Let $p$ be a prime number, and let $a_1, a_2, \dots , a_p$ and $b_1, b_2, \dots , b_p$ be $2p$ (not necessarily distinct) integers chosen from the set $\{1, 2, \dots , p - 1\}$. Prove that there exist integers $i$ and $j$ such that $1 \le i < j \le p$ and $p$ divides $a_ib_j-a_jb_i$.

Let $P,Q,R$ be the polynomials with real or complex coefficients such that at least one of them is not constant. If $P^n+Q^n+R^n = 0$, prove that $n < 3.$

A chord $AB = a$ is drawn in a circle of radius $B$. A circle with center on line $AB$ passes through $A$ and intersects this circle a second time at point $C$. Let $M$ be an arbitrary point of the second circle. Straight lines $MA$ and $MC$ intersect the first circle a second time at points $P$ and $Q$. Find $PQ$.

Suppose three direction on the plane . We draw $ 11$ lines in each direction . Find maximum number of the points on the plane which are on three lines .

In triangle $ABC, O$ is the circumcenter, $H$ is the orthocenter. Construct the circumcircles of triangles $CHB, CHA$ and $AHB$, and let their centers be $A_1, B_1, C_1$, respectively. Prove that triangles $ABC$ and $A_1B_1C_1$ are congruent, and their nine-point circles coincide.

In the figure, points $A$, $B$, $C$ and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$ and segment $CD$ measures $4$. The circumferences woth centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length of segment $AD$? [img]https://cdn.artofproblemsolving.com/attachments/d/4/378243b9f4203e103af266e551eadccfc96adf.png[/img]

The positive integer $n$ is given and for all positive integers $k$, $1\leq k\leq n$, denote by $a_{kn}$ the number of all ordered sequences $(i_1,i_2,\ldots,i_k)$ of positive integers which verify the following two conditions: a) $1\leq i_1<i_2< \cdots i_k \leq n$; b) $i_{r+1}-i_r \equiv 1 \pmod 2$, for all $r \in\{1,2,\ldots,k-1\}$. Compute the number $a(n) = \sum\limits_{k=1}^n a_{kn}$. [i]Ioan Tomescu[/i]

In a simple graph, there are two disjoint set of vertices $A$ and $B$ where $A$ has $k$ and $B$ has $2016$ vertices. Four numbers are written to each vertex using the colors red, green, blue and black. There is no any edge at the beginning. For each vertex in $A$, we first choose a color and then draw all edges from this vertex to the vertices in $B$ having a larger number with the chosen color. It is known that for each vertex in $B$, the set of vertices in $A$ connected to this vertex are different. Find the minimal possible value of $k$.

An $n \times n$ board is given. In the top left corner cell there is a fox, whereas in the bottom left corner cell there is a rabbit. Every minute, the fox and the rabbit jump to a neighbouring cell at the same time. The fox can jump only to neighbouring cells that are below it or on its right, whereas the rabbit can only jump to the cells above it or in its right. They continue like this until they have no possible moves. The fox catches the rabbit if at a certain moment they are in the same cell, otherwise the rabbit gets away. Find all natural numbers $n$ for which the fox has a winning strategy to catch the rabbit. [i](Note: Two squares are considered neighbours if they have a common side.)[/i]

Let $n \ge k \ge 3$ be integers. Show that for every integer sequence $1 \le a_1 < a_2 < . . . < a_k \le n$ one can choose non-negative integers $b_1, b_2, . . . , b_k$, satisfying the following conditions: [list=i] [*] $0 \le b_i \le n$ for each $1 \le i \le k$, [*] all the positive $b_i$ are distinct, [*] the sums $a_i + b_i$, $1 \le i \le k$, form a permutation of the first $k$ terms of a non-constant arithmetic progression. [/list]

A rectangle with sides $ n$ and $p$ is divided into $np$ unit squares. Initially there are m unitary squares painted black and the remaining painted white. The following processoccurs repeatedly: if a unit square painted white has at minus two sides in common with squares painted black then Its color also turns black. Find the smallest integer $m$ that satisfies the property: there exists an initial position of $m$ black unit squares such that the entire $ n \times p$ rectangle is painted black when repeat the process a finite number of times.

Compute the number of nonnegative integer triples $(x,y,z)$ which satisfy $4x+2y+z\leq36$.

Let $p>1$. Show that there exists a constant $K_{p} >0$ such that for every $x,y\in \mathbb{R}$ with $|x|^{p}+|y|^{p}=2$, we have $$(x-y)^{2} \leq K_{p}(4-(x+y)^{2}).$$

A complete graph of $n$ vertices is a set of $n$ vertices and those vertices are connected in pairs by edges. Suppose the graph has $n$ vertices $A_1, A_2, ..., A_n$, the cycle is a set of edges of the form $A_{i_1}A_{i_2}, A_{i_2}A_{i_3},..., A_{i_m}A_{i_1}$ with $i_1, i_2, ..., i_m \in {1, 2, ..., n}$ double one different. We call $m$ the length of this cycle. Find the smallest positive integer$ n$ such that for every way of coloring all edges of a complete graph of $n$ vertices, each edge filled with one of three different colors, there is always a cycle of even length with the same color. PS. The same problem with another wording [url=https://artofproblemsolving.com/community/c6h151391p852296]here [/url].

If $x_1=5, x_2=401$, and \[ x_n=x_{n-2}-\frac 1{x_{n-1}} \] for every $3\leq n \leq m$, what is the largest value of $m$? $ \textbf{(A)}\ 406 \qquad\textbf{(B)}\ 2005 \qquad\textbf{(C)}\ 2006 \qquad\textbf{(D)}\ 2007 \qquad\textbf{(E)}\ \text{None of the above} $

Let $ABC$ be a triangle with side lengths $AB = 2011$, $BC = 2012$, $AC = 2013$. Create squares $S_1 =ABB'A''$, $S_2 = ACC''A'$ , and $S_3 = CBB''C'$ using the sides $AB$, $AC$, $BC$ respectively, so that the side $B'A''$ is on the opposite side of $AB$ from $C$, and so forth. Let square $S_4$ have side length $A''A' $, square $S_5$ have side length $C''C'$, and square $S_6$ have side length $B''B'$. Let $A(S_i)$ be the area of square $S_i$ . Compute $\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}$?

An $L$ is an arrangement of $3$ adjacent unit squares formed by deleting one unit square from a $2 \times 2$ square. a) How many $L$s can be placed on an $8 \times 8$ board (with no interior points overlapping)? b) Show that if any one square is deleted from a $1987 \times 1987$ board, then the remaining squares can be covered with $L$s (with no interior points overlapping).

The pairs $(a,b)$ such that $|a+bt+ t^2 |\leq 1$ for $0\leq t \leq 1$ fill a certain region in the plane. What is the area of this region?

A graph has $N$ vertices. An invisible hare sits in one of the vertices. A group of hunters tries to kill the hare. In each move all of them shoot simultaneously: each hunter shoots at a single vertex, they choose the target vertices cooperatively. If the hare was in one of the target vertices during a shoot, the hunt is finished. Otherwise the hare can stay in its vertex or jump to one of the neighboring vertices. The hunters know an algorithm that allows them to kill the hare in at most $N!$ moves. Prove that then there exists an algorithm that allows them to kill the hare in at most $2^N$ moves.

Fred and George are playing a game, in which Fred flips $2014$ coins and George flips $2015$ coins. Fred wins if he flips at least as many heads as George does, and George wins if he flips more heads than Fred does. Determine the probability that Fred wins.

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$, where $a$, $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that \[ f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} . \]

Let $n$ be a positive integer, $n\geq 4$. $n$ cards are arranged on a circle and the numbers $1$ or $-1$ are written on each of the cards. in a $question$ we may find out the product of the numbers on any $3$ cards. What is the minimum numbers if questions needed to find out the product of all $n$ numbers?

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

A person starting with $64$ cents and making $6$ bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is: $\textbf{(A)}\ \text{a loss of } 27 \qquad \textbf{(B)}\ \text{a gain of }27 \qquad \textbf{(C)}\ \text{a loss of }37 \qquad$ $ \textbf{(D)}\ \text{neither a gain nor a loss} \qquad \textbf{(E)}\ \text{a gain or a loss depending upon the order in which the wins and losses occur}$ Note: Due to the lack of $\LaTeX$ packages, the numbers in the answer choices are in cents ¢