MathDB

1

Part of 2016 China Team Selection Test

Problems(3)

Circumcircles of PDL,PEM,PFN meet at two points

Source: China Team Selection Test 2016 Test 2 Day 1 Q1

3/20/2016
PP is a point in the interior of acute triangle ABCABC. D,E,FD,E,F are the reflections of PP across BC,CA,ABBC,CA,AB respectively. Rays AP,BP,CPAP,BP,CP meet the circumcircle of △ABC\triangle ABC at L,M,NL,M,N respectively. Prove that the circumcircles of △PDL,△PEM,△PFN\triangle PDL,\triangle PEM,\triangle PFN meet at a point TT different from PP.
geometrycircumcirclegeometric transformationreflection
Cyclic hexagon prove equal segments

Source: China Team Selection Test 2016 Test 1 Day 1 Q1

3/16/2016
ABCDEFABCDEF is a cyclic hexagon with AB=BC=CD=DEAB=BC=CD=DE. KK is a point on segment AEAE satisfying ∠BKC=∠KFE,∠CKD=∠KFA\angle BKC=\angle KFE, \angle CKD = \angle KFA. Prove that KC=KFKC=KF.
geometry
The a-th root of 8

Source: China Team Selection Test 2016 Test 3 Day 1 Q1

3/25/2016
Let nn be an integer greater than 11, α\alpha is a real, 0<α<20<\alpha < 2, a1,…,an,c1,…,cna_1,\ldots ,a_n,c_1,\ldots ,c_n are all positive numbers. For y>0y>0, let f(y)=(∑ai≤yciai2)12+(∑ai>yciaiα)1α.f(y)=\left(\sum_{a_i\le y} c_ia_i^2\right)^{\frac{1}{2}}+\left(\sum_{a_i>y} c_ia_i^{\alpha} \right)^{\frac{1}{\alpha}}. If positive number xx satisfies x≥f(y)x\ge f(y) (for some yy), prove that f(x)≤81α⋅xf(x)\le 8^{\frac{1}{\alpha}}\cdot x.
inequalitiesalgebra