2008 AMC 10B Problem 15

Attempt Problem 15 of the 2008 AMC 10B 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 2008 AMC 10B solutions, or check the answer key.

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15.

How many right triangles have integer leg lengths

a

and

b

and a hypotenuse of length

b+1,

where

b\lt 100?

6

7

8

9

10

Set

a^2+b^2=(b+1)^2

and simplify

This gives

a^2=2b+1,

a^2

is an odd perfect square with

a^2\lt 201

From

a^2+b^2=(b+1)^2

we get

a^2=2b+1,

a

is odd and

a^2

is an odd perfect square.

Since

b\lt 100,

we need

a^2=2b+1\lt 201,

and

a^2\ge 9

for

b\ge 4.

The odd squares

9,25,49,81,121,169

give

a=3,5,7,9,11,13,

which is

6

triangles.

Thus, the correct answer is A.

Problem 15 in other years