# Understand the problem

Find all functions such that

holds for all .

holds for all .

##### Source of the problem

Benelux MO 2013

##### Topic

Functional Equations

##### Difficulty Level

Easy

##### Suggested Book

Functional Equations by BJ Venkatachala

# Start with hints

Do you really need a hint? Try it first!

Note that the RHS does not contain . Thus it should be possible to play with different values of to get rid of the inconvenient .

Taking , we get . Now show that . Thus $f(f(f(x)))=f(x)$.

Taking , we get . Now try to show that .

Taking we get the desired inequality from hint 3. Thus for all .

# Watch video

# Connected Program at Cheenta

#### Math Olympiad Program

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.