, ,
Prove that at least one of , , is not greater than .

I assume that all are greater than it, but can't go more.

Let and Prove that

Let and be points on a plane such that , where is a positive integer. Let be the set of all points such that , where is a real number. The path that traces is continuous, and the value of is minimized. Prove that is rational for all positive integers

Suppose and are the incenter and the -excenter of triangle , respectively.
Let be the midpoint of arc on the circumcircle, and be the foot of the
perpendicular from to . The line intersects the circumcircle again at . For
any point on the circumcircle of triangle , let intersect at . Prove
that are concyclic.

On squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.

Function satisfies
for any non-negative integers and . Determine the number of possible sets of integers .

Let be a function from the Euclidean plane to the real numbers such that for any acute triangle with circumcenter , centroid and orthocenter . Prove that is constant.

Given be the positive real number. Prove that

Find all functions , such that for all positive reals , the following is true:


Kazi Aryan Amin

Alice and Bob are playing game on an grid. Alice goes first, and they take turns drawing a black point from the coordinate set
There is a constraint that the distance between any two black points cannot be an integer. The player who cannot draw a black point loses. Find all integers such that Alice has a winning strategy.

Proposed by chengbilly

A unit square is removed from the corner of an grid, where . Prove that the remainder can be covered by copies of the figures of or unit squares depicted in the drawing below.
IMAGE

Note: Every square must be covered once and figures must not go over the bounds of the grid.

Determine all positive integers for which there exists a positive integer with the property that is divisible by and is divisible by .

Let denote the set of real numbers. Find all functions such that

for all .