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International Mathematics Competition
for University Students
2019

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IMC 2025
Information
  Results
  Problems & Solutions
 

IMC2019: Day 1, Problem 1

Problem 1. Evaluate the product

n=3(n3+3n)2n664.

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)2n664.

Notice that

an=(n3+3n)2(n38)(n3+8)=n2(n2+3)2(n2)(n2+2n+4)(n+2)(n22n+4)=nn2nn+2n2+3(n1)2+3n2+3(n+1)2+3.

Hence, for N3 we have

Nn=3an=(Nn=3nn2)(Nn=3nn+2)(Nn=3n2+3(n1)2+3)(Nn=3n2+3(n+1)2+3)=N(N1)1234(N+1)(N+2)N2+322+332+3(N+1)2+3=727N(N1)(N2+3)(N+1)(N+2)((N+1)2+3)=727(11N)(1+3N2)(1+1N)(1+2N)((1+1N)2+3N2),

so

n=3an=lim

IMC
2019

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