1995 AMC 12 Problem 18

Attempt Problem 18 of the 1995 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1995 AMC 12 solutions, or check the answer key.

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18.

Two rays with common endpoint

O

form a

30^\circ

angle. Point

A

lies on one ray, point

B

on the other ray, and

AB=1.

The maximum possible length of

OB

1

\frac{1+\sqrt3}{\sqrt2}

\sqrt3

2

\frac4{\sqrt3}

Apply the Law of Cosines to

\triangle AOB

Treat the resulting relation as a quadratic in

OA

and require a real solution

Let

OA=x

and

OB=y.

The Law of Cosines gives

1=x^2+y^2-\sqrt3xy.

As a quadratic in

x,

this has discriminant

3y^2-4(y^2-1)=4-y^2.

A real

x

requires

y\le2,

and equality is attained when

x=\sqrt3.

Hence the maximum is

2,

and the correct answer is D.

Problem 18 in other years