3

Part of 2008 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Find k where n doesn't divide greatest odd divisor of k^n+1

Source: XVII Olimpíada Matemática Rioplatense (2008)

Find all integers

k\ge 2

such that for all integers

n\ge 2

n

does not divide the greatest odd divisor of

k^n+1

number theory unsolvednumber theory

Dividing stones into two groups with almost equal weights

Source: XVII Olimpíada Matemática Rioplatense (2008)

Consider a collection of stones whose total weight is

65

pounds and each of whose stones is at most

w

pounds. Find the largest number

w

for which any such collection of stones can be divided into two groups whose total weights differ by at most one pound.
Note: The weights of the stones are not necessarily integers.

algebra unsolvedalgebra