Number Theoretic approach please!

Number Theory Level 4

Denote by $\mathbb N$ the set of all positive integers. A function $f : \mathbb N \rightarrow \mathbb N$ is such that for all positive integers $m$ and $n$ , the integer $f(m) + f(n) - mn$ is non-zero and divides $mf(m) + nf(n)$ .

Then find the value of $f(5)$ .

The answer is 25.

1 solution

Jesse Nieminen
Aug 12, 2017

$f(m) + f(n) - mn \mid mf(m) + nf(n) \implies 2f(n) - n^2 \mid 2nf(n) \implies 2f(n) - n^2 \mid n^3 \implies 2f(1) - 1 \mid 1 \implies f(1) = 1$

$f(m) + f(n) - mn \mid mf(m) + nf(n) \implies f(n) - n + 1 \mid nf(n) + 1 \implies f(n) - n + 1 \mid n^2 - n + 1 \implies f(5) - 4 \mid 21 \\ \implies f(5) \in \{7, 11, 25\}$

$2f(n) - n^2 \mid n^3 \implies 2f(5) - 25\mid125 \\ \implies f(5) \in \{13, 15, 25, 50\}$

Hence, $f(5) = \boxed{25}$ .

@Jesse Nieminen ,

This is an IMO 2016 shortlisted problem.

Though i have upvoted it, can you elaborate it more perfectly?

Also you have to prove that $f(n) = n^2$ for every integer., is the only solution.

Sorry, but this is not the elegant solution.

Priyanshu Mishra - 3 years, 9 months ago

For the question that you asked, this is a great solution. It shows how one can isolate the relevant facts to determine the value of $f(5)$ .

Calvin Lin Staff - 3 years, 9 months ago

Number Theoretic approach please!

The answer is 25.

1 solution

0 pending reports