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2024 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2024 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | A | Algebraic Manipulation — Multiplying by 101 means times 100 plus one more copy, and multiplying by 99 means times 100 minus one copy. Both products then come out in one line. |
| 2 | B | Linear Equations & Word Problems — Both trails take 69 minutes, so subtract one equation from the other. That gives a simple link between a and b: a is 1000 times b. |
| 3 | B | Primes — If 2 is one of the five primes, the total is even and cannot be prime. So use five odd primes, and try the smallest totals first. |
| 4 | B | Number Properties — The biggest two-digit number is 99. Twenty of them make only 1980, which is short of 2024, but twenty 99s plus a 44 make exactly 2024. |
| 5 | D | Primes — Break 2024 into primes: 2 times 2 times 2 times 11 times 23. The factorial must include the prime 23, and 23! already includes everything else. |
| 6 | D | Games & Processes — All 15 pairs of letters must switch order, and one swap of neighbours switches only one pair, so at least 15 swaps are needed, and 15 are enough. |
| 7 | B | Divisibility & Factors — The word is integers, so negatives are allowed. Two negative numbers keep the product positive but make the sum small: (−1)(−6)(10) = 60 with sum 3. |
| 8 | A | Linear Equations & Word Problems — Amy, Bomani and Charlie work the whole 105 minutes and pack 350 packages. Daria packs the other 100, which takes her 80 minutes, so she started 80 minutes before 2:45. |
| 9 | B | Basic Counting — Split the juniors into three pairs (15 ways), split the seniors into three pairs (15 ways), then match each junior pair with a senior pair (6 ways). |
| 10 | C | Sequences & Series — Work out the first ten or so steps. From step 6 the values repeat 20, 30, 10, so step 100 matches step 7, which is 30. |
| 11 | D | Diophantine Equations — Square both sides and factor: (n − m)(n + m) = 49. Then list the ways to split 49 into two integer factors. |
| 12 | E | Statistics & Data — Turn the day-to-day changes into actual scores with a running total. Then average how far each score is above 1700 instead of adding six big numbers. |
| 13 | C | Transformations & Symmetry — Write each move as a rule for where (x, y) goes, then do each pair in both orders. Only three pairs give the same point both ways. |
| 14 | D | Circles — The circle sits in the 120 degree corner outside the triangle. The region is a kite minus a 60 degree slice of the circle. |
| 15 | E | Diophantine Equations — The two squares differ by 2560 = (b − a)(b + a). M is biggest when a is biggest, which needs the smallest gap b − a. |
| 16 | D | Similar & Congruent Triangles — Every rectangle has the same long-to-short ratio r. Where rectangle 8 sits exactly on top of rectangles 9 and 1, the widths must match, and that gives r = √2. |
| 17 | E | Basic Probability — List the three ways Team A can win: win-win, win-lose-win, lose-win-win. Add their chances, set the total to one half, and solve the quadratic for p. |
| 18 | D | Bases & Digits — In base b, 2024 means 2(b³ + b + 2). So b³ + b + 2 must be a multiple of 8, which depends only on b's remainder mod 8. |
| 19 | E | Sequences & Series — The middle term squared equals the product of the outer terms, so ab = 720². Make a the largest divisor of 720² below 720. |
| 20 | C | Number Properties — Ten numbers in a row hold at most 3 members; four would force two odd members 6 apart. The pattern 1, 4, 8 in each ten reaches 608. |
| 21 | C | Sequences & Series — Measure from the bottom-right 0. Column 5 goes up in equal steps, and the row steps themselves change by equal amounts, so each entry is P·u + v·(Q + R·u). |
| 22 | B | Triangles: Area & Pythagorean — Every kite corner is 60, 90 or 120 degrees, so all lengths come from 30-60-90 triangles: AB = 3 + 3 and C is 3√3/2 above AB. |
| 23 | D | Systems of Equations — Subtract two equations and factor: (a − c)(b − 1) = 13. Since 13 is prime there are only four cases, and only b = −12 works. |
| 24 | B | Basic Probability — Each roll must use a new direction, and a direction used before must be reversed. Counting roll by roll gives 6 · 4 · 7 = 168. |
| 25 | C | Paths & Grids — An upright middle-row toothpick forces a full top-to-bottom line. A loop has exactly two such lines, and every other middle square picks top or bottom. |
2024 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.