MathDB

2

Part of 2010 South East Mathematical Olympiad

Problems(2)

2010 smo (2)

Source: 2010 China South East Mathematical Olympiad

7/18/2011
For any set A={a1,a2,⋯ ,am}A=\{a_1,a_2,\cdots,a_m\}, let P(A)=a1a2⋯amP(A)=a_1a_2\cdots a_m. Let n=(201099)n={2010\choose99}, and let A1,A2,⋯ ,AnA_1, A_2,\cdots,A_n be all 9999-element subsets of {1,2,⋯ ,2010}\{1,2,\cdots,2010\}. Prove that 2010∣∑i=1nP(Ai)2010|\sum^{n}_{i=1}P(A_i).
algebrapolynomialmodular arithmeticLaTeXnumber theorynumber theory unsolved
2010 smo (6)

Source: 2010 China South East Mathematical Olympiad

7/18/2011
Let N∗\mathbb{N}^* be the set of positive integers. Define a1=2a_1=2, and for n=1,2,…,n=1, 2, \ldots,an+1=min⁡{λ∣1a1+1a2+⋯1an+1λ<1,λ∈N∗} a_{n+1}=\min\{\lambda|\frac{1}{a_1}+\frac{1}{a_2}+\cdots\frac{1}{a_n}+\frac{1}{\lambda}<1,\lambda\in \mathbb{N}^*\} Prove that an+1=an2−an+1a_{n+1}=a_n^2-a_n+1 for n=1,2,…n=1,2,\ldots.
algebra unsolvedalgebra