$10$ people attended a party. For every $3$ people, there exist at least $2$ people who don’t know each other. Prove that there exist $4$ people who don’t know each other.

Let be given two parallel lines $k$ and $l$, and a circle not intersecting $k$. Consider a variable point $A$ on the line $k$. The two tangents from this point $A$ to the circle intersect the line $l$ at $B$ and $C$. Let $m$ be the line through the point $A$ and the midpoint of the segment $BC$. Prove that all the lines $m$ (as $A$ varies) have a common point.

Maria Ivanovna drew on the blackboard a right triangle $ABC$ with a right angle $B$. Three students looked at her and said: $\bullet$ Yura said: "The hypotenuse of this triangle is $10$ cm." $\bullet$ Roma said: "The altitude drawn from the vertex $B$ on the side $AC$ is $6$ cm." $\bullet$ Seva said: "The area of the triangle $ABC$ is $25$ cm$^2$." Determine which of the students was mistaken if it is known that there is exactly one such person.

An automobile starts from rest and ends at rest, traversing a distance of one mile in one minute, along a straight road. If a governor prevents the speed of the car from exceeding $90$ miles per hour, show that at some time of the traverse the acceleration or deceleration of the car was at least $6.6$ ft/sec.

Prove that there are infinitely many positive integers $m$ such that the number of odd distinct prime factor of $m(m+3)$ is a multiple of $3$.

Let $a_1,...,a_n$ be positive real numbers such that $\sum_{i=1}^n a_i =\prod_{i=1}^n a_i $ , and let $b_1,...,b_n$ be positive real numbers such that $a_i \le b_i$ for all $i$. Prove that $\sum_{i=1}^n b_i \le\prod_{i=1}^n b_i $

Let $n > 1$ be an integer. There are $n$ boxes in a row, and there are $n + 1$ identical stones. A [i]distribution [/i] is a way to distribute the stones over the boxes, in which every stone is in exactly one of the boxes. We say that two of such distributions are a [i]stone’s throw away[/i] from each other if we can obtain one distribution from the other by moving exactly one stone from one box to another. The [i]cosiness [/i] of a distribution $a$ is defined as the number of distributions that are a stone’s throw away from $a$. Determine the average cosiness of all possible distributions.

In the five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $\overline{AB}$,$\overline{BC}$,$\overline{CD}$,$\overline{DE}$, and $\overline{EA}$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence? [asy] size(150); defaultpen(linewidth(0.8)); string[] strng = {'A','D','B','E','C'}; pair A=dir(90),B=dir(306),C=dir(162),D=dir(18),E=dir(234); draw(A--B--C--D--E--cycle); for(int i=0;i<=4;i=i+1) { path circ=circle(dir(90-72*i),0.125); unfill(circ); draw(circ); label("$"+strng[i]+"$",dir(90-72*i)); } [/asy] $ \textbf{(A)}\ 9\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 11\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 13$

Let $ ABCD$ be a quadrilateral with $ AB$ extended to $ E$ so that $ \overline{AB} \equal{} \overline{BE}$. Lines $ AC$ and $ CE$ are drawn to form angle $ ACE$. For this angle to be a right angle it is necessary that quadrilateral $ ABCD$ have: $ \textbf{(A)}\ \text{all angles equal}$ $ \textbf{(B)}\ \text{all sides equal}$ $ \textbf{(C)}\ \text{two pairs of equal sides}$ $ \textbf{(D)}\ \text{one pair of equal sides}$ $ \textbf{(E)}\ \text{one pair of equal angles}$

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.)

Let $AB = 10$ be a diameter of circle $P$. Pick point $C$ on the circle such that $AC = 8$. Let the circle with center $O$ be the incircle of $\vartriangle ABC$. Extend line $AO$ to intersect circle $P$ again at $D$. Find the length of $BD$.

Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.

A [i]peacock [/i] is a ten-digit positive integer that uses each digit exactly once. Compute the number of peacocks that are exactly twice another peacock.

Given a circle of radius 2, there are many line segments of length 2 that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments. $\text{(A)}\ \frac \pi 4 \qquad \text{(B)}\ 4 - \pi \qquad \text{(C)}\ \frac \pi 2 \qquad \text{(D)}\ \pi \qquad \text{(E)}\ 2\pi$

You are given a balance and one copy of each of ten weights of $1, 2, 4, 8, 16, 32, 64, 128, 256$ and $512$ grams. An object weighing $M$ grams, where $M$ is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance. (a) Prove that no object may be balanced in more than $89$ ways. (b) Find a value of $M$ such that an object weighing $M$ grams can be balanced in $89$ ways. (A Shapovalov, A Kulakov)

In $1991$ the population of a town was a perfect square. Ten years later, after an increase of $150$ people, the population was $9$ more than a perfect square. Now, in $2011$, with an increase of another $150$ people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period? $ \textbf{(A)}\ 42 \qquad\textbf{(B)}\ 47 \qquad\textbf{(C)}\ 52\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 62 $

Let $ABCD$ be a tetrahedron and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that for every point $P \in (MN)$ with $P \neq M$ and $P \neq N$, there exist unique points $X$ and $Y$ on segments $AB$ and $CD$, respectively, such that $X,P,Y$ are collinear.

Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins. $(a)$ Determine which player has a winning strategy for $N=2005$. $(b)$ Determine which player has a winning strategy for $N=2004$. $(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.

Let $S$ be a finite subset of a straight line. Say that $S$ has the [i]repeated distance property [/i] if every value of the distance between two points of $S$ (except the longest) occurs at least twice. Show that if $S$ has the [i]repeated distance property [/i] then the ratio of any two distances between two points of $S$ is rational.

Prove that for all positive integers $a, b, c$ the following inequality holds: $$\frac{a + b}{a + c}+\frac{b + c}{b + a}+\frac{c + a}{c + b} \le \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$$

A $9 \times 10$ board is divided into $90$ unit cells. There are certain rules for moving a non-standard chess queen from one square to another: [list] [*]The queen can only move along the column or row it is in each step. [*]For any natural number $n$, if $x$ cells move made in $(2n-1)$th step, then $9-x$ cells move will be done in $(2n)$th step. The last cell it stops at during these steps is considered the visited cell. [/list] Is it possible for the queen to move from any square on the board and return to the square where it started after visiting all the squares of the board exactly once? Note: At each step, the queen chooses the right, left, up, and down direction within the above condition can choose.

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $ 32$? $ \textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad \textbf{(E)}\ 2255$

Denote by $a_n$ the last digit of the number $n^{(n^n)}$ (let $n\ne 0$ be a natural number ). Prove that the numbers $a_n$ form a periodic sequence and state this period!