Track / Stage 4 / 202 of 340 #462 of 1964

Problem 462

AMC 12 late, AIME early

Algebra Difficulty 4.8 Find the answer

Given $a f\left(x^{2 n+1}\right)+f\left(-x^{2 n+1}\right)$ $=b x\left(n \in N, a^{2} \neq 1\right)$ , find $f(x)$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Solve: In the original equation, substitute $-x$ for $\hat{x}$ , and then solve the system of equations
$f\left(x^{2 n+1}\right)=\frac{b x}{a-1} \text {. }$

Therefore, $f(x)=\frac{b}{a-1}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.