1992年第9届巴尔干地区数学奥林匹克

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The 9th Balkan Mathematical Olympiad
1992年第9届巴尔干地区数学奥林匹克

Let a(n) = 3⁴ⁿ. For which n is (m^a(n)+6 - m^a(n)+4 - m⁵ + m³) always divisible by 1992 ?
Prove that (2n² + 3n + 1)ⁿ ≥ 6ⁿn! n! for all positive integers.
ABC is a triangle area 1. Take D on BC, E on CA, F on AB, so that AFDE is cyclic. Prove that : area DEF ≤ EF²/(4 AD²).
For each n > 2 find the smallest f(n) such that any subset of {1, 2, 3, ... , n} with f(n) elements must have three which are relatively prime (in pairs).

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