Let $a, b, c, d$ be digits such that $d > c > b > a \geq 0$. How many numbers of the form $1a1b1c1d1$ are multiples of $33$?

A parallelogram $ABCD$ is given. The excircle of triangle $\triangle{ABC}$ touches the sides $AB$ at $L$ and the extension of $BC$ at $K$. The line $DK$ meets the diagonal $AC$ at point $X$; the line $BX$ meets the median $CC_1$ of trianlge $\triangle{ABC}$ at ${Y}$. Prove that the line $YL$, median $BB_1$ of triangle $\triangle{ABC}$ and its bisector $CC^\prime$ have a common point. [i](A. Golovanov)[/i]

Let $ABC$ be an equilateral triangle of side length $a$, and consider $D$, $E$ and $F$ the midpoints of the sides $(AB), (BC)$, and $(CA)$, respectively. Let $H$ be the the symmetrical of $D$ with respect to the line $BC$. Color the points $A, B, C, D, E, F, H$ with one of the two colors, red and blue. [list=1] [*] How many equilateral triangles with all the vertices in the set $\{A, B, C, D, E, F, H\}$ are there? [*] Prove that if points $B$ and $E$ are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set $\{A, B, C, D, E, F, H\}$ and having the same color. [*] Does the conclusion of the second part remain valid if $B$ is blue and $E$ is red? [/list]

For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the dierence between consecutive terms is constant).

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Assume $|G'| = 2$. Prove that $|G : G'|$ is even.

Let $N_0$ be the set of all non-negative integers and let $f : N_0 \times N_0 \to [0, +\infty)$ be a function such that $f(a, b) = f(b, a)$ and $$f(a, b) = f(a + 1, b) + f(a, b + 1),$$ for all $a, b \in N_0$. Denote by $x_n = f(n, 0)$ for all $n \in N_0$. Prove that for all $n \in N_0$ the following inequality takes place $$2^n x_n \ge x_0.$$

We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.

Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.

Pete has $n^3$ white cubes of the size $1\times 1\times 1$. He wants to construct a $n\times n\times n$ cube with all its faces being completely white. Find the minimal number of the faces of small cubes that Basil must paint (in black colour) in order to prevent Pete from fulfilling his task. Consider the cases: a) $n = 2$; [i](2 points)[/i] b) $n = 3$. [i](4 points)[/i]

A little boy takes a $ 12$ in long strip of paper and makes a Mobius strip out of it by tapping the ends together after adding a half twist. He then takes a $ 1$ inch long train model and runs it along the center of the strip at a speed of $ 12$ inches per minute. How long does it take the train model to make two full complete loops around the Mobius strip? A complete loop is one that results in the train returning to its starting point.

Lines that are drawn perpendicular to the faces of a triangular pyramid through the centers of the inscribed circles intersect at one point. Prove that the sums of the opposite edges of such a pyramid are equal to each other.

Let $a$ and $b$ be integers such that the remainder of dividing $a$ by $17$ is equal to the remainder of dividing $b$ by $19$, and the remainder of dividing $a$ by $19$ is equal to the remainder of dividing $b$ by $17$. Determine the possible values of the remainder of $a + b$ when divided by $323$.

You play a game where you and an adversarial opponent take turns writing down positive integers on a chalkboard; the only condition is that, if $m$ and $n$ are written consecutively on the board, $\gcd(m,n)$ must be squarefree. If your objective is to make sure as many integers as possible that are strictly less than $404$ end up on the board (and your opponent is trying to minimize this quantity), how many more such integers can you guarantee will eventually be written on the board if you get to move first as opposed to when your opponent gets to move first?

One side of a triangle is equal to one third of the sum of the other two. Prove that the angle opposite the first side is the smallest angle of the triangle. (AK Tolpygo)

Suppose that $ \frac{2}{3}$ of $ 10$ bananas are worth as much as $ 8$ oranges. How many oranges are worth as much is $ \frac{1}{2}$ of $ 5$ bananas? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ \frac{5}{2} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{7}{2} \qquad \textbf{(E)}\ 4$

There are $2n$ distinct points on a circle. The numbers $1$ through $2n$ are randomly assigned to this one points distributed. Each point is connected to exactly one other point, so that no of the resulting connecting routes intersect. If a segment connects the numbers $a$ and $b$, so we assign the value $ |a - b|$ to the segment . Show that we can choose the routes such that the sum of these values ​​results $n^2$.

Let $ABCD$ be a parallelogram and points $E,F$ be on its exterior. If triangles $BCF$ and $DEC$ are similar, i.e. $\triangle BCF \sim \triangle DEC$, prove that triangle $AEF$ is similar to these two triangles.

ِDo there exist non-zero reals numbers $a_1, a_2, ....., a_{10}$ for which \[(a_1+\frac{1}{a_1})(a_2+\frac{1}{a_2}) \cdots(a_{10}+\frac{1}{a_{10}})= (a_1-\frac{1}{a_1})(a_2-\frac{1}{a_2})\cdots(a_{10}-\frac{1}{a_{10}}) \ ? \]

Find all functions $f : [0,\infty) \to [0,\infty)$ such that $f(f(x)) +f(x) = 12x$, for all $x \ge 0$.

* Given $\vartriangle ABC$, divide it into the minimal number of parts so that after being flipped over these parts can constitute the same $\vartriangle ABC$.

Given a positive integer $n$, consider the sequence $(a_i)$, $1 \leq i \leq 2n$, defined as follows: $a_{2k-1} = -k, 1 \leq k \leq n$ $a_{2k} = n-k+1, 1 \leq k \leq n.$ We call a pair of numbers $(b,c)$ good if the following conditions are met: $i) 1 \leq b < c \leq 2n,$ $ii) \sum_{j=b}^{c}a_j = 0$ If $B(n)$ is the number of good pairs corresponding to $n$, prove that there are infinitely many $n$ for which $B(n) = n$.

Let $ABC$ be a scalene triangle, with circumcircle $\omega$ and incentre $I.{}$ The tangent line at $C$ to $\omega$ intersects the line $AB$ at $D.{}$ The angle bisector of $BDC$ meets $BI$ at $P{}$ and $AI{}$ at $Q{}.$ Let $M{}$ be the midpoint of the segment $PQ.$ Prove that the line $IM$ passes through the middle of the arc $ACB$ of $\omega.$ [i]Dana Heuberger[/i]

Show that if $a$ and $b$ are positive integers, then \[\left( a+\frac{1}{2}\right)^{n}+\left( b+\frac{1}{2}\right)^{n}\] is an integer for only finitely many positive integer $n$.

Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

The graph of the function $f(x)=x^n+a_{n-1}x_{n-1}+\ldots +a_1x+a_0$ (where $n>1$) intersects the line $y=b$ at the points $B_1,B_2,\ldots ,B_n$ (from left to right), and the line $y=c\ (c\not= b)$ at the points $C_1,C_2,\ldots ,C_n$ (from left to right). Let $P$ be a point on the line $y=c$, to the right to the point $C_n$. Find the sum \[\cot (\angle B_1C_1P)+\ldots +\cot (\angle B_nC_nP) \]