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Miklós Schweitzer 2008, Problem 7

Source: Miklós Schweitzer 2008

July 30, 2016
Miklos Schweitzercollege contestsfunctionvectorreal analysis

Problem Statement

Let f ⁣:R1→R2f\colon \mathbb{R}^1\rightarrow \mathbb{R}^2 be a continuous function such that f(x)=f(x+1)f(x)=f(x+1) for all xx, and let t∈[0,14]t\in [0,\frac14]. Prove that there exists x∈Rx\in\mathbb{R} such that the vector from f(x−t)f(x-t) to f(x+t)f(x+t) is perpendicular to the vector from f(x)f(x) to f(x+12)f(x+\frac12).
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