MathDB

Family of segments

Source: 2024 AIME II P12

February 8, 2024

AIMEAIME IAIME II

Problem Statement

Let

O(0,0)

,

A(\tfrac{1}{2},0)

, and

B(0, \tfrac{\sqrt{3}}{2})

be points in the coordinate plane. Let

\mathcal{F}

be the family of segments

\overline{PQ}

of unit length lying in the first quadrant with

P

on the

x

-axis and

Q

on the

y

-axis. There is a unique point

C

on

\overline{AB}

, distinct from

A

and

B

, that does not belong to any segment from

\mathcal{F}

other than

\overline{AB}

. Then

OC^2 = \tfrac{p}{q}

where

p

and

q

are relatively prime positive integers. Find

p + q

.

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