Let and Prove that
Where

Let and . Prove that
Where

Following inequality is given:
Find the range of values that can be taken by :



Where are positive reals.

Let be an integer. In a configuration of an board, each of the cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of , the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey

Consider an acute triangle . Let and be the feet of the altitudes from to and from to respectively.

Define and as the reflections of across lines and , respectively. Let be the circumcircle of . Denote by the second intersection of line with , and by the intersection of ray with .

If is the circumcenter of , prove that , , and are collinear if and only if quadrilateral can be inscribed within a circle.

Kritesh manages traffic on a grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the grid.

Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove cars from the grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.

What is the minimum value of that guarantees that Kritesh's job is possible?

Let the sequence be defined by
Prove that
for all positive integers .