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2021 AMC Fall 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2021 AMC Fall 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | E | Bases & Digits — Each of the digits 1, 2, 3, 4 shows up exactly once in every column, so every column adds to 10 and the total is 10 times 1111. |
| 2 | B | Quadrilaterals & Polygon Areas — Fill in the notch to get a big triangle, then take the notch back out; both triangles sit on the same base of length 4. |
| 3 | E | Algebraic Manipulation — Over a common denominator the top becomes 2021 squared minus 2020 squared, which factors as (1)(4041). Nothing cancels, so p is 4041. |
| 4 | C | Linear Equations & Word Problems — The gap shrinks by 5 + 3 = 8, so it becomes N - 8. That must be 2 or -2, giving N = 10 or N = 6. |
| 5 | E | Exponents, Logarithms & Radicals — Switch to base 2: 8 to the 2022 is 2 to the 6066; dividing by 4 gives 2 to the 6064, which is 4 to the 3032. |
| 6 | B | Divisibility & Factors — 2021 = 43 times 47, so use exponents 46 and 42, the bigger one on 2. The number is 16 times 6 to the 42. |
| 7 | C | Fractions & Decimals — Reduce all 14 special fractions. Two of them can only add to a whole number if both are whole numbers, or both have the same bottom (2 or 4). |
| 8 | C | Primes — 16383 is 128 squared minus 1, so it factors as (128 - 1)(128 + 1) = 127 times 129 = 3 times 43 times 127, and 127 is prime. |
| 9 | C | Ratios, Percents & Averages — Pick 42 knights: 12 red, 30 blue, 7 magical. If a fraction x of blue knights is magical, then 12 times 2x plus 30 times x equals 7. |
| 10 | A | Logic Puzzles — Alice's doubt rules out 1 and 40. Then Bob can only be sure with 1, 2, 39 or 40; prime means 2, and 200 + 25 = 225. |
| 11 | B | Circles — Each reflected arc bends into the hexagon, so the region is the hexagon minus six thin slivers; each sliver is a 60-degree pie slice minus an equilateral triangle. |
| 12 | D | Algebraic Manipulation — Doubled, the left side is the sum of the squared differences of the three numbers. That equals 2 only when two numbers are equal and the third is one away. |
| 13 | B | Similar & Congruent Triangles — The width shrinks steadily going up: 3 wide at height 3, 2 wide at height 5. So the base is 9/2 and the top point is at height 9. |
| 14 | C | Basic Probability — Count failures: the product lacks a factor of 4 only if all dice are odd, or one die shows 2 or 6 and the rest are odd. |
| 15 | D | Similar & Congruent Triangles — Turning the square a quarter turn moves BP exactly onto CQ, so CQ = 13. Then BR is the height to the long side of right triangle QBC. |
| 16 | D | Expected Value — Find the chance one ball ends at home, then multiply by 5. It is home if no swap touches it, or it is swapped away and straight back. |
| 17 | D | Coordinate Geometry — Two flips over lines through the origin make a turn by twice their angle. P to P'' is a quarter turn clockwise, so m is l turned 45 degrees clockwise. |
| 18 | E | Quadrilaterals & Polygon Areas — The outline's 24 corners alternate: a square corner, then a crossing point halfway to the next one. So the shape splits into 24 identical thin triangles. |
| 19 | A | Exponents, Logarithms & Radicals — Write N as a front part times a power of 10 that divides evenly by r, then compare the front part with small r-th powers. |
| 20 | C | Conditional Probability & States — Hugo wins with chance 1/4 by fairness. After rolling 5 he wins with chance 41/96, splitting ties with other 5s evenly. Then divide. |
| 21 | E | Basic Counting — Each side of the polygon with fewer sides crosses the other polygon exactly twice, so each pair meets at twice the smaller side count. |
| 22 | B | Modular Arithmetic — Squaring 1 + 2 + ... + n gives each product twice, leading to a formula with 24 on the bottom. So the top needs a factor of 9. |
| 23 | D | Basic Probability — With no one-color triangle, every corner has 2 red and 2 blue lines, forcing the red lines into a five-sided loop: only 12 of 1024. |
| 24 | A | Arrangements with Restrictions — Sort by the most white cubes on any one face. Four gives 1 cube, three gives 4 (including two mirror-image twists), and two on every face gives 2. Total 7. |
| 25 | E | Quadrilaterals & Polygon Areas — The 1 by 3 rectangle and unit square fix the big square. In units where it is 10 by 10, rectangle R gives a quadratic with two answers. |
2021 AMC Fall 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.