MathDB

4

Part of 2012 Middle European Mathematical Olympiad

Problems(2)

Sequence and prime divisors

Source: Middle European Mathematical Olympiad 2012 - Individuals I-4

9/14/2012
The sequence {an}n≥0 \{ a_n \} _ { n \ge 0 } is defined by a0=2,a1=4 a_0 = 2 , a_1 = 4 and an+1=anan−12+an+an−1 a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} for all positive integers n n . Determine all prime numbers p p for which there exists a positive integer m m such that p p divides the number am−1 a_m - 1 .
number theoryprime numbersnumber theory proposed
Average value of function taken over all permutations

Source: Middle European Mathematical Olympiad 2012 - Team Compt. T-4

9/14/2012
Let p>2 p>2 be a prime number. For any permutation π=(π(1),π(2),⋯ ,π(p)) \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) of the set S={1,2,⋯ ,p} S = \{ 1, 2, \cdots , p \} , let f(π) f( \pi ) denote the number of multiples of p p among the following p p numbers: π(1),π(1)+π(2),⋯ ,π(1)+π(2)+⋯+π(p) \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p) Determine the average value of f(π) f( \pi) taken over all permutations π \pi of S S .
functionmodular arithmeticnumber theorycombinatorics proposedcombinatoricspermutation statistics