MathDB

3

Part of 2019 China Team Selection Test

Problems(4)

Sum of distances power k is constant on unit circle

Source: 2019 China TST Day 1 Q3

3/5/2019
Find all positive integer nn, such that there exists nn points P1,…,PnP_1,\ldots,P_n on the unit circle , satisfying the condition that for any point MM on the unit circle, ∑i=1nMPik\sum_{i=1}^n MP_i^k is a fixed value for \\a) k=2018k=2018 \\b) k=2019k=2019.
algebra
2019 China TST 2 Day 1 Q3

Source: Mar , 2019

3/11/2019
Let nn be a given even number, a1,a2,⋯ ,ana_1,a_2,\cdots,a_n be non-negative real numbers such that a1+a2+⋯+an=1.a_1+a_2+\cdots+a_n=1. Find the maximum possible value of ∑1≤i<j≤nmin⁡{(i−j)2,(n+i−j)2}aiaj.\sum_{1\le i<j\le n}\min\{(i-j)^2,(n+i-j)^2\}a_ia_j .
inequalitiesChinaChina TST
Bijection exist?

Source: 2019 China TST Test 3 P3

3/23/2019
Does there exist a bijection f:N+→N+f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}, such that there exist a positive integer kk, and it's possible to have each positive integer colored by one of kk chosen colors, such that for any x≠yx \neq y , f(x)+yf(x)+y and f(y)+xf(y)+x are not the same color?
Coloringbijectioncombinatorics
20 triangles have intersection

Source: 2019 China TST Test 4 P3

3/29/2019
6060 points lie on the plane, such that no three points are collinear. Prove that one can divide the points into 2020 groups, with 33 points in each group, such that the triangles ( 2020 in total) consist of three points in a group have a non-empty intersection.
combinatoricsgeometrycombinatorial geometry