MathDB

Rational Geometry

Source: 2018 AIME II #7

March 23, 2018

ratiogeometrytrapezoid

Problem Statement

Triangle

ABC

has sides

AB=9,BC = 5\sqrt{3},

and

AC=12

. Points

A=P_0, P_1, P_2, \dots, P_{2450} = B

are on segment

\overline{AB}

with

P_k

between

P_{k-1}

and

P_{k+1}

for

k=1,2,\dots,2449

, and points

A=Q_0, Q_1, Q_2, \dots ,Q_{2450} = C

for

k=1,2,\dots,2449

. Furthermore, each segment

\overline{P_kQ_k}, k=1,2,\dots,2449

, is parallel to

\overline{BC}

. The segments cut the triangle into

2450

regions, consisting of

2449

trapezoids and

1

triangle. Each of the

2450

regions have the same area. Find the number of segments

\overline{P_kQ_k}, k=1,2 ,\dots,2450

, that have rational length.

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