In the numerical triangle $................1..............$ $...........1 ...1 ...1.........$ $......1... 2... 3 ... 2 ... 1....$ $.1...3...6...7...6...3...1$ $...............................$ each number is equal to the sum of the three nearest to it numbers from the row above it; if the number is at the beginning or at the end of a row then it is equal to the sum of its two nearest numbers or just to the nearest number above it (the lacking numbers above the given one are assumed to be zeros). Prove that each row, starting with the third one, contains an even number.

A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$. How many distinct $3$-letter words can be made from a subset of the letters $c, o, m, b, o$, where each letter in the list is used no more than the number of times it appears?

Find all pairs of $a$, $b$ of positive integers satisfying the equation $2a^2 = 3b^3$.

Prove that if a prime is the sum of four perfect squares then the product of two of these is equal to the product of the other two. [i]Gherghe Stoica[/i]

Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.

Petya and Vasya are playing on an $100\times 100$ board. Initially, all the cells of the board are white. With each of his moves, Petya paints one or more white cells standing on the same diagonal in black. With each of his moves, Vasya paints one or more white cells standing on the same column in black. Petya makes the first move. The one who can't make a move loses. Who has a winning strategy? [i]Proposed by M. Didin[/i]

For a given integer $n \ge 4$ we examine whether there exists a table with three rows and $n$ columns which can be filled by the numbers $1, 2,...,, 3n$ such that $\bullet$ each row totals to the same sum $z$ and $\bullet$ each column totals to the same sum $s$. Prove: (a) If $n$ is even, such a table does not exist. (b) If $n = 5$, such a table does exist. (Gerhard J. Woeginger)

[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$. [b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral. [b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms? [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set $\{V, W, X, Y, Z\}$. Using this correspondence, the cryptographer finds that three consecutive integers in increasing order are coded as $VYZ$, $VYX$, $VVW$, respectively. What is the base-10 expression for the integer coded as $XYZ$? $ \textbf{(A)}\ 48 \qquad\textbf{(B)}\ 71 \qquad\textbf{(C)}\ 82 \qquad\textbf{(D)}\ 108 \qquad\textbf{(E)}\ 113$

Is it possible to divide $\mathbb{N}$ into six disjoint sets $A_1, A_2, A_3, A_4, A_5, A_6$, such that $x,y,z$ are not in the same set if $x+2y=5z$?

A diagonal is drawn in an isosceles trapezoid. By the contour of each of the resulting two triangles creeps its own beetle. The velocities of the beetles are constant and identical. Beetles don't change directions around their contours, and along the diagonal of the trapezoid they crawl in different directions. Prove that for any starting positions of the beetles they will ever meet.

a) Four merchants want to travel from Athens to Rome by cart. On the same day, but different times they leave Athens and arrive on another day to Rome, but in reverse order. Every day, when the evening comes, each merchant enters the next inn on the way. When some merchants sleep in the same inn at night, then on the following day at dawn they leave in reverse order of arrival, because they can only park this way on the narrow streets next to the inns. They cannot overtake each other, their order only changes after a night spent together in the same inn. Eventually each merchant arrives in Rome while they sleep with every other merchant in the same inn exactly once. Is it possible, that the number of the inns they sleep in is even every night? b) Is it possible if there are $8$ merchants instead of $4$ and every other condition is the same?

Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $(a^n-1)(b^n-1)$ is not a perfect square. [i]Proposed by Mongolia[/i]

Show that among any 39 consecutive natural numbers, there is a number whose sum of the digits is devisible by 11.

There are 29 unit squares in the diagram below. A frog starts in one of the five (unit) squares on the top row. Each second, it hops either to the square directly below its current square (if that square exists), or to the square down one unit and left one unit of its current square (if that square exists), until it reaches the bottom. Before it reaches the bottom, it must make a hop every second. How many distinct paths (from the top row to the bottom row) can the frog take? [i]Ray Li.[/i]

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Over all integers $1\le n \le 100$, find the maximum value of $\phi(n^2+2n)-\phi(n^2)$. [i]Proposed by Vincent Huang[/i]

In a triangle $ABC$ with a right angle at $A$, $H$ is the foot of the altitude from $A$. Prove that the sum of the inradii of the triangles $ABC$, $ABH$, and $AHC$ is equal to $AH$.

Let $c$ be a given real number. Find all polynomials $P$ with real coefficients such that: $(x + 1)P(x - 1) - (x - 1)P(x) = c$ for all $x \in R$

Determine all positive integer $a$ such that the equation $2x^2 - 30x + a = 0$ has two prime roots, i.e. both roots are prime numbers.

Find all positive integers $n < 589$ for which $589$ divides $n^2 + n + 1$.

Let $I$ be the incenter of the triangle $ABC$. Let $A_1,A_2$ be two distinct points on the line $BC$, let $B_1,B_2$ be two distinct points on the line $CA$, and let $C_1,C_2$ be two distinct points on the line $BA$ such that $AI = A_1I = A_2I$ and $BI = B_1I = B_2I$ and $CI = C_1I = C_2I$. Prove that $A_1A_2+B_1B_2+C_1C_2 = p$ where $p$ denotes the perimeter of $ABC.$ Pierre.

Let $n \in \mathbb N$. Prove that for each factor $m \ge n$ of $n(n + 1)/2$, one can partition the set $\{1,2, 3,\cdots , n\}$ into disjoint subsets such that the sum of elements in each subset is equal to $m$.

The number of ordered pairs of integers $ (m,n)$ for which $ mn \ge 0$ and \[m^3 \plus{} n^3 \plus{} 99mn \equal{} 33^3\] is equal to $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 33\qquad \textbf{(D)}\ 35\qquad \textbf{(E)}\ 99$

The determinant \[\begin{vmatrix}3&-2&5\\ 7&1&-4\\ 5&2&3\end{vmatrix}\] has the same value as the determinant \[\begin{vmatrix}x&1+x&2+x\\ 3&0&1\\ 1&1&0\end{vmatrix}\] Find $x$.

The figure below is a $ 4 \times 4$ grid of points. [asy]unitsize(15); for ( int x = 1; x <= 4; ++x ) for ( int y = 1; y <= 4; ++y ) dot((x, y));[/asy]Each pair of horizontally adjacent or vertically adjacent points are distance 1 apart. In the plane of this grid, how many circles of radius 1 pass through exactly two of these grid points?