2013 AMC 10A Problem 23

Question

In

\triangle ABC,

AB = 86,

and

AC=97.

A circle with center

A

and radius

AB

intersects

\overline{BC}

at points

B

and

X.

Moreover

\overline{BX}

and

\overline{CX}

have integer lengths. What is

BC?

11

28

33

61

72

Use power of a point from

C

Factor

AC^2-AB^2

and use the triangle inequality to choose the valid factor pair

By power of a point from

C

,

CB\cdot CX

=AC^2-AB^2

=97^2-86^2

.

This equals

(97-86)(97+86)

=11\cdot183

=2013

=3\cdot11\cdot61

.

Both

CX

and

BX

are integers, so

BC=BX+CX

is an integer factor paired with

CX

. Also

CX<BC<86+97=183

, so the only possible pair is

CX=33

,

BC=61

.

Thus, D is the correct answer.

Problem 23 in other years