Given five $n$-digit binary numbers. For each two numbers their digits coincide exactly on $m$ places. There is no place with the common digit for all the five numbers. Prove that $$2/5 \le m/n \le 3/5$$

A plane intersects a rectangular parallelepiped in a regular hexagon. Prove that the rectangular parallelepiped is a cube.

If $a$ and $b$ are selected uniformly from $\{0,1,\ldots,511\}$ without replacement, the expected number of $1$'s in the binary representation of $a+b$ can be written in simplest from as $\tfrac{m}{n}$. Compute $m+n$.

For any $n$-tuple $a=(a_1,a_2,\ldots,a_n)\in\mathbb{N}_0^n$ of nonnegative integers, let $d_a$ denote the number of pairs of indices $(i,j)$ such that $a_i-a_j=1.$ Determine the maximum possible value of $d_a$ as $a$ ranges over all elements of $\mathbb{N}_0^n.$

Find the sum of all real solutions to $ (x^2 - 10x - 12)^{x^2+5x+2} = 1 $

Does there exist a positive odd integer $n$ so that there are primes $p_1$, $p_2$ dividing $2^n-1$ with $p_1-p_2=2$?

Let $n$ be a positive integer, the players A and B play the following game: we have $n$ balls with the numbers of $1, 2, 3, 4,...., n$ this balls will be in two boxes with the symbols $\prod$ and $\sum$. In your turn, the player can choose one ball and the player will put this ball in some box, in the final all the balls of the box $\prod$ are multiplied and we will get a number $P$, after this all the balls of the box $\sum$ are added up and we will get a number $Q$(if the box $\prod$ is empty $P = 1$, if the box $\sum$ is empty $Q = 0$). The player(s) play alternately, player A starts, if $P + Q$ is even player A wins, otherwise player B wins. a)If $n= 6$, which player has the winning strategy??? b)If $n = 2012$, which player has the winning strategy???

Let $A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}$ and $S$ be the set of all subsets of $A.$ Find the number of functions $f : S\to B$ satisfying $f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}$ for all $A_1, A_2 \in S.$

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 $a+b$.