A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$. [i]Albania[/i]

Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

What fraction of the square is shaded? [asy] draw((0,0)--(0,3)--(3,3)--(3,0)--cycle); draw((0,2)--(2,2)--(2,0)); draw((0,1)--(1,1)--(1,0)); draw((0,0)--(3,3)); fill((0,0)--(0,1)--(1,1)--cycle,grey); fill((1,0)--(1,1)--(2,2)--(2,0)--cycle,grey); fill((0,2)--(2,2)--(3,3)--(0,3)--cycle,grey);[/asy] $ \text{(A)}\ \frac{1}{3}\qquad\text{(B)}\ \frac{2}{5}\qquad\text{(C)}\ \frac{5}{12}\qquad\text{(D)}\ \frac{3}{7}\qquad\text{(E)}\ \frac{1}{2} $

Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$

A regular $15$-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$? $\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }32\qquad\textbf{(D) }39\qquad\textbf{(E) }54$

Can a figure have a greater than $1$ and finite number of centers of symmetry?

Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|AB|+|BD|=|AC|$ and $m(\widehat{BAD})=m(\widehat{DAC})=30^\circ$. What is $m(\widehat{ACB})$? $ \textbf{(A)}\ 30^\circ \qquad\textbf{(B)}\ 40^\circ \qquad\textbf{(C)}\ 45^\circ \qquad\textbf{(D)}\ 48^\circ \qquad\textbf{(E)}\ 50^\circ $

Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent. [i]Proposed by Eugene Bilopitov, Ukraine[/i]

For all natural $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{th}$ row, such that all the left-most squares in each row are aligned vertically. Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$. (a) Find all $n$ such that $f(n)=n$. (b) Find all $n$ such that $f(n) = n+1$.

Which of the following has the greatest number of line of symmetry? $ \textbf{(A)}\ \text{ Equilateral Triangle}$ $\textbf{(B)}\ \text{Non-square rhombus} $ $\textbf{(C)}\ \text{Non-square rectangle}$ $\textbf{(D)}\ \text{Isosceles Triangle}$ $\textbf{(E)}\ \text{Square} $

In the coordinate space, consider the cubic with vertices $ O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0),\ C(0,\ 1,\ 0),\ D(0,\ 0,\ 1),\ E(1,\ 0,\ 1),\ F(1,\ 1,\ 1),\ G(0,\ 1,\ 1)$. Find the volume of the solid generated by revolution of the cubic around the diagonal $ OF$ as the axis of rotation.

Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$. Prove that the line $AI$, the line perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.

A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region? $\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/3 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 2/3 \qquad \text{(E)}\ 3/4$

Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?

Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$.

Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$.

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$. a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$. b) Give an example of a non-constant function $f$ with property $(P)$. c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.

For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows: - $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$; - $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$. Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.

Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true: i. Either each of the three cards has a different shape or all three of the card have the same shape. ii. Either each of the three cards has a different color or all three of the cards have the same color. iii. Either each of the three cards has a different shade or all three of the cards have the same shade. How many different complementary three-card sets are there?

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

A word of length $n$ is an ordered sequence $x_1x_2\ldots x_n$ where $x_i$ is a letter from the set $\{ a,b,c \}$. Denote by $A_n$ the set of words of length $n$ which do not contain any block $x_ix_{i+1}, i=1,2,\ldots ,n-1,$ of the form $aa$ or $bb$ and by $B_n$ the set of words of length $n$ in which none of the subsequences $x_ix_{i+1}x_{i+2}, i=1,2,\ldots n-2,$ contains all the letters $a,b,c$. Prove that $|B_{n+1}|=3|A_n|$. [i]Vasile Pop[/i]

A hexagon inscribed in a circle has three consecutive sides each of length $3$ and three consecutive sides each of length $5$. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length $3$ and the other with three sides each of length $5$, has length equal to $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. $\text{(A)}\ 309 \qquad \text{(B)}\ 349 \qquad \text{(C)}\ 369 \qquad \text{(D)}\ 389\qquad \text{(E)}\ 409$

$ABCD$ is a square. Describe the locus of points $P$, different from $A, B, C, D$, on that plane for which \[\widehat{APB}+\widehat{CPD}=180^\circ\]

The game of circulate is played with a deck of $kn$ cards each with a number in $1,2,\ldots,n$ such that there are $k$ cards with each number. First, $n$ piles numbered $1,2,\ldots,n$ of $k$ cards each are dealt out face down. The player then flips over a card from pile $1$, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the $k$ cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game? [i]Brian Hamrick.[/i]