AMC 10 · 2021 · #19

Grade 11 algebra
Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination
trigonometric-ratiosperiodic-functionquadratic-equationsestimation convert-to-algebrabound-inequality-then-enumerate ↑ Prerequisites: trigonometric-ratiosquadratic-equations
📏 Medium solution 💡 3 insights
Problem
Angles are measured in degrees. Among all real numbers greater than one, find the smallest whose sine equals the sine of its square, then report the smallest integer at least as large as that number.

Pick an answer.

(A)
10
(B)
13
(C)
14
(D)
19
(E)
20
How to solve
Strategy Convert to Algebra

The equation sin(x)=sin(x²) cannot be attacked by isolating x, because sine has no algebraic inverse. But equal sines are not mysterious: on the unit circle they mean equal heights, and only two points share a height. That turns the trig equation into a plain statement about angles, and the statement carries a whole number k counting full turns. Naming that k converts the problem into a family of quadratics, one per k, which the quadratic formula solves outright. The word "least" then does the rest of the work: the roots grow with k, so only the smallest usable k in each family matters, and the winner is pinned down with a couple of perfect squares.

1STEP 1

Same sine means two cases

Equal sines give two cases.

sin A=sin B ⇔ A = B + 360k or A = 180 - B + 360k, k ∈ Z
2STEP 2

Each case becomes a quadratic

Each becomes a quadratic.

Case 1: x² - x - 360k = 0 Case 2: x² + x - (180+360k) = 0
3STEP 3

Solve both families

Solve both families.

Case 1: x=(1+√(1+1440k))/2 Case 2: x=(-1+√(721+1440k))/2
4STEP 4

Pick the least root above one

Pick the least root above one.

(-1+√(721))/2 ＜ (1+√(1441))/2, x_least=(-1+√(721))/2
5STEP 5

Trap the root between integers

Trapping it between integers gives 13.

26²=676 ＜ 721 ＜ 729=27² → 26 ＜ √(721) ＜ 27 → 12.5 ＜ (-1+√(721))/2 ＜ 13
Answer
13
The winning root satisfies x² + x = 180 exactly, so x² = 180 - x and sin(x²) = sin(180° - x) = sin(x) by the mirror rule. The equation holds by construction, not by luck. Numerically x ≈ 12.926 and x² ≈ 167.074, and those two do add to 180. The choices 19 and 20 are the trap for anyone who only writes Case 1: its first genuine root is (1+√(1441))/2≈ 19.48, a real solution but not the least one. The choice 14 comes from a loose estimate of √(721), yet 27²=729 ＞ 721 already forces x ＜ 13. The choice 10 fails outright, since 10+100=110 is not 180 and 100-10=90 is not a multiple of 360.
💡Key takeaway

Two angles share a sine only when they are the same angle or mirror images about 180°, and here the mirror case x² + x = 180 is reached first, landing between 12 and 13.

  • Same sine means two cases
  • Each case becomes a quadratic
  • Solve both families
  • Pick the least root above one
  • Trap the root between integers

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