MathDB

2

Part of 2000 China National Olympiad

Problems(2)

Closed form expression for f_n for sequence a_n

Source: Chinese Mathematical Olympiad 2000 Problem 2

8/18/2013
A sequence (an)(a_n) is defined recursively by a1=0,a2=1a_1=0, a_2=1 and for n≥3n\ge 3, an=12nan−1+12n(n−1)an−2+(−1)n(1−n2).a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right). Find a closed-form expression for fn=an+2(n1)an−1+3(n2)an−2+…+(n−1)(nn−2)a2+n(nn−1)a1f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1.
algebra unsolvedalgebra
Find all n that can be expressed in such a way

Source: Chinese Mathematical Olympiad 2000 Problem 5

8/20/2013
Find all positive integers nn such that there exists integers n1,…,nk≥3n_1,\ldots,n_k\ge 3, for some integer kk, satisfying n=n1n2⋯nk=212k(n1−1)⋯(nk−1)−1.n=n_1n_2\cdots n_k=2^{\frac{1}{2^k}(n_1-1)\cdots (n_k-1)}-1.
number theory unsolvednumber theory