Question 5
Let $k>1$ be an integer. For each integer $n>1$, define
$$f(n)=kn\prod_{j=1}^r\left(1-\frac1{p_j}\right),$$
where $p_1,\ldots,p_r$ are all the distinct prime divisors of $n$. Determine all $k$ for which, for every integer $a>1$, the sequence $x_0=a$, $x_{m+1}=f(x_m)$ ($m=0,1,2,\ldots$) is bounded.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Importer multipliers: correct 1, incorrect 0, blank 0.
$$f(n)=kn\prod_{j=1}^r\left(1-\frac1{p_j}\right),$$
where $p_1,\ldots,p_r$ are all the distinct prime divisors of $n$. Determine all $k$ for which, for every integer $a>1$, the sequence $x_0=a$, $x_{m+1}=f(x_m)$ ($m=0,1,2,\ldots$) is bounded.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Importer multipliers: correct 1, incorrect 0, blank 0.