International Mathematics Competition for University Students

July 31 – August 6 2017, Blagoevgrad, Bulgaria

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Ivan is Watching You

Day 1
    Problem 1
    Problem 2
    Problem 3
    Problem 4
    Problem 5

Day 2
    Problem 6
    Problem 7
    Problem 8
    Problem 9
    Problem 10

Results
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    Day 1 questions
    Day 1 solutions
    Day 2 questions
    Day 2 solutions
    Closing Ceremony
        Presentation

Official IMC site

Problem 3

3. For any positive integer $m$, denote by $P\left(m\right)$ the product of positive divisors of $m$ (e.g. $P(6)=36$). For every positive integer $n$ define the sequence $$ a_1(n)=n, \qquad a_{k+1}(n)=P(a_k(n)) \quad (k=1,2,\ldots,2016). $$

Determine whether for every set $S\subseteq\{1,2,\ldots,2017\}$, there exists a positive integer $n$ such that the following condition is satisfied:

   For every $k$ with $1\le k\le 2017$, the number $a_k(n)$ is a perfect square if and only if $k\in S$.

Proposed by: Matko Ljulj, University of Zagreb

Hint: Let $n$ be a power of $2$.

  Solution