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2011 MPFG Problem 14

Revision as of 05:07, 23 August 2025 by Cassphe (talk | contribs) (Created page with "==Problem== If <math>0 \leq p \leq 1</math> and <math>0 \leq p \leq 1</math>, define <math>F(p,q)</math> by <math>F(p,q) = -2pq + 3p(1-q) + 3(1-p)q-4(1-p)(1-q)</math>. Define...")
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Problem

If $0 \leq p \leq 1$ and $0 \leq p \leq 1$, define $F(p,q)$ by $F(p,q) = -2pq + 3p(1-q) + 3(1-p)q-4(1-p)(1-q)$. Define $G(p)$ to be the maximum of $F(p,q)$ over all $q$ (in the interval $0 \leq q \leq 1$). What is the value of $p$ (in the interval $0 \leq p \leq 1$) that minimizes $G(p)$? Express your answer as a fraction in simplest form.

Explanation

We can view the problem from a geometric meaning.

[insert picture]

$F(p,q) = -2A + 3B + 3C - 4D = 3(1-A-D) - 2A - 4D = -5A - 7D + 3$

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