MathDB

2

Part of 2022 China Team Selection Test

Problems(4)

Mean-free subset

Source: 2022 China TST, Test 1, P2 (posting for better LaTeX)

3/24/2022
Let pp be a prime, AA is an infinite set of integers. Prove that there is a subset BB of AA with 2p−22p-2 elements, such that the arithmetic mean of any pairwise distinct pp elements in BB does not belong to AA.
combinatoricsnumber theoryprime numbers
Intersection of Simson lines lies on NPC

Source: 2022 China TST, Test 2, P2

3/28/2022
Given a non-right triangle ABCABC with BC>AC>ABBC>AC>AB. Two points P1≠P2P_1 \neq P_2 on the plane satisfy that, for i=1,2i=1,2, if APi,BPiAP_i, BP_i and CPiCP_i intersect the circumcircle of the triangle ABCABC at Di,EiD_i, E_i, and FiF_i, respectively, then DiEi⊥DiFiD_iE_i \perp D_iF_i and DiEi=DiFi≠0D_iE_i = D_iF_i \neq 0. Let the line P1P2P_1P_2 intersects the circumcircle of ABCABC at Q1Q_1 and Q2Q_2. The Simson lines of Q1Q_1, Q2Q_2 with respect to ABCABC intersect at WW.
Prove that WW lies on the nine-point circle of ABCABC.
geometrySimson lineNine Point Circle
Disjoint sequence implies "Beatty"

Source: 2022 China TST, Test 3 P2

4/30/2022
Two positive real numbers α,β\alpha, \beta satisfies that for any positive integers k1,k2k_1,k_2, it holds that ⌊k1α⌋≠⌊k2β⌋\lfloor k_1 \alpha \rfloor \neq \lfloor k_2 \beta \rfloor, where ⌊x⌋\lfloor x \rfloor denotes the largest integer less than or equal to xx. Prove that there exist positive integers m1,m2m_1,m_2 such that m1α+m2β=1\frac{m_1}{\alpha}+\frac{m_2}{\beta}=1.
floor functionnumber theoryalgebra
Concurrency in a quadrilateral implies equal length

Source: 2022 China TST, Test 4 P2

4/30/2022
Let ABCDABCD be a convex quadrilateral, the incenters of △ABC\triangle ABC and △ADC\triangle ADC are I,JI,J, respectively. It is known that AC,BD,IJAC,BD,IJ concurrent at a point PP. The line perpendicular to BDBD through PP intersects with the outer angle bisector of ∠BAD\angle BAD and the outer angle bisector ∠BCD\angle BCD at E,FE,F, respectively. Show that PE=PFPE=PF.
geometryincenterangle bisector