The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

Let $VA_1A_2\ldots A_n$ be a pyramid, where $n\ge 4$. A plane $\Pi$ intersects the edges $VA_1,VA_2,\ldots, VA_n$ at the points $B_1,B_2,\ldots,B_n$ respectively such that the polygons $A_1A_2\ldots A_n$ and $B_1B_2\ldots B_n$ are similar. Prove that the plane $\Pi$ is parallel to the plane containing the base $A_1A_2\ldots A_n$. [i]Laurentiu Panaitopol[/i]

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

A table $10 \times 10$ was filled according to the rules of the game “Bomb Squad”: several cells contain bombs (one bomb per cell) while each of the remaining cells contains a number, equal to the number of bombs in all cells adjacent to it by side or by vertex. Then the table is rearranged in the “reverse” order: bombs are placed in all cells previously occupied with numbers and the remaining cells are filled with numbers according to the same rule. Can it happen that the total sum of the numbers in the table will increase in a result?

On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected, respectively, so that $PN = NC$, the point $Q$ Is a point on the segment $AN$ for which $\angle NCB = \angle QPN$. Prove that $\angle BCQ = \tfrac {1} {2} \angle PQA$.

Four points $ A, B, C, D$ are chosen on a circle in such a way that arcs $ AB, BC,$ and $ CD$ are of the same length and the $ arc DA$ is longer than these three. Line $ AD$ and the line tangent to the circle at $ B$ intersect at $ E$. Let $ F$ be the other endpoint of the diameter starting at $ C$ of the circle. Prove that triangle $ DEF$ is equilateral.

The numbers $2, 2, ..., 2$ are written on a blackboard (the number $2$ is repeated $n$ times). One step consists of choosing two numbers from the blackboard, denoting them as $a$ and $b$, and replacing them with $\sqrt{\frac{ab + 1}{2}}$. $(a)$ If $x$ is the number left on the blackboard after $n - 1$ applications of the above operation, prove that $x \ge \sqrt{\frac{n + 3}{n}}$. $(b)$ Prove that there are infinitely many numbers for which the equality holds and infinitely many for which the inequality is strict.

For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$ $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Let $ABC$ be an equilateral triangle with $AB=1.$ Let $M$ be the midpoint of $BC,$ and let $P$ be on segment $AM$ such that $AM/MP=4.$ Find $BP.$

Solve the equation $$x(2^{1-2x}-1)=2^{x-2x^2}-1$$

On a $5 \times 5$ board, $n$ white markers are positioned, each marker in a distinct $1 \times 1$ square. A smart child got an assignment to recolor in black as many markers as possible, in the following manner: a white marker is taken from the board, it is colored in black, and then put back on the board on an empty square such that none of the neighboring squares contains a white marker (two squares are called neighboring if they share a common side). If it is possible for the child to succeed in coloring all the markers black, we say that the initial positioning of the markers was [i]good[/i]. a) Prove that if $n = 20$, then a [i]good [/i] initial positioning exists. b) Prove that if $n = 21$, then a [i]good [/i] initial positioning does not exist.

We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$. Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively intersect at a point $X$, and the lines $AR,BS,CT$ intersect at a point $Y$, such that $O,X,Y$ are collinear. [i]Proposed by Sammy Luo[/i]

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

Determine all finite nonempty sets $S$ of positive integers satisfying \[ {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, \] where $(i,j)$ is the greatest common divisor of $i$ and $j$.

Let $S$ be a subset with $673$ elements of the set $\{1,2,\ldots ,2010\}$. Prove that one can find two distinct elements of $S$, say $a$ and $b$, such that $6$ divides $a+b$.

In the acute-angled triangle $ABC$, let $AD$, $D \in BC$ be the $A$-angle bisector. The perpenducular to $BC$ through $D$ and the perpendicular to $AD$ through $A$ meet at $I$. The circle with center $I$ and radius $ID$, intersects $AB$ and $AC$ at $F$ and $E$ respectively. On the arc $FE$, which does not contain $A$, of the circumcircle of $AFE$, consider a point $X$, such that $\frac{XF}{XE}=\frac{AF}{AE}$. Prove that the circumcircles of triangles $AFE$ and $BXC$ are tangent.

Suppose $ABCD$ is a cyclic quadrilateral with $BC = CD$. Let $\omega$ be the circle with center $C$ tangential to the side $BD$. Let $I$ be the centre of the incircle of triangle $ABD$. Prove that the straight line passing through $I$, which is parallel to $AB$, touches the circle $\omega$.

A matrix $\begin{bmatrix} x & y \\ z & w \end{bmatrix}$ has square root $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ if $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}^2 = \begin{bmatrix} a^2 + bc &ab + bd \\ ac + cd & bc + d^2 \end{bmatrix} = \begin{bmatrix} x & y \\ z & w \end{bmatrix}$$ Determine how many square roots the matrix $\begin{bmatrix} 2 & 2 \\ 3 & 4 \end{bmatrix}$ has (complex coefficients are allowed).

Let $A$ be an integer and $A>1$. Let $a_{1}=A^{A}$, $a_{n+1}=A^{a_{n}}$ and $b_{1}=A^{A+1}$, $b_{n+1}=2^{b_{n}}$, $n=1, 2, 3, ...$. Prove that $a_{n}<b_{n}$ for each $n$.

Let $M,N, P$ be the midpoints of the sides $BC,CA,AB$ of the triangle $ABC$, respectively, and let $G$ be the centroid of the triangle. Prove that if $BMGP$ is cyclic and $2BN = \sqrt3 AB$ , then triangle $ABC$ is equilateral.

Solve in the interval $ (2,\infty ) $ the following equation: $$ 1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right) . $$

Suppose that $ m$ and $ n$ are positive integers such that $ 75m \equal{} n^{3}$. What is the minimum possible value of $ m \plus{} n$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 5700$

A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples. What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }24\qquad \textbf{(E) }25$