7

Part of 2018 Tuymaada Olympiad

Problems(2)

Inequality with x, y, z >= 1

Source: Tuymaada 2018 Senior League/Problem 6, Junior League/Problem 7

Prove the inequality

(x^3+2y^2+3z)(4y^3+5z^2+6x)(7z^3+8x^2+9y)\geq720(xy+yz+xz)

x, y, z \geq 1

.
Proposed by K. Kokhas

inequalitiesalgebra

Sending non-intersecting teams to an olympiad

Source: Tuymaada 2018 Senior League/Problem 7

A school has three senior classes of

M

students each. Every student knows at least

\frac{3}{4}M

people in each of the other two classes. Prove that the school can send

M

non-intersecting teams to the olympiad so that each team consists of

3

students from different classes who know each other.
Proposed by C. Magyar, R. Martin