MathDB

2

Part of 2012 China Team Selection Test

Problems(6)

prove three points collinear

Source: 2012 China TST P2

3/15/2012
Given a scalene triangle ABCABC. Its incircle touches BC,AC,ABBC,AC,AB at D,E,FD,E,F respectvely. Let L,M,NL,M,N be the symmetric points of DD with EFEF,of EE with FDFD,of FF with DEDE,respectively. Line ALAL intersects BCBC at PP,line BMBM intersects CACA at QQ,line CNCN intersects ABAB at RR. Prove that P,Q,RP,Q,R are collinear.
geometrygeometry unsolved
finite good numbers is not divisible by $k$

Source: 2012 China TST - Quiz 1 - Day 2 - P5

3/15/2012
For a positive integer nn, denote by τ(n)\tau (n) the number of its positive divisors. For a positive integer nn, if τ(m)<τ(n)\tau (m) < \tau (n) for all m<nm < n, we call nn a good number. Prove that for any positive integer kk, there are only finitely many good numbers not divisible by kk.
number theory proposednumber theory
find the maximum

Source: 2012 China TST Test 2 p5

3/20/2012
Given two integers m,nm,n which are greater than 11. r,sr,s are two given positive real numbers such that r<sr<s. For all aij≥0a_{ij}\ge 0 which are not all zeroes,find the maximal value of the expression f=(∑j=1n(∑i=1maijs)rs)1r(∑i=1m)∑j=1naijr)sr)1s.f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.
functioninequalitiesvectorinequalities proposed
n-element set

Source: 2012 China TST Test 2 p2

3/19/2012
Prove that there exists a positive real number CC with the following property: for any integer n≥2n\ge 2 and any subset XX of the set {1,2,…,n}\{1,2,\ldots,n\} such that ∣X∣≥2|X|\ge 2, there exist x,y,z,w∈Xx,y,z,w \in X(not necessarily distinct) such that 0<∣xy−zw∣<Cα−40<|xy-zw|<C\alpha ^{-4} where α=∣X∣n\alpha =\frac{|X|}{n}.
pigeonhole principlefloor functionceiling functioninequalitiesfunctionalgebradifference of squares
exist k integers

Source: 2012 China TST,Test 3,Problem 2

3/25/2012
Given an integer k≥2k\ge 2. Prove that there exist kk pairwise distinct positive integers a1,a2,…,aka_1,a_2,\ldots,a_k such that for any non-negative integers b1,b2,…,bk,c1,c2,…,ckb_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k satisfying a1≤bi≤2ai,i=1,2,…,ka_1\le b_i\le 2a_i, i=1,2,\ldots,k and ∏i=1kbici<∏i=1kbi\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i, we have k∏i=1kbici<∏i=1kbi.k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.
inductioninequalitiesinequalities proposed
find all k

Source: 2012 China TST Test 3 p5

3/26/2012
Find all integers k≥3k\ge 3 with the following property: There exist integers m,nm,n such that 1<m<k1<m<k, 1<n<k1<n<k, gcd⁡(m,k)=gcd⁡(n,k)=1\gcd (m,k)=\gcd (n,k) =1, m+n>km+n>k and k∣(m−1)(n−1)k\mid (m-1)(n-1).
number theory proposednumber theory