1

Part of 2003 Turkey MO (2nd round)

Problems(2)

n cars participating in a rally

Source: Turkey NMO 2003 Problem 1

n\geq 2

cars are participating in a rally. The cars leave the start line at different times and arrive at the finish line at different times. During the entire rally each car takes over any other car at most once , the number of cars taken over by each car is different and each car is taken over by the same number of cars. Find all possible values of

n

inductioncombinatorics unsolvedcombinatorics

Turkey NMO 2003 Problem 4

Source: Turkey NMO 2003 Problem 4

2^{2n+1}+ 2^{n}+1=x^{k}

k\geq2

n

are positive integers. Find all possible values of

n

number theorygreatest common divisormodular arithmeticDiophantine equationnumber theory unsolved