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4

Part of 2022 Tuymaada Olympiad

Problems(2)

Tuymaada 2022

Source: Tuymaada 2022 Junior P-4

9/12/2022
Several goodgood points, several badbad points and several segments are drawn in the plane. Each segment connects a goodgood point and a badbad one; at most 100100 segments begin at each point. We have paint of 200200 colors. One half of each segment is painted with one of these colors, and the other half with another one. Is it always possible to do it so that every two segments with common end are painted with four different colors? (M. Qi, X. Zhang)
graph theoryColoring
6 variable inequality on 4th roots

Source: Tuymaada Olympiad 2022 Senior Category Problem 4

7/7/2022
For every positive a1,a2,…,a6a_1, a_2, \dots, a_6, prove the inequality a1a2+a3+a44+a2a3+a4+a54+⋯+a6a1+a2+a34≥2 \sqrt[4]{\frac{a_1}{a_2 + a_3 + a_4}} + \sqrt[4]{\frac{a_2}{a_3 + a_4 + a_5}} + \dots + \sqrt[4]{\frac{a_6}{a_1 + a_2 + a_3}} \ge 2
Inequalityinequalities