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2022 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2022 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 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 — 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.