Each day $289$ students are divided into $17$ groups of $17$. No two students are ever in the same group more than once. What is the largest number of days that this can be done?

Two players play alternately. The first player is given a pair of positive integers $(x_1, y_1)$. Each player must replace the pair $(x_n, y_n)$ that he is given by a pair of non-negative integers $(x_{n+1}, y_{n+1})$ such that $x_{n+1} = min(x_n, y_n)$ and $y_{n+1} = max(x_n, y_n)- k\cdot x_{n+1}$ for some positive integer $k$. The first player to pass on a pair with $y_{n+1} = 0$ wins. Find for which values of $x_1/y_1$ the first player has a winning strategy.

For a positive integer $n$, we dene $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$. 1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer. 2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.

A river has a right-angle bend. Except at the bend, its banks are parallel lines of distance $a$ apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length $c$ and negligible width which can pass through the bend?

Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.

A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?

Let $x\in\mathbb{R}$. What is the maximum value of the following expression: $\sqrt{x-2018} + \sqrt{2020-x}$ ?

Let $a, b, c$ positive reals such that $a+b+c=1$. Prove that $$\min\{a(1-b),b(1-c),c(1-a)\}\leq \frac{1}{4}$$ $$\max\{a(1-b),b(1-c),c(1-a)\}\geq \frac{2}{9}$$

Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.

16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.

For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$. Find the largest possible value of $T_A$.

Define the maximal value of the ratio of a three-digit number to the sum of its digits.

Let the sum of positive numbers $x_1, x_2, ... , x_n$ be $1$. Let $s$ be the greatest of the numbers $$\left\{\frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, ..., \frac{x_n}{1+x_1+...+x_n}\right\}$$ What is the minimal possible $s$? What $x_i $correspond it?

Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.

For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$. Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)

In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$. What is the largest possible value of $\frac{R}{C}$? [i]Proposed by Paulius Aleknavičius, Lithuania, and Anghel David Andrei, Romania[/i]

Delete $100$ digits from the number $1234567891011... 9899100$ so that the remaining number were as big as possible.

In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).

Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\ | x - 2y + 3z| \le 6 \\ | x - 2y - 3z| \le 6 \\ | x + 2y + 3z| \le 6 \end{cases}$ Determine the greatest value of $M = |x| + |y| + |z|$.

The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon. (a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.) (b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter. (c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths. It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle. (d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon can be increased without changing the perimeter. (e) Prove that the polygon of maximum area must be equiangular, or have all angles equal. (f ) Prove that the polygon of maximum area is a regular polygon. PS. You had better use hide for answers.

For positive numbers $a, b, c$ holds $(a + c) (b^2 + a c) = 4a$. Determine the maximum value of $b + c$ and find all triplets of numbers $(a, b, c)$ for which expression takes this value

In two bowls there are in total ${N}$ balls, numbered from ${1}$ to ${N}$. One ball is moved from one of the bowls into the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount, ${x}$. Determine the largest possible value of ${x}$.

Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.

$A$ and $B$ wish to divide a cake into two pieces. Each wants the largest piece he can get. The cake is a triangular prism with the triangular faces horizontal. $A$ chooses a point $P$ on the top face. $B$ then chooses a vertical plane through the point $P$ to divide the cake. $B$ chooses which piece to take. Which point $P$ should $A $ choose in order to secure as large a slice as possible?

Given regular $1976$-gon. The midpoints of all the sides and diagonals are marked. What is the greatest number of the marked points lying on one circumference?