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March 9, 2020

Functions and Equations |Pre-RMO, 2019

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


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

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