Select Page

# Understand the problem

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$for all positive integers $a,b$.

APMO 2019

Number Theory
Medium
##### Suggested Book
An Excursion in Mathematics

Do you really need a hint? Try it first!

Considering prime numbers might be a good idea.
Show that, for any prime $p$, $f(p)|p^2$. Hence or otherwise show that $f(p)=p$.
If $gcd(p,b)=1$ for a prime $p$, then show that $p+b|p+f(b)$. This is equivalent to $p+b|f(b)-b$. Note that the RHS is independent of $p$.
As in $p+b|f(b)-b$ the RHS is fixed and the LHS is arbitrarily large, the RHS has to be zero. Hence $f(b)=b$ for every natural number $b$.

# Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

# Similar Problems

## RMO 2019 (Maharashtra Goa) Adding GCDs

Can you add GCDs? This problem from RMO 2019 (Maharashtra region) has a beautiful solution. We also give some bonus questions for you to try.

## Number Theory, Ireland MO 2018, Problem 9

This problem in number theory is an elegant applications of the ideas of quadratic and cubic residues of a number. Try with our sequential hints.

## Number Theory, France IMO TST 2012, Problem 3

This problem is an advanced number theory problem using the ideas of lifting the exponents. Try with our sequential hints.

## Algebra, Austria MO 2016, Problem 4

This algebra problem is an elegant application of culminating the ideas of polynomials to give a simple proof of an inequality. Try with our sequential hints.

## Number Theory, Cyprus IMO TST 2018, Problem 1

This problem is a beautiful and simple application of the ideas of inequality and bounds in number theory. Try with our sequential hints.

## Number Theory, South Africa 2019, Problem 6

This problem in number theory is an elegant applciations of the modulo technique used in the diophantine equations. Try with our sequential hints

## Number Theory, Korea Junior MO 2015, Problem 7

This problem in number theory is an elegant application of the ideas of the proof of infinitude of primes from Korea. Try with our sequential hints.

## Inequality, Israel MO 2018, Problem 3

This problem is a basic application of triangle inequality along with getting to manipulate the modulus function efficently. Try with our sequential hints.

## Number Theory, Greece MO 2019, Problem 3

This problem is a beautiful application of prime factorization theorem, and reveal how important it is. Try with our sequential hints.

## Algebra, Germany MO 2019, Problem 6

This problem is a beautiful application of algebraic manipulations, ideas of symmetry, and vieta’s formula in polynomials. Try with our sequential hints.