AMC 10 · 2008 · #15
Grade 8 number-theoryPick an answer.
AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The three sides are already named a, b, and b+1, so tool #4 (Introduce a Variable) leads: writing the Pythagorean relation a²+b²=(b+1)² and cleaning it up collapses the whole problem into one tidy equation, a²=2b+1. That equation ties b directly to a, turning 'count the triangles' into 'count the allowed values of a'. Tool #3 (Eliminate Possibilities) then throws out every even a, because 2b+1 is odd, so a must be odd for b to be a whole number. Finally tool #2 (Make a Systematic List) walks the odd values of a that keep b a positive integer under 100 and counts them.
Write the Pythagorean equation
Legs a and b with hypotenuse b+1 make the Pythagorean rule read a²+b²=(b+1)² — one equation for every allowed triangle.
In any right triangle the squares on the two legs together equal the square on the hypotenuse.
8.G.B.7Introduce A VariableSimplify to a clean form
Expanding cancels both b² terms and leaves a²=2b+1, so b=(a²-1)/2 — pick the leg a and b is fixed.
Because the two b² terms match, they wash out and the messy square shrinks to a simple line.
6.EE.A.3Introduce A VariableKeep only odd legs a
Since 2b+1 is odd, a² is odd, so a must be odd; a=1 would give b=0 and no triangle, so the smallest usable leg is a=3.
Half of an even number is whole, so a²-1 must be even, which only happens when a is odd.
Half of the leftover must be a whole number, which happens only when the leg is odd.
▸ Why?
An odd number squared is odd, so one less than it is even and halves cleanly.
▸ Why?
Otherwise the halving leaves a remainder, so no whole-number side can come out of it.
List the odd legs under the limit
b ＜ 100 forces a² ＜ 201, so odd a runs 3,5,7,9,11,13 with b=4,12,24,40,60,84 — 6 triangles, choice (A).
The limit b ＜ 100 caps a at 13, and counting the odd legs from 3 up gives the total directly.
7.EE.B.4Make A Systematic ListWhen the hypotenuse is just one more than a leg, the Pythagorean equation collapses to a²=2b+1, so you only have to count the odd legs that stay in range.
- Write the Pythagorean equation
- Simplify to a clean form
- Keep only odd legs a
- List the odd legs under the limit