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Romania National Olympiad Grade 11 P1

Source:

April 28, 2021
real analysis

Problem Statement

Let f:[a,b]→Rf:[a,b] \rightarrow \mathbb{R} a function with Intermediate Value property such that f(a)∗f(b)<0f(a) * f(b) < 0. Show that there exist α\alpha, β\beta such that a<α<β<ba < \alpha < \beta < b and f(α)+f(β)=f(α)∗f(β)f(\alpha) + f(\beta) = f(\alpha) * f(\beta).
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