2022 AMC 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 124 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | D | Fractions & Decimals | Work from the inside out. Each layer is 3 plus a fraction: add, then flip the result upside down to get the fraction for the next layer up. | |
| 2 | B | Ratios, Percents & Averages | At a steady speed, laps go up in step with time. 27 minutes is 9/19 of the ride, so he did 9/19 of the 15 laps, a little over 7. | |
| 3 | E | Linear Equations & Word Problems | Both other numbers are described using the third one, so call the third number c. Then the first is 6c and the second is c + 40. | |
| 4 | E | Ratios, Percents & Averages | Follow one trip of 100 kilometers: change it to miles, then to gallons of fuel used, then to liters. | |
| 5 | C | Triangles: Area & Pythagorean | The corner triangle at B has two equal legs, 1 - s, so its slanted side is (1 - s) times the square root of 2, which must equal s. | |
| 6 | A | Absolute Value & Inequalities | For negative a, the square root of (a - 1) squared is 1 - a, not a - 1. The inside becomes 2a - 3, which is negative. | |
| 7 | B | GCD & LCM | Break every number into primes. The lcm condition says n has two 2s and one 5; the gcd condition says n has exactly one 3. | |
| 8 | D | Statistics & Data | The mean is (20 + X)/6. Set it equal to each number in the list, including X itself, and keep the answers where X is a positive whole number. | |
| 9 | D | Arrangements with Restrictions | The wide bottom-middle region touches all four others. Paint it first, then go around the other four in order: 4 choices for the first, then 3 for each. | |
| 10 | E | Quadrilaterals & Polygon Areas | Call the sides a and b. The two measurements give a squared plus b squared = 64 and a + b = 10; the area ab follows directly. | |
| 11 | C | Exponents, Logarithms & Radicals | 4096 is 2 to the 12th power, so write both expressions as powers of 2. Equal exponents give a quadratic with solutions 3 and 4. | |
| 12 | A | Logic Puzzles | Write down how each kind of child answers the three questions. Truth-tellers say yes, no, no; liars say yes, yes, no; alternaters say no, no, no or yes, yes, yes. | |
| 13 | C | Triangle Centers & Cevians | Fold B across the bisector AP onto a point E of AC, so AE = AB. Then EC is half of AE, and similar triangles make AD twice BC. | |
| 14 | E | Arrangements with Restrictions | Numbers 8 to 14 can never be the smaller one in a pair, so they are the seven larger numbers. Pair 7 first, then 6, 5, 4. | |
| 15 | D | Circles | 7, 24, 25 and 15, 20, 25 are both right-triangle side sets with long side 25. So the diagonal AC is 25 and is a diameter of the circle. | |
| 16 | D | Polynomials | The new volume is (r + 2)(s + 2)(t + 2), where r, s, t are the roots. Plugging x = -2 into the polynomial gives this product almost directly. | |
| 17 | D | Fractions & Decimals | A repeating block abc equals abc/999 and a repeating digit a equals a/9. The condition becomes 7a = 3b + 4c, and then it is a short digit search. | |
| 18 | A | Transformations & Symmetry | Track only the point's angle. Each step turns angle A into 180 - A - k, so every two steps together turn the point back 1 degree. | |
| 19 | C | Modular Arithmetic | Multiply through by L17. Then h is L17/1 + L17/2 + ... + L17/17, and every term except the last is a multiple of 17. | |
| 20 | E | Sequences & Series | Subtracting neighboring terms twice removes the arithmetic part and leaves b(r - 1) squared = 28. So the ratio r is 2 or 3. | |
| 21 | B | Solid Geometry | The rim is level, so its area equals its shadow seen from above. A shared hexagon edge shows each top edge sits 1 unit outside the square. | |
| 22 | D | Basic Counting | Two passes means cards 1 to k appear in increasing order and so do cards k+1 to 13. Choosing which k spots hold the first group decides the whole row. | |
| 23 | B | Quadrilaterals & Polygon Areas | Put the trapezoid's mirror line on the y-axis. Then PD squared minus PA squared, and PC squared minus PB squared, are each a base times the same unknown. | |
| 24 | E | Recursive Counting | Count the bad strings by the first condition they break. At that point the string splits into a smaller good string of low digits plus a block of high digits. | |
| 25 | B | Coordinate Geometry | A lattice square of side n has (n + 1) squared points. Every condition becomes a count, giving one whole-number equation plus a remainder condition. |
2022 AMC 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.