MathDB

2

Part of 2016 China Team Selection Test

Problems(3)

Geometric inequality with 12 points

Source: China Team Selection Test 2016 Test 2 Day 1 Q2

3/20/2016
Find the smallest positive number λ\lambda, such that for any 1212 points on the plane P1,P2,…,P12P_1,P_2,\ldots,P_{12}(can overlap), if the distance between any two of them does not exceed 11, then ∑1≤i<j≤12∣PiPj∣2≤λ\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda.
geometryinequalitiesgeometric inequality
China Team Selection Test 2016 TST 1 Day 1 Q2

Source: China Changsha

3/15/2016
Find the smallest positive number λ\lambda , such that for any complex numbers z1,z2,z3∈{z∈C∣∣z∣<1}{z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\} ,if z1+z2+z3=0z_1+z_2+z_3=0, then ∣z1z2+z2z3+z3z1∣2+∣z1z2z3∣2<λ.\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .
inequalitiescomplex numbersChina TST
Colouring rational points

Source: China Team Selection Test 2016 Test 3 Day 1 Q2

3/25/2016
In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer nn, is there a way to use nn colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?
combinatoricsnumber theoryChina TSTChinaTST