HMMT February

Algebra Difficulty 5.0 AIME Find the answer

$r$ and $s$ are integers such that $3 r \geq 2 s-3 \text { and } 4 s \geq r+12$ . What is the smallest possible value of $r / s$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We simply plot the two inequalities in the $s r$ -plane and find the lattice point satisfying both inequalities such that the slope from it to the origin is as low as possible. We find that this point is $(2,4)$ (or $(3,6))$ , as circled in the figure, so the answer is $2 / 4=1 / 2$ .

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Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.