2021 AMC Fall 10B
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 127 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 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. 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.