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2021 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2021 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | D | Absolute Value & Inequalities — 3π is about 9.42, so x can be any whole number from −9 up to 9. That is 9 negatives, 9 positives, and zero. |
| 2 | D | Exponents, Logarithms & Radicals — The square root of a square is never negative. Since 2√3 (about 3.46) is bigger than 3, the first bracket is negative and becomes 2√3 − 3. |
| 3 | C | Ratios, Percents & Averages — A quarter of the juniors equals a tenth of the seniors, so for every 2 juniors there are 5 seniors. Split 28 into 2 and 5 parts: 8 juniors. |
| 4 | B | Linear Equations & Word Problems — The 23 blue–blue pairs use 46 blue students. The other 11 blue students each sit with a yellow, which leaves 64 yellows to pair with each other: 32 pairs. |
| 5 | B | Divisibility & Factors — Using single digits, 30 can only be 5 × 6. Since 6 is now taken, 24 has to be 3 × 8, not 4 × 6. |
| 6 | C | Statistics & Data — Only the 3 to 4 ratio matters, so pretend there are 3 morning and 4 afternoon students. Their scores total 532 over 7 students, which is 76. |
| 7 | D | Circles — Circles on one side sit inside each other, so a side counts only its biggest circle minus its second biggest. Put 7 alone: 49 + 16 = 65. |
| 8 | A | Paths & Grids — Odd squares sit at the top-right corners of the rings. 169 ends the 13 by 13 ring in row 2, so row 2 holds 157 to 170, plus 210. |
| 9 | D | Coordinate Geometry — Work backward and undo the last move first: reflecting (−6, 3) over y = −x gives (−3, 6). Then turn that point a quarter turn clockwise around (1, 5). |
| 10 | A | Solid Geometry — Pouring keeps the amount of water the same. Set the cone's volume, (1/3)π · 12² · 18, equal to π · 24² · h, so h = 1.5. |
| 11 | D | Diophantine Equations — The inside pieces must be half of all pieces: 2(m − 2)(n − 2) = mn. Only two pans work, 5 by 12 and 6 by 8. |
| 12 | C | Divisibility & Factors — Every even divisor is an odd divisor times 2, 4, or 8. So the even divisors add up to 2 + 4 + 8 = 14 times the odd ones. |
| 13 | B | Bases & Digits — Both base-n numbers start 3n² + 2n. Subtracting the two equations cancels that part and gives d = 2, and then 3n² + 2n = 261 gives n = 9. |
| 14 | B | Circles — Equal chords are equally far from the center, so it sits halfway between the 38-chords. Then 19² + (d/2)² = 17² + (3d/2)². |
| 15 | B | Algebraic Manipulation — Square twice to get x⁴ + 1/x⁴ = 7. Multiply that by x⁷ and you get x¹¹ + x³ = 7x⁷, so the expression is 0. |
| 16 | C | Divisibility & Factors — Uphill numbers cannot end in 0, so a multiple of 5 ends in 5. Then choose digits from 1 to 4 so the digit sum is a multiple of 3. |
| 17 | C | Logic Puzzles — Start with the smallest score. 4 can only be 1 + 3. Then 7 must be 2 + 5, and 11 must be 4 + 7. |
| 18 | C | Basic Probability — Rolls that repeat a number already seen change nothing, so skip them. Then the first three different numbers to appear must be 2, 4, 6: (1/2)(2/5)(1/4) = 1/20. |
| 19 | D | Statistics & Data — Removing the greatest or the least leaves n − 1 numbers either way. Those two totals differ by 72 and the averages by 8, so n − 1 = 9. |
| 20 | D | Quadrilaterals & Polygon Areas — Triangles ABF and BCF are equilateral, so the diagonal AC is 2√3, and the same holds for AD. Cut the pentagon into triangles ABC, ACD, ADE. |
| 21 | A | Transformations & Symmetry — Folding keeps lengths, so the small right triangle at D has sides 1/3, 4/9, 5/9. Triangle AEC' has the same shape, 3/2 times as big, so its perimeter is 2. |
| 22 | D | Inclusion-Exclusion — Ang's order can be taken as fixed. A box is all one color when Ben and Jasmin both match Ang there. Count with add, subtract, add: 2556 of 14400. |
| 23 | C | Geometric Probability — Follow the coin's center. It lands in a 7 by 7 square, and the coin touches black exactly when the center is within 1/2 of a black shape. |
| 24 | B | Games & Processes — If the walls come in identical pairs, the second player wins by copying each move on the twin wall. In every wrong choice, Arjun can make such pairs at once. |
| 25 | E | Coordinate Geometry — The area under the line is 450m, so try slope 2/3: exactly 300 points. The count stays 300 until the line reaches the point (28, 19). |
2021 AMC 10B — 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.