MathDB

8

Part of 2018 Caucasus Mathematical Olympiad

Problems(2)

Easy but nice inequality

Source: III Caucasus Mathematical Olympiad

3/17/2018
Let a,b,ca, b, c be the lengths of sides of a triangle. Prove the inequality (a+b)ab+(a+c)ac+(b+c)bc≥(a+b+c)2/2.(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.
inequalities
intresting estimate, hard example

Source: III Caucasus Mathematical Olympiad

3/17/2018
In the cells of an 8×88\times 8 board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
combinatorics