Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Find the answer

4. For a function $f(x)$ defined on the interval $[a, b]$ , if there exists a constant $c$ , such that for any $x_{1} \in[a, b]$ there is a unique $x_{2} \in[a, b]$ , satisfying $\frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2}=c$ , then the function $f(x)$ is said to have a "mean" of $c$ on $[a, b]$ . Then, the mean of the function $f(x)=\lg x$ on $[10,100]$ is:

Pick one

Solution

4. C.

Let for any $x_{1} \in [10,100]$ , there exists a unique $x_{2} \in$ $[10,100]$ , such that $x_{1} x_{2}=10^{3}$ . Therefore,
$\begin{array}{l} \frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2}=\frac{\lg x_{1}+\lg x_{2}}{2} \\ =\frac{\lg x_{1} x_{2}}{2}=\frac{\lg 10^{3}}{2}=\frac{3}{2} . \end{array}$

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