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PrintTHE 68th ROMANIAN MATHEMATICAL OLYMPIAD
Romania number theory
Problem
Let .
a) Solve in positive integers the equation .
b) Prove that there exist infinitely many positive integers for which the equation has solutions in positive integers.
a) Solve in positive integers the equation .
b) Prove that there exist infinitely many positive integers for which the equation has solutions in positive integers.
Solution
a) Rewrite the equation as which is equivalent to , and the solutions are all pairs , with .
b) Observe that , , hence for , , the equation has the solution .
b) Observe that , , hence for , , the equation has the solution .
Final answer
a) All solutions are x = y with x, y positive integers. b) For every positive integer k, n = 11k + 3 has a solution with (x, y) = (6k + 1, 1), so infinitely many n of the form 11k + 3 work.
Techniques
Techniques: modulo, size analysis, order analysis, inequalitiesFractions