2
Part of 1997 Vietnam National Olympiad
Problems(2)
sequences
Source: Vietnam NMO 1997, problem 2
9/5/2008
Let n be an integer which is greater than 1, not divisible by 1997.
Let a_m\equal{}m\plus{}\frac{mn}{1997} for all m=1,2,..,1996
b_m\equal{}m\plus{}\frac{1997m}{n} for all m=1,2,..,n-1
We arrange the terms of two sequence in the ascending order to form a new sequence c_1\le c_2\le ...\le c_{1995\plus{}n}
Prove that c_{k\plus{}1}\minus{}c_k<2 for all k=1,2,...,1994+n
inequalitiesnumber theoryrelatively primealgebra proposedalgebra
2^n | 19^k-97
Source: Vietnam NMO 1997, Problem 5
9/5/2008
Prove that for evey positive integer n, there exits a positive integer k such that 2^n | 19^k \minus{} 97
inductionnumber theory proposednumber theory