AMC 10 · 2024 · #7
Grade 12 geometry-2dPick an answer.
Adding 2024 vectors one after another is hopeless, so the entire problem is about choosing a better order to add them in. Tool #15 (Organize Information in More Ways) is that move: instead of sweeping from P₁ to P₂₀₂₄, pair the outermost two points, then the next two inward, and so on. Each such pair straddles the midpoint M of AC, so each pair adds up to the very same vector 2BM, and 2024 different vectors collapse into many copies of one. Tool #1 (Draw a Diagram) supplies coordinates with B at the origin, which is what makes the symmetry checkable instead of merely believable. Tool #4 (Introduce a Variable) names the index i so the pairing rule can be stated once and reused. Tool #9 (Solve an Easier Related Problem) is the safety net: rerun the argument with 2 points instead of 2024 and confirm the pattern before trusting it at scale.
Put the right angle at the origin
With B at the origin, A and C sit on the axes.
With B at the origin, "the vector BP" and "the coordinates of P" are the same object, so adding vectors becomes adding coordinates.
10.G-GPE.B.4Draw A DiagramThe hypotenuse has length 2
Pythagoras gives AC equals 2.
The legs were chosen as √(2) precisely so the hypotenuse comes out a clean 2 — a hint that half of it, 1, is about to matter.
8.G.B.8Draw A Diagram2024 points make 2025 gaps
There is one more gap than point.
Fence posts and gaps: 2024 posts strictly inside a fence leave 2025 gaps, and every wrong answer choice here is built out of getting that count off by one.
7.RP.A.2Introduce A VariablePair the ends inward
Every pair sums to twice the midpoint.
Two vectors reaching to two points that straddle a midpoint always add to twice the vector reaching that midpoint — the sideways wobble cancels.
Two vectors reaching points that straddle a midpoint always add to twice the vector reaching that midpoint.
▸ Why?
Points placed symmetrically about the middle always pair to the same total.
▸ Why?
That shared total is twice the midpoint, because the midpoint is the two positions shared equally.
1012 identical pairs
The total becomes 2024 times the median vector.
Regrouping the sum rewrites 2024 different vectors as one known vector scaled by 2024 — the count of pairs is not the answer, the count of terms is.
9.A-SSE.A.2Organize Information In More WaysThe median to the hypotenuse is 1
That median is half the hypotenuse, namely 1.
The right angle at B puts B on the circle with diameter AC, so BM is a radius — half the hypotenuse, every time.
10.G-CO.C.10Draw A DiagramTake the length
2024 times 1 is 2024.
A vector's length scales with a positive scalar, so once the sum is a multiple of BM the answer is just that multiple times 1.
12.N-VM.A.1Organize Information In More WaysNever add 2024 vectors one at a time — pair the first with the last, because every such pair lands on twice the midpoint vector, and here that midpoint vector has length exactly 1.
- Put the right angle at the origin
- The hypotenuse has length 2
- 2024 points make 2025 gaps
- Pair the ends inward
- 1012 identical pairs
- The median to the hypotenuse is 1
- Take the length