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2017 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2017 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Sequences & Series — Start with the innermost parentheses and work outward. Each layer doubles the number and adds 1, giving 3, 7, 15, 31, 63, 127. |
| 2 | D | Linear Equations & Word Problems — The 5-popsicle box is the cheapest per popsicle. Buy two of them for $6, then spend the last $2 on a 3-popsicle box. |
| 3 | B | Quadrilaterals & Polygon Areas — Find the size of the whole garden, including every walkway strip, then subtract the six flower beds. What is left over is the walkway. |
| 4 | B | Games & Processes — The box gains 1 toy per half-minute overall, but it reaches 30 right when Mom drops in 3 toys, before Mia can take any out. |
| 5 | C | Algebraic Manipulation — Adding the two reciprocals gives the sum of the numbers divided by their product. The problem says that sum is 4 times the product, so the answer is 4. |
| 6 | B | Logic Puzzles — All right means an A. So no A means not all right: at least one answer was wrong. That flipped-and-reversed statement is the only one that must be true. |
| 7 | A | Triangles: Area & Pythagorean — Pick an easy side length, 1. Jerry walks 2, Silvia walks the diagonal, about 1.41. She saves about 0.59 out of Jerry's 2, which is about 30%. |
| 8 | B | Basic Counting — A pair shakes hands exactly when at least one of the two is one of the 10 strangers. Count stranger-with-group pairs, then stranger-with-stranger pairs. |
| 9 | C | Linear Equations & Word Problems — Time each stretch as distance divided by speed. Penny rides the loop backward, so Minnie's uphill stretch is Penny's downhill one, and the other way round. |
| 10 | B | Triangles: Area & Pythagorean — The longest side must be shorter than the other three combined. That allows lengths 6 through 24, minus the rods 7 and 15 already used. |
| 11 | D | Solid Geometry — The region is a capsule: a cylinder of radius 3 along the segment, with a half-ball on each end. The two half-balls together make one whole ball. |
| 12 | E | Coordinate Geometry — Make each pair of quantities equal. Each gives a line, and 'the third is not bigger' keeps only half of it. All three halves start at (1, 7). |
| 13 | D | Modular Arithmetic — The list repeats every 8 terms, so any 8 terms in a row make one full cycle, which adds up to 9. |
| 14 | D | Ratios, Percents & Averages — Write both sentences as equations: ticket is a fifth of (A minus soda), soda is a twentieth of (A minus ticket). With A = 99, costs are 19 and 4. |
| 15 | C | Geometric Probability — Half the time Laurent picks above 2017 and wins for sure. The other half, both pick from the same range, so he wins half of those. |
| 16 | B | Divisibility & Factors — Horse k is at the start at multiples of k minutes. So T is the smallest number divisible by at least five of 1 to 10, which is 12. |
| 17 | D | Coordinate Geometry — Make PQ as long as possible without being the diameter (7 times root 2), and RS as short as possible (root 2). |
| 18 | D | Conditional Probability & States — If both players miss in a round (chance 2/5), the game starts over. So Amelia's chance P satisfies P = 1/3 + (2/5)P. |
| 19 | C | Arrangements with Restrictions — Alice can sit only next to Derek or Eric, so place Alice first. Derek and Eric can end up side by side only when Alice sits at an end. |
| 20 | D | Bases & Digits — Adding 1 raises the digit sum by 1, but each 9 that turns into 0 takes away 9. So the new digit sum is 1275 minus a multiple of 9. |
| 21 | D | Similar & Congruent Triangles — Each square cuts off small right triangles shaped like the 3-4-5 triangle. Their matching side ratios let you write a side of the big triangle using the square's side. |
| 22 | E | Circles — Radii to B and C meet the sides at right angles, so the center angle is 120 degrees. The part inside the circle is the slice cut off by BC. |
| 23 | B | Basic Counting — Count all ways to pick 3 of the 25 points, then subtract picks lying on one line. Sort those lines by direction so none are missed. |
| 24 | C | Polynomials — Since f has all three roots of g, f equals g times (x - r). Matching x terms gives r = -90, so f(1) = g(1) times 91. |
| 25 | A | Basic Counting — A three-digit number is a multiple of 11 when first minus middle plus last is 0 or 11. So one digit is the others' sum, or that sum minus 11. |
2017 AMC 10A — answer key. Problems © Mathematical Association of America, American Mathematics Competitions (maa.org/amc), licensed CC BY-NC-SA 4.0. Typeset and annotated by AMC 10 Step by Step (amcstep.com) and shared under the same licence. Not endorsed by the MAA.