AMC 10 · 2024 · #11

Grade 11 algebra
Archetype Pair-Sum Invariant └ Sub-archetype Uniform Pair-Sum Recovery
pythagorean-identitytrigonometric-ratiosmean-median-mode-rangesymmetry-argument symmetry-argumenteasier-related-problemidentify-subproblems ↑ Prerequisites: trigonometric-ratiosmean-median-mode-range
📏 Medium solution 💡 3 insights
Problem
For each whole number n from 1 to 90, define x_n = sin²(n^°). Find the average of the 90 numbers x₁, x₂, x₃, …, x₉₀.

Pick an answer.

(A)
$\frac{11}{45}$
(B)
$\frac{22}{45}$
(C)
$\frac{89}{180}$
(D)
$\frac{1}{2}$
(E)
$\frac{91}{180}$
How to solve
Strategy Organize Information in More Ways

Almost none of the 90 terms can be computed individually, so the sum has to be attacked as a whole. Tool #15 (Organize Information in More Ways) is the move: instead of reading the list left to right, read it from both ends inward, matching x_n with x₉₀-n. Each such pair collapses to 1 because sin(90^° - n^°) = cos(n^°) and sin² + cos² = 1. Tool #7 (Identify Subproblems) then handles the terms the pairing leaves over — x₄₅, which would pair with itself, and x₉₀, whose partner x₀ is not in the list. Those two leftovers are the entire difficulty of this problem, and every wrong answer choice comes from mishandling them. Tool #9 (Solve an Easier Related Problem) rehearses the pairing on a short list first so the bookkeeping is trustworthy, and Tool #3 (Eliminate Possibilities) checks at the end that the leftover terms push the mean just above 1/2, not below.

1STEP 1

Find the collapsing pair

Complementary pairs always add to 1.

x_n + x₉₀-n = sin²(n^°) + sin²(90^°-n^°) = sin²(n^°) + cos²(n^°) = 1
2STEP 2

Rehearse on a short list

A toy list confirms the pairing scheme.

(x₁+x₅)₁ + (x₂+x₄)₁ + x₃ + x₆
3STEP 3

Pair the ends inward

That makes 44 pairs.

Σ_n=1⁴⁴(x_n + x₉₀-n) = 44 × 1 = 44
4STEP 4

Account for the two leftovers

Forty-five and ninety contribute three halves.

x₄₅ = (√(2)/2)² = 1/2, x₉₀ = 1² = 1, x₄₅ + x₉₀ = 3/2
5STEP 5

Add the sum, then divide

Ninety-one halves over ninety is 91 over 180.

Σ_n=1⁹⁰ x_n = 44 + 3/2 = 91/2, mean = 91/2 ÷ 90 = 91/180 → (E)
Answer
91/180
The mean 91/180 ≈ 0.5056 sits just barely above 1/2, which is exactly what the structure predicts: the 44 pairs average 1/2 per term, the self-paired x₄₅ also contributes 1/2, and only the unpartnered x₉₀ = 1 pushes the average up — by 1/180, a tiny nudge. The wrong choices are all bookkeeping slips on the leftovers, which is why Tool #3 (Eliminate Possibilities) settles this fast: dropping x₉₀ entirely leaves a sum of 89/2 and gives (C) 89/180; dropping both leftovers leaves 44 and gives (B) 22/45; forcing the sum to a flat 45 by treating x₄₅ as a full pair gives (D) 1/2. Any correct count must land slightly above 1/2, and (E) is the only choice that does. A direct numerical sum of the 90 values confirms 91/2 exactly.
💡Key takeaway

Read the list from both ends inward — every pair of angles adding to 90^° contributes exactly 1 — then count the two leftovers, 45^° and 90^°, one time each.

  • Find the collapsing pair
  • Rehearse on a short list
  • Pair the ends inward
  • Account for the two leftovers
  • Add the sum, then divide

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