MathDB

2

Part of 1995 Turkey Team Selection Test

Problems(2)

Number of permutations such that \sigma(j) \geq j for 2 j

Source: Turkey IMO TST 1995 #2

7/8/2011
Let nn be a positive integer. Find the number of permutations σ\sigma of the set {1,2,...,n}\{1, 2, ..., n\} such that σ(j)≥j\sigma(j) \geq j holds for exactly two values of jj.
combinatorics unsolvedcombinatorics
Prove the two conditions are equivalent

Source: Turkey IMO TST 1995 #5

7/8/2011
Let n∈Nn\in\mathbb{N} be given. Prove that the following two conditions are equivalent:
 (\text{i})\: n|a^n-a for any positive integer aa;  (\text{ii})\: For any prime divisor pp of nn, p2∤np^2 \nmid n and p−1∣n−1p-1|n-1.
number theory unsolvednumber theory