Prove that for any integer the number $2n^3+3n^2+7n$ is divisible by $6$.

Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $

2) Let $a, b, c$ be real numbers such that $$a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1$$a) Prove that $ab+bc+ca=0$ and $a+b+c=3$. b) Prove that $|a|+|b|+|c|< 5$

Let $ABCD$ be a quadrilateral with $\angle ADC = 70^{\circ}$, $\angle ACD = 70^{\circ}$, $\angle ACB = 10^{\circ}$ and $\angle BAD = 110^{\circ}$. The measure of $\angle CAB$ (in degrees) is:

Let $ABC$ be a non-isosceles triangle. The altitude from $A$, the bisector from $B$ and the median from $C$ concur at point $K$. a) Which of the sidelengths of the triangle is medial (intermediate in length)? b) Which of the lengths of segments $AK, BK, CK$ is medial (intermediate in length)?

Find for which natural numbers $n$ one can color the sides and diagonals of a regular $n$-gon with $n$ colors in such a way that for each triplet in pairs of different colors, a triangle can be found, the sides of which are sides or diagonals of $n$-gon and which is colored with exactly these three colors.

Suppose $a$ and $b$ are integers such that $$x^2+ax+b+1=0$$ has $2$ positive integer solutions. Show that $a^2+b^2$ is not prime.

Given a real number $A$ and an integer $n$ with $2 \leq n \leq 19$, find all polynomials $P(x)$ with real coefficients such that $P(P(P(x))) = Ax^n +19x+99$.

Find all $n>1$ and integers $a_1,a_2,\dots,a_n$ satisfying the following three conditions: (i) $2<a_1\le a_2\le \cdots\le a_n$ (ii) $a_1,a_2,\dots,a_n$ are divisors of $15^{25}+1$. (iii) $2-\frac{2}{15^{25}+1}=\left(1-\frac{2}{a_1}\right)+\left(1-\frac{2}{a_2}\right)+\cdots+\left(1-\frac{2}{a_n}\right)$

Let ABC be a triangle and $\omega_1$ its incircle. Let points $D$ and $E$ be on segments $AB$, $AC$ respectively such that $DE$ is parallel to $BC$ and tangent to $\omega_1$ . Now let $\omega_2$ be the incircle of $\vartriangle ADE$ and let points $F$ and $G$ be on segments $AD,$ $AE$ respectively such that F G is parallel to $DE$ and tangent to $\omega_2$. Given that $\omega_2$ is tangent to line $AF$ at point X and line $AG$ at point $Y$ , the radius of $\omega_1$ is $60$, and $$4(AX) = 5(F G) = 4(AY),$$ compute the radius of $\omega_2$.

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Given nine positive integers, is it always possible to choose four different numbers $a,b,c,d$ such that $a+b$ and $c+d$ are congruent modulo $20$?

A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the following game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers $a, b$, neither of which was chosen earlier by any player and move the marker by $a$ units in the horizontal direction and $b$ units in the vertical direction. Hobbes wins if the marker is back at the origin any time after the first turn. Prove or disprove that Calvin can prevent Hobbes from winning. Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).

A cube is made by soldering twelve $ 3$-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is: $ \textbf{(A)}\ 24\text{ in.}\qquad \textbf{(B)}\ 12\text{ in.}\qquad \textbf{(C)}\ 30\text{ in.}\qquad \textbf{(D)}\ 18\text{ in.}\qquad \textbf{(E)}\ 36\text{ in.}$

Inside the triangle $DD_1D_3$ the cevian $DD_2$ is constructed. Perpendiculars from $D_1, D_2$ and $D_3$ to lines $DD_1, DD_2$ and $DD_3$, respectively, intersect in points $A,B$ and $C$ such that $AB\perp DD_1, AC\perp DD_2, BC\perp DD_3$. Prove that $\frac{AC}{DD_2}=\frac{AB}{DD_1}+\frac{BC}{DD_3}$.

Prove that if the feet of the altitudes of a tetrahedron are the incenters of the corresponding faces, then the tetrahedron is regular.

Let $O$ be the circumcenter of the triangle $ABC, A'$ be a point symmetric of $A$ wrt line $BC, X$ is an arbitrary point on the ray $AA'$ ($X \ne A$). Angle bisector of angle $BAC$ intersects the circumcircle of triangle $ABC$ at point $D$ ($D \ne A$). Let $M$ be the midpoint of the segment $DX$. A line passing through point $O$ parallel to $AD$, intersects $DX$ at point $N$. Prove that angles $BAM$ and $CAN$ angles are equal.

In the space are given $n$ points and no four of them belongs to a common plane. Some of the points are connected with segments. It is known that four of the given points are vertices of tetrahedron which edges belong to the segments given. It is also known that common number of the segments, passing through vertices of tetrahedron is $2n$. Prove that there exists at least two tetrahedrons every one of which have a common face with the first (initial) tetrahedron. [i]N. Nenov, N. Hadzhiivanov[/i]

Given a square $ABCD$ with side $10$. On sides BC and $AD$ of this square are selected respectively points $E$ and $F$ such that formed a rectangle $ABEF$. Rectangle $KLMN$ is located so that its the vertices $K, L, M$ and $N$ lie one on each segments $CD, DF, FE$ and $EC$, respectively. It turned out that the rectangles $ABEF$ and $KLMN$ are equal with $AB = MN$. Find the length of segment $AL$.

Show how to divide space into (a) congruent tetrahedra, (b) congruent “equifaced” tetrahedra. (A tetrahedron is called equifaced if all its faces are congruent triangles.) (NB Vassiliev)

Given two finite collections of pairs of real numbers It turned out that for any three pairs $(a_1, b_1)$, $(a_2, b_2)$ and $(a_3, b_3)$ from the first collection there is a pair $(c, d)$ from the second collection, such that the following three inequalities hold: \[ a_1c + b_1d \geq 0,a_2c + b_2c \geq 0 \text{ and } a_3c + b_3d \geq 0 \] Prove that there is a pair $(\gamma, \delta)$ in the second collection, such that for any pair $(\alpha, \beta)$ from the first collection inequality $\alpha \gamma + \beta \delta \geq 0$ holds.

A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is ${\text{(A)}\ 8 \qquad \text{(B)}\ 9 \qquad \text(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 18$

Suppose that each point on a plane is colored with one of the colors red or yellow. Show that exist a convex pentagon with three right angles and all vertices are with same color.

In an acute triangle $ABC$, the points $H$, $G$, and $M$ are located on $BC$ in such a way that $AH$, $AG$, and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$, $AB=10$, and $AC=14$. Find the area of triangle $ABC$.

A ‘magic square’ of size \(n\) is an \(n\times n\) array of real numbers such that all the rows, all the columns and the two main diagonals have the same sum. Determine the dimension, over \(\mathbb{R}\), of the vector space of \(n\times n\) magic squares.\\