Let be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of . It is known that the graph of contains exactly chords, one of which has length 2025. Find the minimum possible value of .

Let and be monic quadratic trinomials, and let and be the vertices of the parabolas and , respectively. Let denote the minimum value of the function . It is known that the differences and are equal positive numbers. Find the angle between the line and the -axis.

The numbers are written in a row in that exact order. Igor and Ruslan take turns inserting the signs between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by , Igor wins; otherwise, Ruslan wins. Which player has a winning strategy regardless of the opponent’s moves?

Let be real numbers such that Find the maximum value of the expression

A quadrilateral with no parallel sides is inscribed in a circle . Circles are inscribed in triangles , respectively. Common external tangents are drawn between and , and , and , and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle . Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.

ones are written in a circle. Petya and Vasya take turns making moves each. In each move, Petya chooses 9 consecutive numbers and decreases each by . Vasya chooses consecutive numbers and increases each by . They alternate turns, starting with Petya. Prove that Vasya can act in such a way that after each of his moves, there are always at least five positive numbers, regardless of how Petya plays.

A competition consists of sports, each awarding one gold medal to a winner. athletes participate, each in all sports. There are also experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to . An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number such that the experts can make their predictions so that at least of them are guaranteed to be competent regardless of the outcome?

What is the smallest value of such that for any polynomial of degree with real coefficients, there exists a polynomial of degree at most with real coefficients such that the graphs of and intersect at exactly points?

Let positive real numbers such that . Prove that :

Let be an isosceles triangle with and circumcircle . The line through perpendicular to is denoted by . Furthermore, let be any point on . The circle with center and radius intersects once more at point and the circumcircle once more at point . Prove that the points and lie on a straight line.

(Karl Czakler)