MathDB

1

Part of 2012 China Team Selection Test

Problems(6)

Complex numbers

Source: 2012 China TST Quiz 1 Day 1 P1

3/14/2012
Complex numbers xi,yi{x_i},{y_i} satisfy ∣xi∣=∣yi∣=1\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1 for i=1,2,…,ni=1,2,\ldots ,n. Let x=1n∑i=1nxix=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}, y=1n∑i=1nyiy=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}} and zi=xyi+yxi−xiyiz_i=x{y_i}+y{x_i}-{x_i}{y_i}. Prove that ∑i=1n∣zi∣⩽n\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n.
inequalitiescomplex numberstriangle inequalityalgebra proposedalgebra
Centroid is a fixed point

Source: 2012 China TST - Quiz 1 - Day 2 - P4

3/15/2012
Given two circles ω1,ω2{\omega _1},{\omega _2}, SS denotes all ΔABC\Delta ABC satisfies that ω1{\omega _1} is the circumcircle of ΔABC\Delta ABC, ω2{\omega _2} is the AA- excircle of ΔABC\Delta ABC , ω2{\omega _2} touches BC,CA,ABBC,CA,AB at D,E,FD,E,F. SS is not empty, prove that the centroid of ΔDEF\Delta DEF is a fixed point.
geometrycircumcirclegeometry proposedInversion
t-group and central point

Source: 2012 China TST Test 2 p1

3/19/2012
In a simple graph GG, we call tt pairwise adjacent vertices a tt-clique. If a vertex is connected with all other vertices in the graph, we call it a central vertex. Given are two integers n,kn,k such that 32≤12n<k<n\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n. Let GG be a graph on nn vertices such that (1) GG does not contain a (k+1)(k+1)-clique; (2) if we add an arbitrary edge to GG, that creates a (k+1)(k+1)-clique. Find the least possible number of central vertices in GG.
algorithminductioncombinatorics proposedcombinatorics
a finite number of n-tuples

Source: 2012 China TST Test 2 p4

3/20/2012
Given an integer n≥2n\ge 2. Prove that there only exist a finite number of n-tuples of positive integers (a1,a2,…,an)(a_1,a_2,\ldots,a_n) which simultaneously satisfy the following three conditions:
[*] a1>a2>…>ana_1>a_2>\ldots>a_n; [*] gcd⁡(a1,a2,…,an)=1\gcd (a_1,a_2,\ldots,a_n)=1; [*] a1=∑i=1ngcd⁡(ai,ai+1)a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1}),where an+1=a1a_{n+1}=a_1.
inequalitiesinductionnumber theory proposednumber theory
Prove that H, O and D are collinear

Source: Chinese TST, Test 3, Problem 1 2012

3/25/2012
In an acute-angled ABCABC, ∠A>60∘\angle A>60^{\circ}, HH is its orthocenter. M,NM,N are two points on AB,ACAB,AC respectively, such that ∠HMB=∠HNC=60∘\angle HMB=\angle HNC=60^{\circ}. Let OO be the circumcenter of triangle HMNHMN. DD is a point on the same side with AA of BCBC such that △DBC\triangle DBC is an equilateral triangle. Prove that H,O,DH,O,D are collinear.
geometrycircumcircletrigonometryChina
ab+1 is a perfect square

Source: 2012 China TST Test 3 p4

3/26/2012
Given an integer n≥4n\ge 4. S={1,2,…,n}S=\{1,2,\ldots,n\}. A,BA,B are two subsets of SS such that for every pair of (a,b),a∈A,b∈B,ab+1(a,b),a\in A,b\in B, ab+1 is a perfect square. Prove that min⁡{∣A∣,∣B∣}≤log⁡2n.\min \{|A|,|B|\}\le\log _2n.
logarithmsfloor functionfunctionnumber theory proposednumber theory