MathDB

2

Part of 2014 China Team Selection Test

Problems(3)

China Team Selection Test 2014 TST 1 Day 1 Q2

Source: China Nanjing , 12 Mar 2014

3/12/2014
Let AA be a finite set of positive numbers , B={a+bc+d∣a,b,c,d∈A}B=\{\frac{a+b}{c+d} |a,b,c,d \in A \}. Show that: ∣B∣≥2∣A∣2−1\left | B \right | \ge 2\left | A \right |^2-1 , where ∣X∣|X| be the number of elements of the finite set XX. (High School Affiliated to Nanjing Normal University )
logarithmsanalytic geometrygraphing linesslopecombinatorics proposedcombinatoricsChina TST
Relation of no., sum, product of factors of n

Source: 2014 China TST 2 Day 1 Q2

3/20/2014
Given a fixed positive integer a≥9a\geq 9. Prove: There exist finitely many positive integers nn, satisfying: (1)τ(n)=a\tau (n)=a (2)n∣ϕ(n)+σ(n)n|\phi (n)+\sigma (n) Note: For positive integer nn, τ(n)\tau (n) is the number of positive divisors of nn, ϕ(n)\phi (n) is the number of positive integers ≤n\leq n and relatively prime with nn, σ(n)\sigma (n) is the sum of positive divisors of nn.
functioninductionnumber theoryrelatively primenumber theory proposed
triangle with colorued vertices on 101-gon

Source: 2014 China TST 3 Day 1 Q2

4/5/2014
Let A1A2...A101A_1A_2...A_{101} be a regular 101101-gon, and colour every vertex red or blue. Let NN be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the 101101-gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour. (1)(1) Find the largest possible value of NN. (2)(2) Find the number of ways to colour the vertices such that maximum NN is acheived. (Two colourings a different if for some AiA_i the colours are different on the two colouring schemes).
combinatorics proposedcombinatorics