2017 AMC 10A

All 25 problems, 75 minutes, scored 6 / 1.5 / 0. The answer key and the idea behind each problem are on this page too, folded away until you ask for them.

Take it timed (75 min)Print the paper PDF 177 KB Blank answer sheet Enter answers from paper

The problems

Show the answer key, topics and key insights

This gives away all 25 answers and the idea behind each one. Sit the paper first if you mean to.

The same key as a PDF 125 KB — a separate file from the paper, so one can be handed over without the other.

#	Answer	Topic	Key insight
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. 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.