MathDB

2

Part of 2011 India Regional Mathematical Olympiad

Problems(2)

Indian RMO 2011: Question 2

Source:

12/4/2011
Let (a1,a2,a3,...,a2011)(a_1,a_2,a_3,...,a_{2011}) be a permutation of the numbers 1,2,3,...,20111,2,3,...,2011. Show that there exist two numbers j,kj,k such that 1≤j<k≤20111\leq{j}<k\leq2011 and ∣aj−j∣=∣ak−k∣|a_j-j|=|a_k-k|
2n+1, 3n+1 perfect squares [RMO2-2011, India]

Source:

12/31/2011
Let nn be a positive integer such that 2n+12n+1 and 3n+13n+1 are both perfect squares. Show that 5n+35n+3 is a composite number.