(a) Prove that $x^4+3x^3+6x^2+9x+12$ cannot be expressed as product of two polynomials of degree 2 with integers coefficients. (b) $2n+1$ segments are marked on a line. Each of these segments intersects at least $n$ other segments. Prove that one of these segments intersects all other segments.

Given a set of 102 elements. Is it possible to choose 102 17-element subsets so that the intersection of any two subsets contains no more than 3 elements?

Prove that for all $\displaystyle a,b \in \left( 0 ,\frac{\pi}{4} \right)$ and $\displaystyle n \in \mathbb N^\ast$ we have \[ \frac{\sin^n a + \sin^n b}{\left( \sin a + \sin b \right)^n} \geq \frac{\sin^n 2a + \sin^n 2b}{\left( \sin 2a + \sin 2b \right)^n} . \]

Determine for what $n\ge 3$ integer numbers, it is possible to find positive integer numbers $a_1 < a_2 < ...< a_n$ such $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$ and $a_1 a_2\cdot\cdot\cdot a_n$ is a perfect square.

Let the roots of $x^2-3x+1=0$ be $r$ and $s$. Then the expression $r^2+s^2$ is: $ \textbf{(A)}\ \text{a positive integer} \qquad\textbf{(B)}\ \text{a positive fraction greater than 1}\qquad\textbf{(C)}\ \text{a positive fraction less than 1}$ $\textbf{(D)}\ \text{an irrational number}\qquad\textbf{(E)}\ \text{an imaginary number}$

Define the sequence of positive integers $f_n$ by $f_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n$. Show that $f_n =\frac{ (a^{n+1} - b^{n+1})}{\sqrt5}$, where $a, b$ are real numbers such that $a + b = 1, ab = -1$ and $a > b$.

Let \(\triangle ABC\) have side lengths \(AB=30\), \(BC=32\), and \(AC=34\). Point \(X\) lies in the interior of \(\overline{BC}\), and points \(I_1\) and \(I_2\) are the incenters of \(\triangle ABX\) and \(\triangle ACX\), respectively. Find the minimum possible area of \(\triangle AI_1I_2\) as \( X\) varies along \(\overline{BC}\).

Fow which real values of $m$ does the polynomial $x^3+1995x^2-1994x+m$ have all three roots integers?

Give an isosceles triangle $ABC$ at $A$. Draw ray $Cx$ being perpendicular to $CA, BE$ perpendicular to $Cx$ ($E \in Cx$).Let $M$ be the midpoint of $BE$, and $D$ be the intersection point of $AM$ and $Cx$. Prove that $BD \perp BC$.

Let $P$ units be the increase in the circumference of a circle resulting from an increase in $\pi$ units in the diameter. Then $P$ equals: $\textbf{(A)}\ \dfrac{1}{\pi} \qquad \textbf{(B)}\ \pi \qquad \textbf{(C)}\ \dfrac{\pi^2}{2} \qquad \textbf{(D)}\ \pi^2 \qquad \textbf{(E)}\ 2\pi $

Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$ For which positive integers $m$ is $f(m)$ rational?

A $m \times n$ table is filled with signs $"+"$ and $"-"$. A table is called irreducible if one cannot reduce it to the table filled with $"+"$, applying the following operations (as many times as one wishes). $a)$ It is allowed to flip all the signs in a row or in a column. Prove that an irreducible table contains an irreducible $2\times 2$ sub table. $b)$ It is allowed to flip all the signs in a row or in a column or on a diagonal (corner cells are diagonals of length $1$). Prove that an irreducible table contains an irreducible $4\times 4$ sub table.

Let $ABC$ be an acute-angled triangle with $AB<AC<BC$ and $c(O,R)$ the circumscribed circle. Let $D, E$ be points in the small arcs $AC, AB$ respectively. Let $K$ be the intersection point of $BD,CE$ and $N$ the second common point of the circumscribed circles of the triangles $BKE$ and $CKD$. Prove that $A, K, N$ are collinear if and only if $K$ belongs to the symmedian of $ABC$ passing from $A$.

Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$

[b]a)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ and positive integer $M$ there exists $n\in\mathbb N$ such that set $\left\{k\in \mathbb N : f(n,k)=g(k)\right\}$ has at least $M$ elements? [b]b)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ there exists $n\in \mathbb N$ such that set $\left\{k\in\mathbb N : f(n,k)=g(k)\right\}$ has an infinite number of elements?

Let $a_1, a_2, \dots$ be a sequence of real numbers satisfying $a_{i+j} \leq a_i+a_j$ for all $i,j=1,2,\dots$. Prove that \[ a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \cdots + \frac{a_n}{n} \geq a_n \] for each positive integer $n$.

Three sidelines of on acute-angled triangle are drawn on the plane. Fyodor wants to draw the altitudes of this triangle using a ruler and a compass. Ivan obstructs him using an eraser. For each move Fyodor may draw one line through two markeed points or one circle centered at a marked point and passing through another marked point. After this Fyodor may mark an arbitrary number of points (the common points of drawn lines, arbitrary points on the drawn lines or arbitrary points on the plane). For each move Ivan erases at most three of marked point. (Fyodor may not use the erased points in his constructions but he may mark them for the second time). They move by turns, Fydors begins. Initially no points are marked. Can Fyodor draw the altitudes?

In the beginnig, each unit square of a $m\times n$ board is colored white. We are supposed to color all the squares such that one of two adjacent squares having a common side is black and the other is white. At each move, we choose a $2 \times 2$ square, and we color each of $4$ squares inversely such that if the square is black then it is colored white or vice versa. For which of the following ordered pair $(m, n)$, can the board be colored in this way? $ \textbf{a)}\ (3,3) \qquad\textbf{b)}\ (2,6) \qquad\textbf{c)}\ (4,8) \qquad\textbf{d)}\ (5,5) \qquad\textbf{e)}\ \text{None of above} $

You flip a fair coin which results in heads ($\text{H}$) or tails ($\text{T}$) with equal probability. What is the probability that you see the consecutive sequence $\text{THH}$ before the sequence $\text{HHH}$?

There are $2015$ coins on a table. For $i = 1, 2, \dots , 2015$ in succession, one must turn over exactly $i$ coins. Prove that it is always possible either to make all of the coins face up or to make all of the coins face down, but not both.

The inscribed circle of an acute triangle $ABC$ meets the segments $AB$ and $BC$ at $D$ and $E$ respectively. Let $I$ be the incenter of the triangle $ABC$. Prove that the intersection of the line $AI$ and $DE$ is on the circle whose diameter is $AC$(passing through A, C).

Let $n$ and $k$ are integers with $n>0$. Prove that \[-\frac{1}{2n}\sum^{n-1}_{m=1}\cot \frac{\pi m}{n}\sin \frac{2\pi km}{n}= \begin{cases}\tfrac{k}{n}-\lfloor\tfrac{k}{n}\rfloor-\frac12 & \text{if }k|n \\ 0 & \text{otherwise}\end{cases}.\]

How many positive integers $N$ in the segment $\left[10, 10^{20} \right]$ are such that if all their digits are increased by $1$ and then multiplied, the result is $N+1$? [i](F. Bakharev)[/i]

$n>1$ is odd number. There are numbers $n,n+1,n+2,...,2n-1$ on the blackboard. Prove that we can erase one number, such that the sum of all numbers will be not divided any number on the blackboard.

Let $P$ be a point and $\omega$ be a circle with center $O$ and radius $r$ . We define the power of the point $P$ with respect to the circle $\omega$ to be $OP^2 - r^2$ , and we denote this by pow $(P, \omega)$. We define the radical axis of two circles $\omega_1$ and $\omega_2$ to be the locus of all points P such that pow $(P,\omega_1) =$ pow $(P,\omega_2)$. It turns out that the pairwise radical axes of three circles are either concurrent or pairwise parallel. The concurrence point is referred to as the radical center of the three circles. In $\vartriangle ABC$, let $I$ be the incenter, $\Gamma$ be the circumcircle, and $O$ be the circumcenter. Let $A_1,B_1,C_1$ be the point of tangency of the incircle of $\vartriangle ABC$ with side $BC,CA, AB$, respectively. Let $X_1,X_2 \in \Gamma$ such that $X_1,B_1,C_1,X_2$ are collinear in this order. Let $M_A$ be the midpoint of $BC$, and define $\omega_A$ as the circumcircle of $\vartriangle X_1X_2M_A$. Define $\omega_B$ ,$\omega_C$ analogously. The goal of this problem is to show that the radical center of $\omega_A$, $\omega_B$, $\omega_C$ lies on line $OI$. (a) Let$ A'_1$ denote the intersection of $B_1C_1$ and $BC$. Show that $\frac{A_1B}{A_1C}=\frac{A'_1B}{A'_1C}$. (b) Prove that $A_1$ lies on $\omega_A$. (c) Prove that $A_1$ lies on the radical axis of $\omega_B$ and $\omega_C$ . (d) Prove that the radical axis of $\omega_B$ and $\omega_C$ is perpendicular to $B_1C_1$. (e) Prove that the radical center of $\omega_A$, $\omega_B$, $\omega_C$ is the orthocenter of $\vartriangle A_1B_1C_1$. (f ) Conclude that the radical center of $\omega_A$, $\omega_B$, $\omega_C$ , $O$, and $I$ are collinear. PS. You had better use hide for answers.