MathDB

2

Part of 2014 China National Olympiad

Problems(2)

Difference of factors of integers

Source: China Mathematical Olympiad 2014 Q2

12/21/2013
For the integer n>1n>1, define D(n)={a−b∣ab=n,a>b>0,a,b∈N}D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}. Prove that for any integer k>1k>1, there exists pairwise distinct positive integers n1,n2,…,nkn_1,n_2,\ldots,n_k such that n1,…,nk>1n_1,\ldots,n_k>1 and ∣D(n1)∩D(n2)∩⋯∩D(nk)∣≥2|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2.
number theory proposednumber theory
relation between x and f(x)

Source: China Mathematical Olympiad 2014 Q5

12/22/2013
Let f:X→Xf:X\rightarrow X, where X={1,2,…,100}X=\{1,2,\ldots ,100\}, be a function satisfying: 1) f(x)≠xf(x)\neq x for all x=1,2,…,100x=1,2,\ldots,100; 2) for any subset AA of XX such that ∣A∣=40|A|=40, we have A∩f(A)≠∅A\cap f(A)\neq\emptyset. Find the minimum kk such that for any such function ff, there exist a subset BB of XX, where ∣B∣=k|B|=k, such that B∪f(B)=XB\cup f(B)=X.
functioncombinatorics proposedcombinatorics