(V.Protasov, 8) For a given pair of circles, construct two concentric circles such that both are tangent to the given two. What is the number of solutions, depending on location of the circles?

In a triangle $ABC$, the lengths of the sides are consecutive integers and median drawn from $A$ is perpendicular to the bisector drawn from $B$. Find the lengths of the sides of triangle $ABC$.

For every positive integer $n$, determine the greatest possible value of the quotient $$\frac{1-x^{n}-(1-x)^{n}}{x(1-x)^n+(1-x)x^n}$$ where $0 < x < 1$.

Suppose $j$, $x$, and $u$ are positive real numbers such that $jxu=20$ and $x+u=24$. Find the minimum possible value of $j\max(x,u)$.

Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

Using $600$ cards, $200$ of them having written the number $5$, $200$ having a $2$, and the other $200$ having a $1$, a student wants to create groups of cards such that the sum of the card numbers in each group is $9$. What is the maximum amount of groups that the student may create?

Show that $ \det \left( I_n+A \right)\ge 1, $ for any $ n\times n $ antisymmetric real matrix $ A. $

A bitstring of length $\ell$ is a sequence of $\ell$ $0$'s or $1$'s in a row. How many bitstrings of length $2014$ have at least $2012$ consecutive $0$'s or $1$'s?

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

Let $n\geq 1$ be an integer. Show that there exists an integer between $\sqrt{2n}$ and $\sqrt{5n}$, exclusive.

Let $x_1, x_2, \ldots , x_n \ (n \geq 1)$ be real numbers such that $0 \leq x_j \leq \pi, \ j = 1, 2,\ldots, n.$ Prove that if $\sum_{j=1}^n (\cos x_j +1) $ is an odd integer, then $\sum_{j=1}^n \sin x_j \geq 1.$

Write $102$ as the sum of the largest number of distinct primes.

Find the fundamental group of the topology of $ \text{SL}_2\left(\mathbb{R}\right) $ on $ \mathbb{R}^4. $

Let $a$ be a positive integer. Find all prime numbers $ p $ with the following property: there exist exactly $ p $ ordered pairs of integers $ (x, y)$, with $ 0 \leq  x, y \leq p - 1 $, such that $ p $ divides $ y^2 - x^3 - a^2x $.

For any pair of coprime positive integers $a$ and $b$, define $f(a, b)$ to be the smallest nonnegative integer $k$ such that $b \mid ak+1$. Prove that if a and b are coprime positive integers satisfying $$f(a, b) - f(b, a) = 2,$$ then there exists a prime number $p$ such that $p^2\mid a + b$. [i]Proposed by usjl[/i]

We call a triple of positive integers $(a, b, c)$ [i]harmonic [/i] if $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$. Prove that, for any given positive integer $c$, the number of harmonic triples $(a, b, c)$ is equal to the number of positive divisors of $c^2$.

A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false? (1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$. (2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.

Let $a, b, c$ be positive real numbers satisfying \[a+b+c = a^2 + b^2 + c^2.\] Let \[M = \max\left(\frac{2a^2}{b} + c, \frac{2b^2}{a} + c \right) \quad \text{ and } \quad N = \min(a^2 + b^2, c^2).\] Find the minimum possible value of $M/N$.

Let $\bigoplus$ and $\bigotimes$ be two binary boolean operators, i.e. functions that send $\{\text{True}, \text{False}\}\times \{\text{True}, \text{False}\}$ to $\{\text{True}, \text{False}\}$. Find the number of such pairs $(\bigoplus, \bigotimes)$ such that $\bigoplus$ and $\bigotimes$ distribute over each other, that is, for any three boolean values $a, b, c$, the following four equations hold: 1) $c \bigotimes (a \bigoplus b) = (c \bigotimes a) \bigoplus (c \bigotimes b);$ 2) $(a \bigoplus b) \bigotimes c = (a \bigotimes c) \bigoplus (b \bigotimes c);$ 3) $c \bigoplus (a \bigotimes b) = (c \bigoplus a) \bigotimes (c \bigoplus b);$ 4) $(a \bigotimes b) \bigoplus c = (a \bigoplus c) \bigotimes (b \bigoplus c).$ [i]Proposed by Yannick Yao

Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$

Let $x_n$ be the $n$-th non-square positive integer. Thus $x_1=2, x_2=3, x_3=5, x_4=6,$ etc. For a positive real number $x$, denotes the integer closest to it by $\langle x\rangle$. If $x=m+0.5$, where $m$ is an integer, then define $\langle x\rangle=m$. For example, $\langle 1.2\rangle =1, \langle 2.8 \rangle =3, \langle 3.5\rangle =3$. Show that $x_n=n+\langle \sqrt{n}\rangle$

Let $n>d>0$ integers. Batman, Joker, Clark play the following game in an infinite checkered board. Initially, Batman and Joker are in cells with distance $n$ and a candy is in a cell with distance $d$ to Batman. Batman is blindfold, and can only see his cell. Clark and Joker can see the whole board. The following two moves go alternately. 1 - Batman goes to an adjacent cell. If he touches Joker, Batman loses. If he touches the candy, Batman wins. If the cell is empty, Clark chooses to say loudly one of the following two words [b]hot[/b] or [b]cold[/b]. 2 - Joker goes to an adjacent cell. If he touches Batman or candy, Joker wins. Otherwise, the game continues. Determine for each $d$, the least $n$, such that Batman, and Clark can plan an strategy to ensure the Batman's win, regardless of initial positions of the Joker and of the candy. Note: Two cells are adjacent if its have a common side. The distance between two cells $X$ and $Y$ is the least $p$ such that there exist cells $X=X_0,X_1,X_2,\dots, X_p=Y$ with $X_i$ adjacent to $X_{i-1}$ for all $i=1,2,\dots,p$.

Some students on the faculty speak several languages and some - Russian only. $50$ of them know English, $50$ -- French and $50$ -- Spanish. Prove that it is possible to divide them onto $5$ groups, not necessary equal, to get $10$ of them knowing English, $10$ -- French and $10$ -- Spanish in each group.

Find the minimum value of $\int_1^e \left|\ln x-\frac{a}{x}\right|dx\ (0\leq a\leq e)$

For $n\in\mathbb{N}$, $n\geq 2$, $a_{i}, b_{i}\in\mathbb{R}$, $1\leq i\leq n$, such that \[\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0. \] Prove that \[\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n. \] [i]Cezar Lupu & Tudorel Lupu[/i]