The numbers $x, y, z, t, a$ and $b$ are positive integers, so that $xt-yz = 1$ and $$\frac{x}{y} \ge \frac{a}{b} \ge \frac{z}{t} .$$Prove that $$ab \le (x + z) (y +t)$$

Alex and Bob play a game 2015 x 2015 checkered board by the following rules.Initially the board is empty: the players move in turn, Alex moves first. By a move, a player puts either red or blue token into any unoccopied square. If after a player's move there appears a row of three consecutive tokens of the same color( this row may be vertical,horizontal, or dioganal), then this player wins. If all the cells are occupied by tokens, but no such row appears, then a draw is declared.Determine whether Alex, Bob, or none of them has winning strategy.

Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.

Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$.We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$ Suppose $ n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places. Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/i]

There are three candidates in the Hundilaane forest governor elections: $A, B$ and $C$. For each of the $20$ forest dwellers, the names of all three candidates were written on the ballot paper in the order of their preference. Examination of the ballots revealed that $11$ forest dwellers prefer $A$, $12$ $B$ and $14$ $C$. Which of the candidates will be marked first on the largest number of ballot papers when it is known that each possible the order of the candidates appears on at least one ballot?

$\boxed{N1}$Let $n$ be a positive integer,$g(n)$ be the number of positive divisors of $n$ of the form $6k+1$ and $h(n)$ be the number of positive divisors of $n$ of the form $6k-1,$where $k$ is a nonnegative integer.Find all positive integers $n$ such that $g(n)$ and $h(n)$ have different parity.

Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that \[\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.\]

The surface area and the volume of a cube are numerically equal. Find the cube’s volume.

Divide each side of a triangle into $50$ equal parts, and each point of the division is joined to the opposite vertex by a segment. Calculate the number of intersection points determined by these segments. Clarification : the vertices of the original triangle are not considered points of intersection or division.

Regular hexagon $ABCDEF$ is inscribed in rectangle $PQRS$ with $AB = 1$, A and $B$ on side $PQ$, $C$ on side $QR$, $D$ and $E$ on side $RS$, and $F$ on side $SP$. What is the area of $PQRS$?

The number of real quadruples $(x,y,z,w)$ satisfying $x^3+2=3y, y^3+2=3z, z^3+2=3w, w^3+2=3x$ is $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \text{None}$

Determine all integers $n \ge 3$ whose decimal expansion has less than $20$ digits, such that every quadratic non-residue modulo $n$ is a primitive root modulo $n$. [i]An integer $a$ is a quadratic non-residue modulo $n$, if there is no integer $b$ such that $a - b^2$ is divisible by $n$. An integer $a$ is a primitive root modulo $n$, if for every integer $b$ relatively prime to n there is a positive integer $k$ such that $a^k - b$ is divisible by $n$.[/i]

Let $\{a_n\}$ be a sequence of positive terms such that $a_{n+1}=a_n+ \frac{n^2}{a_n}$ . Let $b_n =a_n-n$ . (1) Are there infinitely many $n$ such that $b_n \ge 0$ ? (2) Prove that there is a positive number $M$ such that $\sum^{\infty}_{n=3} \frac{b_n}{n+1}<M$.

Define $f(x)$ as $\frac{x^2-x-2}{x^2+x-6}$. $f(f(f(f(1))))$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p,q$. Find $10p+q$.

$ a_1, a_2, \ldots, a_n$ are integers, not all equal. Prove that there exist infinitely many prime numbers $ p$ such that for some $ k$ \[ p\mid a_1^k \plus{} \dots \plus{} a_n^k.\]

Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$

A $2.00 \text{L}$ balloon at $20.0^{\circ} \text{C}$ and $745 \text{mmHg}$ floats to an altitude where the temperature is $10.0^{\circ} \text{C}$ and the air pressure is $700 \text{mmHg}$. What is the new volume of the balloon? $ \textbf{(A)}\hspace{.05in}0.94 \text{L}\qquad\textbf{(B)}\hspace{.05in}1.06 \text{L}\qquad\textbf{(C)}\hspace{.05in}2.06 \text{L}\qquad\textbf{(D)}\hspace{.05in}2.20 \text{L}\qquad $

Let $\displaystyle ABC$ and $\displaystyle DBC$ be isosceles triangle with the base $\displaystyle BC$. We know that $\displaystyle \measuredangle ABD = \frac{\pi}{2}$. Let $\displaystyle M$ be the midpoint of $\displaystyle BC$. The points $\displaystyle E,F,P$ are chosen such that $\displaystyle E \in (AB)$, $\displaystyle P \in (MC)$, $\displaystyle C \in (AF)$, and $\displaystyle \measuredangle BDE = \measuredangle ADP = \measuredangle CDF$. Prove that $\displaystyle P$ is the midpoint of $\displaystyle EF$ and $\displaystyle DP \perp EF$.

Let us denote successively $r$ and $r_a$ the radii of the inscribed circle and the exscribed circle wrt to side BC of triangle $ABC$. Prove that if it is true that $r+r_a=|BC|$ , then the triangle $ABC$ is a right one

Let $ f_1,f_2,f_3,f_4 $ be four polynomials with real coefficients, having the property that $$ f_1 (1) =f_2 (0), \quad f_2 (1) =f_3 (0),\quad f_3 (1) =f_4 (0),\quad f_4 (1) =f_1 (0) . $$ Prove that there exists a polynomial $ f\in\mathbb{R}[X,Y] $ such that $$ f(X,0)=f_1(X),\quad f(1,Y) =f_2(Y) ,\quad f(1-X,1) =f_3(X),\quad f(0,1-Y)=f_4(Y) . $$

Evaluate \[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]

Moe uses a mower to cut his rectangular $ 90$-foot by $ 150$-foot lawn. The swath he cuts is $ 28$ inches wide, but he overlaps each cut by $ 4$ inches to make sure that no grass is missed. He walks at the rate of $ 5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn? $ \textbf{(A)}\ 0.75 \qquad \textbf{(B)}\ 0.8 \qquad \textbf{(C)}\ 1.35 \qquad \textbf{(D)}\ 1.5 \qquad \textbf{(E)}\ 3$

Two circles $\omega_1$ and $\omega_2$ are said to be $\textit{orthogonal}$ if they intersect each other at right angles. In other words, for any point $P$ lying on both $\omega_1$ and $\omega_2$, if $\ell_1$ is the line tangent to $\omega_1$ at $P$ and $\ell_2$ is the line tangent to $\omega_2$ at $P$, then $\ell_1\perp \ell_2$. (Two circles which do not intersect are not orthogonal.) Let $\triangle ABC$ be a triangle with area $20$. Orthogonal circles $\omega_B$ and $\omega_C$ are drawn with $\omega_B$ centered at $B$ and $\omega_C$ centered at $C$. Points $T_B$ and $T_C$ are placed on $\omega_B$ and $\omega_C$ respectively such that $AT_B$ is tangent to $\omega_B$ and $AT_C$ is tangent to $\omega_C$. If $AT_B = 7$ and $AT_C = 11$, what is $\tan\angle BAC$?

We will say that a plane is [i]well-colored[/i] in several colors if it is divided into convex polygons with an area of at least $1/1000$ and each polygon is colored in one color. Points lying on the border of several polygons can be colored in any of their colors. Are there convex is a $9$-gon $R$ and a good coloring of the plane in $7$ colors such that in any polygon obtained from $R$ by a translate to any vector, all colors occupy the same area ($1/7$ of the area of $R$)?