2021 AMC Fall 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.
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 | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Algebraic Manipulation | 91 is 7 times 13, and 169 is 13 times 13. So the fraction is 7 squared over 1, which is 49. Simplify before squaring. | |
| 2 | E | Linear Equations & Word Problems | Try both sides. Only cutting the 4-inch side gives 3 by 6 = 18, so the other side is the 6-inch side, and 4 by 5 = 20. | |
| 3 | D | Solid Geometry | The clay can be reshaped, so only the amount of clay matters. Divide the cube's volume 216 by one ball's volume, about 33.5, to get about 6.4. | |
| 4 | B | Linear Equations & Word Problems | Find the time for each piece of each route in minutes. Route A takes 12 minutes. Route B takes 6.75 + 1.5 = 8.25 minutes. Subtract. | |
| 5 | E | Primes | Cross out answers with quick tests: 5 ends in 5, digits 1 and 7 make the digit sum a multiple of 3, and 202103 is a multiple of 11. | |
| 6 | B | Linear Equations & Word Problems | 41 poles make 40 gaps, so each gap is 132 feet. A stride is 132 / 44 = 3 feet and a leap is 132 / 12 = 11 feet. | |
| 7 | D | Angles & Polygons | Around D, 360 - 90 - 110 leaves 160 degrees inside triangle DEF. Its two equal sides make the other angles 10 each, so angle AFE is 170. | |
| 8 | B | Bases & Digits | Write the number as 10a + b. Cuddly means 9a = b(b - 1). Checking all ten units digits, only b = 9, a = 8 works. | |
| 9 | E | Basic Probability | Each roll is even with probability 3/4 and odd with probability 1/4. The sum is even when both rolls match, so the chance is 9/16 + 1/16 = 5/8. | |
| 10 | B | Statistics & Data | t is the plain average, 20. For s, each student reports their own class size, so the big class counts 50 times: s = 33.5. | |
| 11 | A | Linear Equations & Word Problems | Walking with the ship, Emily gains one ship length at the difference of the speeds; walking against it, they close one ship length at the sum. | |
| 12 | D | Modular Arithmetic | Powers of 9 leave remainders 1, 4, 1, 4, ... when divided by 5. Multiply each nonzero digit by the remainder for its place and add. | |
| 13 | D | Basic Probability | Each ball needs at least 3 of the other color among the other 5. That forces exactly 3 black and 3 white: 20 of 64 colorings. | |
| 14 | D | Absolute Value & Inequalities | The second equation is two diamonds, |x| + |y| = 3 or 5. The parabola touches the small one at 3 corners and crosses the big one twice. | |
| 15 | C | Circles | Tangent radii make right angles at B and C, so right triangles ABO and ACO share side AO. The circle through A, B, C has AO as diameter. | |
| 16 | D | Functions | Plot some values: f is 0 between whole numbers, 1 at 1, 2, 3, ... and -1 at 0, -1, -2, .... The dots pair up around (1/2, 0). | |
| 17 | D | Solid Geometry | On a flat panel, equal steps give equal height changes. B to A adds 3, so the center is 13; center to E adds 4, so E is 17. | |
| 18 | C | Arrangements with Restrictions | Diagonal sections do not touch, so fill them first. Each other section then has 3 choices if the diagonal crops match, 2 if not. | |
| 19 | A | Circles | Inside, the disk sweeps a band 2 wide but misses a piece of 1 - pi/4 at each corner. Outside, it sweeps four strips and four quarter circles. | |
| 20 | B | Quadratics | Both discriminants must be zero or negative: b squared at most 4c, c squared at most 4b. That forces b and c to be at most 4. | |
| 21 | E | Basic Counting | Both probabilities share the bottom 5 to the 20th, so compare counts. Most cancels, leaving 20 ways to pick the special bins times 4/5. | |
| 22 | B | Solid Geometry | The centers form an equilateral triangle of side 2r at height r, so each is 2r over root 3 from the axis. Then slice through one center. | |
| 23 | D | Divisibility & Factors | For n up to 50, f1(n) is an even number at most 20. Only 12, 18 and 20 end at 12, so count n with 6, 9 or 10 divisors. | |
| 24 | E | Arrangements with Restrictions | Exactly six edges are 1s, and top and bottom use four, so exactly 2 upright edges are 1s. Opposite uprights give 12; neighboring uprights give 8. | |
| 25 | A | Quadratics | p(p(x)) = 0 means p(x) equals one root or the other. Three solutions means one root of p equals the lowest value of p. |
2021 AMC Fall 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.