How to solve problems in elementary number theory?

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I have studied and solved almost all of Elementary Number Theory by David M. Burton. Yet, tough problems in NT from Olympiads seen unapproachable to me. What should I do? What should I study? I feel so frustrated! Please help me! Thank you! :)

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This Kind of problems need something more, a lot of special training, there are some trics and constructions, it's no easy to everyone. Somethimes I get stuck for 3-4 days in a problem, somethimes i can't solve, there are a problem that i can't make a elegant solution: https://math.stackexchange.com/questions/1225548/maxa-1-in-x-n-n-in-mathbbn