MathDB

1

Part of 2015 China Team Selection Test

Problems(4)

A circle through another's centre

Source: 2015 China TST 1 Day 1 Q1

3/14/2015
The circle Γ\Gamma through AA of triangle ABCABC meets sides AB,ACAB,AC at EE,FF respectively, and circumcircle of ABCABC at PP. Prove: Reflection of PP across EFEF is on BCBC if and only if Γ\Gamma passes through OO (the circumcentre of ABCABC).
geometryTST
Addition and subtraction of subsets

Source: 2015 China TST 2 Day 1 Q1

3/19/2015
For a positive integer nn, and a non empty subset AA of {1,2,...,2n}\{1,2,...,2n\}, call AA good if the set {u±v∣u,v∈A}\{u\pm v|u,v\in A\} does not contain the set {1,2,...,n}\{1,2,...,n\}. Find the smallest real number cc, such that for any positive integer nn, and any good subset AA of {1,2,...,2n}\{1,2,...,2n\}, ∣A∣≤cn|A|\leq cn.
combinatoricscombinatorics proposed
Concyclic with B,C

Source: China Team Selection Test 3 Day 1 P1

3/25/2015
△ABC\triangle{ABC} is isosceles with AB=AC>BCAB = AC >BC. Let DD be a point in its interior such that DA=DB+DCDA = DB+DC. Suppose that the perpendicular bisector of ABAB meets the external angle bisector of ∠ADB\angle{ADB} at PP, and let QQ be the intersection of the perpendicular bisector of ACAC and the external angle bisector of ∠ADC\angle{ADC}. Prove that B,C,P,QB,C,P,Q are concyclic.
geometryperpendicular bisectorangle bisectorgeometry proposed
China Team Selection Test 2015 TST 3 Day 2 Q1

Source: China Hangzhou

3/24/2015
Let x1,x2,⋯ ,xnx_1,x_2,\cdots,x_n (n≥2)(n\geq2) be a non-decreasing monotonous sequence of positive numbers such that x1,x22,⋯ ,xnnx_1,\frac{x_2}{2},\cdots,\frac{x_n}{n} is a non-increasing monotonous sequence .Prove that ∑i=1nxin(∏i=1nxi)1n≤n+12n!n \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}
inequalitiesinequalities proposedChina TST