MathDB

3

Part of 2011 China National Olympiad

Problems(2)

union of the subsets 2011 China national Olmypaid 3

Source:

1/17/2011
Let AA be a set consist of finite real numbers,A1,A2,⋯ ,AnA_1,A_2,\cdots,A_n be nonempty sets of AA, such that
(a) The sum of the elements of AA is 0,0,
(b) For all xi∈Ai(i=1,2,⋯ ,n)x_i \in A_i(i=1,2,\cdots,n),we have x1+x2+⋯+xn>0x_1+x_2+\cdots+x_n>0.
Prove that there exist 1≤k≤n,1\le k\le n, and 1≤i1<i2<⋯<ik≤n1\le i_1<i_2<\cdots<i_k\le n, such that ∣Ai1⋃Ai2⋃⋯⋃Aik∣<kn∣A∣.|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.
Where ∣X∣|X| denote the numbers of the elements in set XX.
ceiling functioninequalitiesintegrationcalculuscombinatorics proposedcombinatorics
infinite couple of a,b 2011 China national Olmypiad 6

Source:

1/16/2011
Let m,nm,n be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers (a,b)(a,b) such that a+b| am^a+bn^b ,  \gcd(a,b)=1.
modular arithmeticalgorithmnumber theoryprime numbersDivisibility