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

## Lines and Angles | PRMO 2019 | Question 7

Try this beautiful problem from the Pre-RMO, 2019 based on Lines and Angles. You may use sequential hints to solve the problem.

## Logarithm and Equations | AIME I, 2012 | Question 9

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 based on Logarithm and Equations.

## Cross section of solids and volumes | AIME I 2012 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Cross section of solids and volumes.

## Angles of Star | AMC 8, 2000 | Problem 24

Try this beautiful problem from GeometryAMC-8, 2000 ,Problem-24, based triangle. You may use sequential hints to solve the problem.

## Unit digit | Algebra | AMC 8, 2014 | Problem 22

Try this beautiful problem from Algebra about unit digit from AMC-8, 2014. You may use sequential hints to solve the problem.

## Problem based on Integer | PRMO-2018 | Problem 6

Try this beautiful problem from Algebra based on Quadratic equation from PRMO 8, 2018. You may use sequential hints to solve the problem.

## Number counting | ISI-B.stat Entrance | Objective from TOMATO

Try this beautiful problem Based on Number counting .You may use sequential hints to solve the problem.

## Area of a Triangle | AMC-8, 2000 | Problem 25

Try this beautiful problem from Geometry: Area of the triangle from AMC-8, 2000, Problem-25. You may use sequential hints to solve the problem.

## Trapezium | Geometry | PRMO-2018 | Problem 5

Try this beautiful problem from Geometry based on Trapezium from PRMO , 2018. You may use sequential hints to solve the problem.

## Probability Problem | AMC 8, 2016 | Problem no. 21

Try this beautiful problem from Probability from AMC-8, 2016 Problem 21. You may use sequential hints to solve the problem.