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IMC2019: Day 1, Problem 1Problem 1. Evaluate the product ∞∏n=3(n3+3n)2n6−64. Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan Solution. Let an=(n3+3n)2n6−64. Notice that an=(n3+3n)2(n3−8)(n3+8)=n2(n2+3)2(n−2)(n2+2n+4)⋅(n+2)(n2−2n+4)=nn−2⋅nn+2⋅n2+3(n−1)2+3⋅n2+3(n+1)2+3. Hence, for N≥3 we have N∏n=3an=(N∏n=3nn−2)(N∏n=3nn+2)(N∏n=3n2+3(n−1)2+3)(N∏n=3n2+3(n+1)2+3)=N(N−1)1⋅2⋅3⋅4(N+1)(N+2)⋅N2+322+3⋅32+3(N+1)2+3=727⋅N(N−1)(N2+3)(N+1)(N+2)((N+1)2+3)=727⋅(1−1N)(1+3N2)(1+1N)(1+2N)((1+1N)2+3N2), so ∞∏n=3an=lim | |||||||||
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