2012 AMC 12B Problem 20

Question

A trapezoid has side lengths

3,

5,

7,

and

11.

The sum of all the possible areas of the trapezoid can be written in the form of

r_1\sqrt{n_1} + r_2\sqrt{n_2} + r_3,

where

r_1,

r_2,

and

r_3

are rational numbers and

n_1

and

n_2

are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to

r_1 + r_2 + r_3 + n_1 + n_2?

57

59

61

63

65

Slide one parallel side over: a trapezoid with parallel sides

a\lt c

and legs

b,d

gives a triangle with sides

b,d,c-a

The triangle inequality forces the longer parallel side to be

11;

apply Heron’s formula in the cases

a=3,5,7

For a trapezoid with parallel sides

a\lt c

and legs

b,d,

translating a leg forms a triangle with sides

b,

d,

and

c-a.

The triangle inequality forces the longer parallel side to be

c=11.

If

a=3,

the triangle has sides

5,7,8

with area

10\sqrt3,

and the trapezoid has area

\tfrac{35}{2}\sqrt3.

If

a=5,

the triangle has sides

3,6,7

with area

4\sqrt5,

giving trapezoid area

\tfrac{32}{3}\sqrt5.

If

a=7,

the triangle has sides

3,4,5,

a right triangle, giving trapezoid area

27.

The total is

\tfrac{35}{2}\sqrt3+\tfrac{32}{3}\sqrt5+27,

so

\begin{gathered} r_1+r_2+r_3+n_1+n_2 \\ = \tfrac{35}{2}+\tfrac{32}{3}+27+3+5 \\ = 63+\tfrac16. \end{gathered}

The greatest integer at most this value is

63.

Thus, the correct answer is D.

Problem 20 in other years