The natural numbers $x, y > 1$, are such that $x^2 + xy -y$ is the square of a natural number. Prove that $x + y + 1$ is a composite number.

Let $S=\{a_1,a_2,\ldots,a_n\}$ of nenul vectors in a plane. Show that $S{}$ can be partitioned in nenul subsets $B_1, B_2,\ldots, B_m$ with the properties: 1) each vector from $S{}$ is part of only on subset; 2) if $a_i\in B_j$ then the angle between vectors $a_i$ and $c_j$, which is the sum of all vectors from $B_j$ is not greater than $\frac{\pi}{2}$; 3) if $i\neq j$ then the angle between vectors $c_i$ and $c_j$, which is the sum of all vectors from $B_i$ and $B_j$, respectively, is greater than $\frac{\pi}{2}$. What are the possible values of $m$?

In triangle $ ABC$, points $ M,N$ lie on line $ AC$ such that $ MA\equal{}AB$ and $ NB\equal{}NC$. Also $ K,L$ lie on line $ BC$ such that $ KA\equal{}KB$ and $ LA\equal{}LC$. It is know that $ KL\equal{}\frac12{BC}$ and $ MN\equal{}AC$. Find angles of triangle $ ABC$.

Let $p$ be a prime number. Find all subsets $S\subseteq\mathbb Z/p\mathbb Z$ such that 1. if $a,b\in S$, then $ab\in S$, and 2. there exists an $r\in S$ such that for all $a\in S$, we have $r-a\in S\cup\{0\}$. [i]Proposed by Harun Khan[/i]

The equations $ 2x \plus{} 7 \equal{} 3$ and $ bx\minus{}10 \equal{} \minus{}\!2$ have the same solution for $ x$. What is the value of $ b$? $ \textbf{(A)}\minus{}\!8 \qquad \textbf{(B)}\minus{}\!4 \qquad \textbf{(C)}\minus{}\!2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that (a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$ (b) $\sum_{k=1}^n (a_k+b_k)=2n;$ (c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$

Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.

Suppose that every integer is colored using one of $4$ colors. Let $m, n$ be distinct odd integers such that $m + n \ne 0$. Prove that there exist integers $a$, $ b$ of the same color such that $ a - b$ equals one of the numbers $m$, $n$, $m - n$, $m + n$.

p1. Is there any natural number n such that $n^2 + 5n + 1$ is divisible by $49$ ? Explain. p2. It is known that the parabola $y = ax^2 + bx + c$ passes through the points $(-3,4)$ and $(3,16)$, and does not cut the $x$-axis. Find all possible abscissa values ​​for the vertex point of the parabola. p3. It is known that $T.ABC$ is a regular triangular pyramid with side lengths of $2$ cm. The points $P, Q, R$, and $S$ are the centroids of triangles $ABC$, $TAB$, $TBC$ and $TCA$, respectively . Determine the volume of the triangular pyramid $P.QRS$ . p4. At an event invited $13$ special guests consisting of $ 8$ people men and $5$ women. Especially for all those special guests provided $13$ seats in a special row. If it is not expected two women sitting next to each other, determine the number of sitting positions possible for all those special guests. p5. A table of size $n$ rows and $n$ columns will be filled with numbers $ 1$ or $-1$ so that the product of all the numbers in each row and the product of all the numbers in each column is $-1$. How many different ways to fill the table?

As the graph, a pond is divided into 2n (n $\geq$ 5) parts. Two parts are called neighborhood if they have a common side or arc. Thus every part has three neighborhoods. Now there are 4n+1 frogs at the pond. If there are three or more frogs at one part, then three of the frogs of the part will jump to the three neighborhoods repsectively. Prove that for some time later, the frogs at the pond will uniformily distribute. That is, for any part either there are frogs at the part or there are frogs at the each of its neighborhoods. [img]http://www.mathlinks.ro/Forum/files/china2005_2_214.gif[/img]

Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

Let $O$ be the center of the circumcircle, and $AD$ be the angle bisector of the acute triangle $ABC$. The perpendicular drawn from point $D$ on the line $AO$ ​​intersects the line $AC$ at the point $P$. Prove that $AP = AB$.

A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.

Do either $(1)$ or $(2)$ $(1)$ $S$ is a thin spherical shell of constant thickness and density with total mass $M$ and center $O.$ $P$ is a point outside $S.$ Prove that the gravitational attraction of $S$ at $P$ is the same as the gravitational attraction of a point mass $M$ at $O.$ $(2)$ $K$ is the surface $z = xy$ in Euclidean $3-$space. Find all straight lines lying in $S$. Draw a diagram to illustrate them.

Line $\ell_1$ has equation $3x-2y=1$ and goes through $A=(-1,-2)$. Line $\ell_2$ has equation $y=1$ and meets line $\ell_1$ at point $B$. Line $\ell_3$ has positive slope, goes through point $A$, and meets $\ell_2$ at point $C$. The area of $\triangle ABC$ is $3$. What is the slope of $\ell_3$? $ \textbf{(A)}\ \frac{2}{3}\qquad\textbf{(B)}\ \frac{3}{4}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{4}{3}\qquad\textbf{(E)}\ \frac{3}{2} $

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers for which \[ a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right]. \] Prove that there exists a positive integer $m$ such that $a_m=4$ and $a_{m+1} \in\{3,4\}$. [b]Note.[/b] $[x]$ is the greatest integer not exceeding $x$.

A circle $O$ with radius of $R$ is drawn on a piece of paper. $A$ is a fixed point inside circle $O$, and $OA=a$. Fold the paper, so that a point $A'$ on the circle is coincident with $A$. For all such foldings, a kink mark is remained. Find the set of points on a certain kink mark.

Prove that for an arbitrary $\Delta ABC$ the following inequality holds: $\frac{l_a}{m_a}+\frac{l_b}{m_b}+\frac{l_c}{m_c} >1$, Where $l_a,l_b,l_c$ and $m_a,m_b,m_c$ are the lengths of the bisectors and medians through $A$, $B$, and $C$.

Let $ABC$ be an acute triangle and let $B'$ and $C'$ be the feet of the heights $B$ and $C$ of triangle $ABC$ respectively. Let $B_A'$ and $B_C'$ be reflections of $B'$ with respect to the lines $BC$ and $AB$, respectively. The circle $BB_A'B_C'$, centered in $O_B$, intersects the line $AB$ in $X_B$ for the second time. The points $C_A', C_B', O_C, X_C$ are defined analogously, by replacing the pair $(B, B')$ with the pair $(C, C')$. Show that $O_BX_B$ and $O_CX_C$ are parallel.

2 squares are painted in blue and 2 squares are painted in red on a $ 3\times 3$ board in such a way that two square with same color is neither at same row nor at same column. In how many different ways can these four squares be painted? $\textbf{(A)}\ 198 \qquad\textbf{(B)}\ 288 \qquad\textbf{(C)}\ 396 \qquad\textbf{(D)}\ 576 \qquad\textbf{(E)}\ 792$

Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$

The trapezium $[ABCD]$ has bases $[AB]$ and $[CD]$ (with $[AB]$ being the largest base). Knowing that $BC = 2 DA$ and that $\angle DAB + \angle ABC =120^o$ , determines the measure of $\angle DAB$.

Let $n$ and $k$ be given integers such that $n\ge k\ge 2$. Alice and Bob play a game on an $n$ by $n$ table with white cells. They take turns to pick a white cell and color it black. Alice moves first. The game ends as soon as there is at least one black cell in every $k$ by $k$ square after a player moves, who is declared the winner of the game. Who has the winning strategy?

Solve in positive integers the equation $$m^{\frac{1}{n}}+n^{\frac{1}{m}}=2+\frac{2}{mn(m+n)^{\frac{1}{m}+\frac{1}{n}}}.$$

The positive real numbers $a, b, c$ satisfy the equality $a + b + c = 1$. For every natural number $n$ find the minimal possible value of the expression $$E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}$$