AMC 10 · 2025 · #3
Grade 7 geometry-2dPick an answer.
AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
An isosceles triangle has an equal pair A, A and a third side B. The number 2025 is the longest side, and it can play one of two roles: it is one of the equal sides, or it is the single unequal side. Those two roles never happen at the same time, so splitting the count into these two separate subproblems lets us count each cleanly and add. Inside each case, naming the sides with a variable and pushing to the triangle-inequality boundary turns the count into counting a run of consecutive integers.
Name the sides
Write the sides as A, A, B with A and B positive integers; one of them is 2025 and no side exceeds it.
Every isosceles triangle is just an equal pair plus a third side, so two letters capture all of them.
7.G.A.2Introduce A VariableSplit by the role of 2025
2025 is either one of the two equal sides or the single odd side — two exclusive cases, count each and add.
The biggest side has to be somewhere in the pattern A, A, B, and there are only two spots it can sit.
7.G.A.2Identify SubproblemsCase 1: equal sides are 2025
For 2025, 2025, B any B from 1 to 2025 keeps 2025 longest, so there are 2025 triangles (B = 2025 is the equilateral one).
With two long equal sides, any positive base up to 2025 still closes into a real triangle.
6.EE.B.8Extreme PrincipleCase 2: base is 2025
For A, A, 2025 the legs must reach across: A + A greater than 2025 gives A from 1013 to 2024, so 1012 triangles.
The two equal legs must together stretch past the base, which forces each leg to be more than half of 2025.
The two equal legs must together stretch past the base, which forces each to be more than half of it.
▸ Why?
Two sides must together outreach the third, or the ends never meet.
▸ Why?
That comparison puts a hard floor under the leg length, so only the values above it survive.
Add the two cases
The two cases share no triangle, so add them: 2025 + 1012 = 3037, choice (D).
Two non-overlapping piles of triangles just add together.
4.OA.A.3Make A Systematic ListAsk where the longest side can sit in the pattern A, A, B, count each spot with the triangle inequality, and add.
- Name the sides
- Split by the role of 2025
- Case 1: equal sides are 2025
- Case 2: base is 2025
- Add the two cases
More like this
Same archetype — closest grade level first.
