find all pairs of such that

Let be three non-negative real numbers satisfying
Prove that

2201. Proposed by Leonard Giugiuc, Drobeta-Turnu Severin, Romania. Find all real numbers such that
for all nonnegative real numbers , , and with .

Given an acute triangle ABC, where H and O are the orthocenter and circumcenter, respectively. Point K is the midpoint of segment AH, and ℓ is a line through O. Points P and Q are the projections of B and C onto ℓ. Prove that KP + KQ ≥BC

A board has some of its cells coloured red, blue, or green. Each cell is coloured with at most one colour, and some cells may remain uncoloured. Additionally, there is at least one cell of each colour. Two coloured cells are said to be friends if they have different colours and lie in the same row or in the same column. The following conditions are satisfied:

1. Each coloured cell has exactly three friends.
2. All three friends of any given coloured cell lie in the same row or in the same column.

Determine the maximum number of cells that can be coloured on the board.

Find all functions such that for any ,
Proposed by hyay

Find all functions such that any real numbers and satisfy
Proposed by EmilXM

Let be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like which .