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2021 AMC Fall 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2021 AMC Fall 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 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 — 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.