AMC 10 · 2012 · #25

Grade 10 geometry-2d

Archetype Pair-Sum Invariant └ Sub-archetype Uniform Pair-Sum Recovery

symmetry-argumentreflection-symmetrytrigonometric-ratioscoordinate-geometry symmetry-argumentsystematic-enumerationidentify-subproblems ↑ Prerequisites: trigonometric-ratioscoordinate-geometry

📏 Long solution 💡 4 insights

Problem

A ratio is taken over every right triangle on a punctured grid and all of them are multiplied. Find the product.

Pick an answer.

(A)

(B)

$\frac{625}{144}$

(C)

$\frac{125}{24}$

(D)

(E)

$\frac{625}{24}$

How to solve

Strategy Visualize Spatial Relationships

Computing 846 tangents is hopeless, so the product must collapse. Tool #17 (Visualize Spatial Relationships) supplies the collapse: reflect a triangle in a mirror. A mirror keeps both leg lengths but reverses orientation, which forces the counter-clockwise rule to swap B and C — so the reflected triangle's f-value is the reciprocal, and any mirror-matched pair multiplies to 1. Tool #16 (Change Focus / Count the Complement) then changes the question from "what is the product?" to "which triangles have no mirror partner inside S?", because those are the only ones that survive. Since (0,0) was deleted, the up-down mirror y = 5/2 still works for everything that avoids the opposite corner (0,5), and the diagonal mirror x+y=5 fixes (0,5) and still works for everything that avoids the bottom row — Tool #7 (Identify Subproblems) splits T into those two clean pieces plus a small residue. Tool #4 (Introduce a Variable) names the residue's vertices in coordinates so perpendicularity becomes a dot-product equation, and Tool #2 (Make a Systematic List) solves that equation exhaustively over the finite grid, which is what turns a claimed configuration list into a proved-complete one. Tool #1 (Draw a Diagram) keeps the two mirrors and the surviving triangles concrete throughout.

1STEP 1

f is just a ratio of the two legs

The quantity is a ratio of the two legs.

f(t)=tan∠ CBA=opposite/adjacent=AC/AB

2STEP 2

Any mirror turns f upside down

Any mirror turns it upside down.

ψ a reflection → f(ψ(t))=AB/AC=1/f(t), so f(t) · f(ψ(t))=1; and ψ(t)=t → f(t)²=1 → f(t)=1

3STEP 3

Mirror 1: flip across y=5/2

One mirror kills every triangle missing the odd point.

σ(x,y)=(x,5-y); σ(S ∖ {(0,5)})=S ∖ {(0,5)}, hence Π_t ∈ T₁ f(t)=1 where T₁={t ∈ T : (0,5) ∉ t}

4STEP 4

Mirror 2: flip across x+y=5

A second mirror kills most of the rest.

τ(x,y)=(5-y, 5-x), τ(0,5)=(0,5), τ(S')=S' for S'={(x,y): 0 ≤ x ≤ 4, 1 ≤ y ≤ 5}, hence Π_t ∈ T₂ f(t)=1

5STEP 5

The survivors: (0,5) plus exactly one bottom point

Only a narrow family of triangles survives.

Bottom-row pair impossible: -a(b-a)=0 → a=0 ∉ S, and ab+25=0 has no solution with a,b ≥ 1. Survivor shape: P=(0,5), Q=(q,0), R=(r,s), 1 ≤ q ≤ 4, 0 ≤ r ≤ 4, 1 ≤ s ≤ 5

6STEP 6

Find every survivor by dot products

A dot-product search finds eight of them.

Right angle at R: r(r-q)=s(5-s). Solutions: (R,Q)=((q,5),(q,0)) for q=1,2,3,4; ((4,1),(3,0)); ((4,4),(3,0)); ((3,2),(1,0)); ((3,3),(1,0))

7STEP 7

Evaluate f on the eight survivors

Each survivor's ratio is easy to read.

f-values: 5/1, 5/2, 5/3, 5/4 (axis-parallel); 2/3 at R=(3,2); 1/4 at R=(4,1); 1 at R=(3,3); 1 at R=(4,4)

8STEP 8

Multiply the survivors

Multiplying gives 625/144, choice (B).

Π_t ∈ T f(t) = 1 · 1 · (5/1 · 5/2 · 5/3 · 5/4) · (1 · 1) · (2/3 · 1/4) = 625/24 · 1/6 = 625/144 → (B)

Answer

625/144

Each cancellation step rests on a map that was checked to be a reflection sending its set exactly onto itself, so no triangle was silently dropped or double-counted, and the two families genuinely partition everything outside the eight survivors: T₁ is "no vertex at (0,5)", T₂ is "uses (0,5) but misses the bottom row", and the survivors are "uses (0,5) and meets the bottom row". A full sweep of all C(29, 3)=3654 vertex triples confirms the bookkeeping exactly: T contains 846 right triangles, of which 776 lie in T₁ and multiply to 1, 62 lie in T₂ and multiply to 1, and the remaining 8 are precisely the list in Step 6, with product 625/144. The distractors mark the two natural slips. Stopping after the four axis-parallel survivors gives 625/24, which is exactly choice (E) — that is the trap for anyone who assumes the leftover triangles must have horizontal and vertical legs. Choice (A) 1 is the trap for believing the mirrors clean up everything, forgetting that deleting (0,0) leaves a residue with no partner. It is also worth noting that the two isosceles survivors R=(3,3) and R=(4,4) are easy to miss when enumerating; they happen to contribute 1 each, so the answer survives the omission, but the enumeration is only honest with them included — and the 2/3 and 1/4 triangles, which look similar, are the ones that actually move the answer.

💡Key takeaway

Mirror a right triangle and its two legs stay the same, but which one counts as "opposite" flips — so its tangent turns upside down and the pair cancels; find enough mirrors and only a few stubborn triangles are left to multiply.

f is just a ratio of the two legs
Any mirror turns f upside down
Mirror 1: flip across y=5/2
Mirror 2: flip across x+y=5
The survivors: (0,5) plus exactly one bottom point
Find every survivor by dot products
Evaluate f on the eight survivors
Multiply the survivors