MathDB

2

Part of 2017 Middle European Mathematical Olympiad

Problems(2)

Inequality in three variables with sum -1

Source: MEMO 2017 T2

8/25/2017
Determine the smallest possible real constant CC such that the inequality ∣x3+y3+z3+1∣≤C∣x5+y5+z5+1∣|x^3 + y^3 + z^3 + 1| \leq C|x^5 + y^5 + z^5 + 1| holds for all real numbers x,y,zx, y, z satisfying x+y+z=−1x + y + z = -1.
algebrainequalities
Memorable labellings of a regular n-gon

Source: MEMO 2017 I2

8/25/2017
Let n≥3n \geq 3 be an integer. A labelling of the nn vertices, the nn sides and the interior of a regular nn-gon by 2n+12n + 1 distinct integers is called memorable if the following conditions hold: (a) Each side has a label that is the arithmetic mean of the labels of its endpoints. (b) The interior of the nn-gon has a label that is the arithmetic mean of the labels of all the vertices. Determine all integers n≥3n \geq 3 for which there exists a memorable labelling of a regular nn-gon consisting of 2n+12n + 1 consecutive integers.
combinatorics