MathDB

4

Part of 2020 South East Mathematical Olympiad

Problems(2)

2019 CSMO Grade 10 Problem 4

Source: 2019 CSMO Grade 10 Problem 4

8/10/2020
Let a1,a2,…,a17a_1,a_2,\dots, a_{17} be a permutation of 1,2,…,171,2,\dots, 17 such that (a1−a2)(a2−a3)…(a17−a1)=n17(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17} .Find the maximum possible value of nn .
number theorymaximum valuecombinatorics proposed
China South East Mathematical Olympiad 2020 Grade11 Q4

Source: China Zhuji

8/5/2020
Let 0≤a1≤a2≤⋯≤an−1≤an0\leq a_1\leq a_2\leq \cdots\leq a_{n-1}\leq a_n and a1+a2+⋯+an=1.a_1+a_2+\cdots+a_n=1. Prove that: For any non-negative numbers x1,x2,⋯ ,xn;y1,y2,⋯ ,ynx_1,x_2,\cdots,x_n ; y_1, y_2,\cdots, y_n , have (∑i=1naixi−∏i=1nxiai)(∑i=1naiyi−∏i=1nyiai)≤an2(n∑i=1nxi∑i=1nyi−∑i=1nxi∑i=1nyi)2.\left(\sum_{i=1}^n a_ix_i - \prod_{i=1}^n x_i^{a_i}\right) \left(\sum_{i=1}^n a_iy_i - \prod_{i=1}^n y_i^{a_i}\right) \leq a_n^2\left(n\sqrt{\sum_{i=1}^n x_i\sum_{i=1}^n y_i} - \sum_{i=1}^n\sqrt{x_i} \sum_{i=1}^n\sqrt{y_i}\right)^2.
inequalitiesChinaalgebran-variable inequality