Let $n\ge 2$ be a natural number. An $n \times n$ grid is drawn on a blackboard and each field with one of the numbers $-1$ or $+1$ labeled. Then the $n$ row and also the $n$ column sums calculated and the sum $S_n$ of all these $2n$ sums determined. (a) Show that for no odd number $n$ there is a label with $S_n = 0$. (b) Show that if $n$ is an even number, there are at least six different labels with $S_n = 0$.

The inscribed circle of the triangle $A_1A_2A_3$ touches the sides $A_2A_3,A_3A_1,A_1A_2$ at points $S_1,S_2,S_3$, respectively. Let $O_1,O_2,O_3$ be the centres of the inscribed circles of triangles $A_1S_2S_3, A_2S_3S_1,A_3S_1S_2$, respectively. Prove that the straight lines $O_1S_1,O_2S_2,O_3S_3$ intersect at one point.

A function $f$ is given by $f(x) = x^2 - 2x$ . Prove that there exists a number a which satisfies $f(f(a)) = a$ without satisfying $f(a) = a$ .

The following grid represents a mountain range; the number in each cell represents the height of the mountain located there. Moving from a mountain of height $a$ to a mountain of height $b$ takes $(b - a)^2$ time. Suppose that you start on the mountain of height $1$ and that you can move up, down, left, or right to get from one mountain to the next. What is the minimum amount of time you need to get to the mountain of height $49$? [img]https://cdn.artofproblemsolving.com/attachments/0/6/10b07a2b2ae4ba750cfffc3dc678880333c2de.png[/img]

Find all functions $f: R \to R$, such that equality $f (xf (y) - yf (x)) = f (xy) - xy$ holds for all $x, y \in R$.

A quadrilateral $ABCD$ is inscribed in a circle with center $O$. It's diagonals meet at $M$.The circumcircle of $ABM$ intersects the sides $AD$ and $BC$ at $N$ and $K$ respectively. Prove that areas of $NOMD$ and $KOMC$ are equal.

There are 2023 natural written in a row. The first number is 12, and each number starting from the third is equal to the product of the previous two numbers, or to the previous number increased by 4. What is the largest number of perfect squares that can be among the 2023 numbers? [i]Based on the British Mathematical Olympiad[/i]

Points on a square with side length $ c$ are either painted blue or red. Find the smallest possible value of $ c$ such that how the points are painted, there exist two points with same color having a distance not less than $ \sqrt {5}$. $\textbf{(A)}\ \frac {\sqrt {10} }{2} \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \sqrt {5} \qquad\textbf{(D)}\ 2\sqrt {2} \qquad\textbf{(E)}\ \text{None}$

On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ?

Find all the pairs positive integers $(x, y)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}$ , where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$.

A recipe that makes $ 5$ servings of hot chocolate requires $ 2$ squares of chocolate, $ \frac{1}{4}$ cup sugar, $ 1$ cup water and $ 4$ cups milk. Jordan has $ 5$ squares of chocolate, $ 2$ cups of sugar, lots of water and $ 7$ cups of milk. If she maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate she can make? $ \textbf{(A)}\ 5 \frac18 \qquad \textbf{(B)}\ 6\frac14 \qquad \textbf{(C)}\ 7\frac12 \qquad \textbf{(D)}\ 8 \frac34 \qquad \textbf{(E)}\ 9\frac78$

Positive integers are colored in black and white. We know that the sum of two numbers of different colors is always black, and that there are infinitely many numbers that are white. Prove that the sum and product of two white numbers are also white numbers.

Let $a$ and $b$ be relatively prime positive integers such that \[\dfrac ab=\dfrac1{2^1}+\dfrac2{3^2}+\dfrac3{2^3}+\dfrac4{3^4}+\dfrac5{2^5}+\dfrac6{3^6}+\cdots,\] where the numerators always increase by $1$, and the denominators alternate between powers of $2$ and $3$, with exponents also increasing by $1$ for each subsequent term. Compute $a+b$.

In $\triangle ABC$, $c-a$ is equal to height on side $AC$. Then, the value of $\sin\frac{C-A}{2}+\cos\frac{C+A}{2}$ is $\text{(A)}1\qquad\text{(B)}\frac{1}{2}\qquad\text{(C)}\frac{1}{3}\qquad\text{(D)}-1$

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?

Prove that for all integers $ a > 1$ and $ b > 1$ there exists a function $ f$ from the positive integers to the positive integers such that $ f(a\cdot f(n)) \equal{} b\cdot n$ for all $ n$ positive integer.

In a triangle $ABC$, let $H$ be the point of intersection of altitudes, $I$ the center of incircle, $O$ the center of circumcircle, $K$ the point where the incircle touches $BC$. Given that $IO$ is parallel to $BC$, prove that $AO$ is parallel to $HK$.

The increasing sequence $2,3,5,6,7,10,11,\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence.

Let circles $ \Gamma $ and $ \omega $ are circumcircle and incircle of the triangle $ABC$, the incircle touches sides $BC,CA,AB$ at the points $A_1,B_1,C_1$. Let $A_2$ and $B_2$ lies the lines $A_1I$ and $B_1I$ ($A_1$ and $A_2$ lies different sides from $I$, $B_1$ and $B_2$ lies different sides from $I$) such that $IA_2=IB_2=R$. Prove that : (a) $AA_2=BB_2=IO$; (b) The lines $AA_2$ and $BB_2$ intersect on the circle $ \Gamma ;$

Opposite sides of a regular hexagon are $12$ inches apart. The length of each side, in inches, is $\textbf{(A) }7.5\qquad\textbf{(B) }6\sqrt{2}\qquad\textbf{(C) }5\sqrt{2}\qquad\textbf{(D) }\frac{9}{2}\sqrt{3}\qquad \textbf{(E) }4\sqrt{3}$

If two points are picked randomly on the perimeter of a square, what is the probability that the distance between those points is less than the side length of the square?

We consider the following triangular array \[ \begin{array}{cccccccc} 0 & 1 & 1 & 2 & 3 & 5 & 8 & \ldots \\ \ & 0 & 1 & 1 & 2 & 3 & 5 & \ldots \\ \ & \ & 2 & 3 & 5 & 8 & 13 & \ldots \\ \ & \ & \ & 4 & 7 & 11 & 18 & \ldots \\ \ & \ & \ & \ & 12 & 19 & 31 & \ldots \\ \end{array} \] which is defined by the conditions i) on the first two lines, each element, starting with the third one, is the sum of the preceding two elements; ii) on the other lines each element is the sum of the two numbers found on the same column above it. a) Prove that all the lines satisfy the first condition i); b) Let $a,b,c,d$ be the first elements of 4 consecutive lines in the array. Find $d$ as a function of $a,b,c$.

Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$ are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$

A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line. (a) Do there exist two congruent simple $7$-gons in the plane such that all the seven vertices of one of the $7$-gons are the vertices of the other one and yet these two $7$-gons have no common sides? (b) Do there exist three such $7$-gons? (V Proizvolov)