove that the number of good insertions is at least $\lambda^{st}n^{s+t}.$ [i]Proposed by Kada Williams, Cambridge[/i]

If $ n$ is a real number, then the simultaneous system $ nx \plus{} y \equal{} 1$ $ ny \plus{} z \equal{} 1$ $ x \plus{} nz \equal{} 1$ has no solution if and only if $ n$ is equal to $ \textbf{(A)}\ \minus{}1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \text{ or } 1 \qquad \textbf{(E)}\ \frac12$

Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer).

Find all $f(x)\in \mathbb Z (x)$ that satisfies the following condition, with the lowest degree. [b]Condition[/b]: There exists $g(x),h(x)\in \mathbb Z (x)$ such that $$f(x)^4+2f(x)+2=(x^4+2x^2+2)g(x)+3h(x)$$.

At Andover, $35\%$ of students are lowerclassmen and the rest are upperclassmen. Given that $26\%$ of lowerclassmen and $6\%$ of upperclassmen take Latin, what percentage of all students take Latin? [i]Proposed by Anthony Yang[/i]

Calculate the area of the intersection of the interior of the ellipse $\frac{x^2}{16}+ \frac{y^2}{4}= 1$ with the circle bounded by the circumference $(x -2)^2 + (y - 1)^2 = 5$.

A set consisting of $n$ points of a plane is called an isosceles $n$-point if any three of its points are located in vertices of an isosceles triangle. Find all natural numbers for which there exist isosceles $n$-points.

Investigate whether a chess knight can traverse a $4 \times 4$ mini-chessboard so that it reaches each of the $16$ squares only once. Note: the drawing below shows the endpoints of the eight possible moves of the knight $(C)$ on a chessboard of size $8 \times 8$. [asy] unitsize(0.4 cm); int i; fill(shift((2,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((4,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((1,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((5,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((1,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((5,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((2,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); fill(shift((4,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7)); for (i = 0; i <= 8; ++i) { draw((i,0)--(i,8)); draw((0,i)--(8,i)); } label("C", (3.5,4.5), fontsize(8)); [/asy]

A board $n$×$n$ is coloured black and white like a chessboard. The following steps are permitted: Choose a rectangle inside the board (consisting of entire cells)whose side lengths are both odd or both even, but not both equal to $1$, and invert the colours of all cells inside the rectangle. Determine the values of $n$ for which it is possible to make all the cells have the same colour in a finite number of such steps.

Ali and Shayan are playing a turn-based game on an infinite grid. Initially, all cells are white. Ali starts the game, and in the first turn, he colors one unit square black. In the following turns, each player must color a white square that shares at least one side with a black square. The game continues for exactly 2808 turns, after which each player has made 1404 moves. Let $A$ be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape $A$ by playing optimally. (The perimeter of shape $A$ is defined as the total length of the boundary segments between a black and a white cell.) What are the possible values of the perimeter of $A$, assuming both players play optimally?

[b]rolling cube[/b] $a$,$b$ and $c$ are natural numbers. we have a $(2a+1)\times (2b+1)\times (2c+1)$ cube. this cube is on an infinite plane with unit squares. you call roll the cube to every side you want. faces of the cube are divided to unit squares and the square in the middle of each face is coloured (it means that if this square goes on a square of the plane, then that square will be coloured.) prove that if any two of lengths of sides of the cube are relatively prime, then we can colour every square in plane. time allowed for this question was 1 hour.

Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$. If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.

Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$, incenter $I$ and $(I)$ tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. Suppose that $EF$ cuts $(O)$ at $P, Q$. Prove that $(PQD)$ bisects segment $BC$.