Track / Stage 4 / 252 of 340 #512 of 1964

Problem 512

AMC 12 late, AIME early

Combinatorics Difficulty 4.9 Find the answer

3. Let $f(x)$ be a function defined on $\mathbf{R}$ . If $f(x) + x^2$ is an odd function, and $f(x) + 2^x$ is an even function, then the value of $f(1)$ is $\qquad$

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Official solution

According to the problem, $\left\{\begin{array}{l}f(-1)+1=-f(1)-1, \\ f(-1)+\frac{1}{2}=f(1)+2\end{array} \Rightarrow\left\{\begin{array}{l}f(-1)+f(1)=-2, \\ f(-1)-f(1)=\frac{3}{2}\end{array} \Rightarrow f(1)=-\frac{7}{4}\right.\right.$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.