Problem #1

Week of April 7, 2026·Deadline: Monday, April 13, 2026

Problem Statement

Let $f : [0, 1] \to \mathbb{R}$ be a continuous function satisfying

\int_0^1 f(x)\,dx = 1.

Prove that there exists a point $c \in (0, 1)$ such that

f(c) = 2c.

Show Hint

Consider the auxiliary function $g(x) = \int_0^x f(t)\,dt - x^2$ and examine its values at the endpoints of $[0, 1]$ .

Write out your complete proof below. We review all submissions and publish a solution article the following week.

Problem of the Week

Undergraduate