MathDB

1

Part of 2019 China Second Round Olympiad

Problems(2)

Surprisingly Easy Geo

Source: 2019 China Second Round P1

9/8/2019
In acute triangle △ABC\triangle ABC, MM is the midpoint of segment BCBC. Point PP lies in the interior of △ABC\triangle ABC such that APAP bisects ∠BAC\angle BAC. Line MPMP intersects the circumcircles of △ABP,△ACP\triangle ABP,\triangle ACP at D,ED,E respectively. Prove that if DE=MPDE=MP, then BC=2BPBC=2BP.
geometryChina second round
Inequality with 100 variables.

Source: 2019 China Second Round P1 (Paper B)

9/8/2019
Suppose that a1,a2,⋯ ,a100∈R+a_1,a_2,\cdots,a_{100}\in\mathbb{R}^+ such that ai≥a101−i (i=1,2,⋯ ,50).a_i\geq a_{101-i}\,(i=1,2,\cdots,50). Let xk=kak+1a1+a2+⋯+ak (k=1,2,⋯ ,99).x_k=\frac{ka_{k+1}}{a_1+a_2+\cdots+a_k}\,(k=1,2,\cdots,99). Prove that x1x22⋯x9999≤1.x_1x_2^2\cdots x_{99}^{99}\leq 1.
inequalities