Try this beautiful problem from Pre-RMO, 2019 based on **Functions and Equations**.

## Functions and Equations – PRMO, 2019

Let f(x) = $x^{2}+ax+b$, if for all non zero real x, f(x+$\frac{1}{x})$=f(x)+f($\frac{1}{x}$) and the roots of f(x)=0 are integers, find the value of $a^{2}+b^{2}$.

- 10
- 20
- 30
- 13

**Key Concepts**

Functions

Algebra

Polynomials

## Check the Answer

But try the problem first…

Answer: 13.

Pre-RMO, 2019

Functional Equation by Venkatchala .

## Try with Hints

First hint

f(x+$\frac{1}{x})$=f(x)+f($\frac{1}{x}$)

Second hint

then $(x+\frac{1}{x})^{2}+a(x+\frac{1}{x})+b$=$x^{2}+ax+b+\frac{1}{x^{2}}+\frac{a}{x}+b$ then b=2, product of roots is 2 then roots are (1,2),(-1,-2) and a=3or-3

Final Step

then $a^{2}+b^{2}$=4+9=13

## Other useful links

- https://www.cheenta.com/geometry-of-plane-figures-pre-rmo-2019/
- https://www.youtube.com/watch?v=65RRPvbATsk

Google