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2024 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2024 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | B | Basic Counting — The person in the middle is counted once from each end. Count the people on each side of that person, then add that person back once. |
| 2 | B | Algebraic Manipulation — 10 times 9 times 8 is 720, which is also 6!. So 10! and 7! times 6! are the same number, and their difference is zero. |
| 3 | E | Absolute Value & Inequalities — 7 times pi is just under 22, so 2x can go from -21.99 to 21.99. That lets x be every whole number from -10 to 10. |
| 4 | D | Sequences & Series — Step 63 ends at ball 2016, so ball 2024 is in step 64. Bins repeat every 5 steps, and step 64 lands on bin D. |
| 5 | B | Sequences & Series — The full sum is 2500. Flipping a plus to a minus takes away twice that number, so the flipped numbers must add up to more than 1250. |
| 6 | B | Divisibility & Factors — The perimeter is smallest when the two sides are as close as possible. Since 2024 is one less than 45 squared, the sides 44 and 46 work. |
| 7 | A | Modular Arithmetic — Pull out the common factor 7 to the power 2024. What is left is 1 + 7 + 49 = 57, which is 3 times 19. |
| 8 | D | Divisibility & Factors — The divisors of 42 pair up so each pair multiplies to 42. Four pairs make N equal to 42 to the 4th power, which ends in 6. |
| 9 | A | Algebraic Manipulation — Squaring the sum of the numbers gives their squares plus twice their pairwise products. So 0 = 30 plus twice the products, and the products add to -15. |
| 10 | A | Similar & Congruent Triangles — Triangle AFE is a half-size copy of triangle CFB, so its area is one quarter as big. Give AFE area 1 and find every other piece from there. |
| 11 | C | Similar & Congruent Triangles — Equal areas make XM twice ZA. The right angle at M makes the two corner triangles WXM and MYA the same shape, which pins down ZA = 1. |
| 12 | A | Basic Counting — All students speak the same number of languages, so the rule just says no two students speak exactly the same group. Choosing 4 of 9 languages gives 126 groups. |
| 13 | B | Exponents, Logarithms & Radicals — The square root of 1183 is 13 root 7, so root x and root y are whole multiples of root 7 that add to 13. |
| 14 | B | Geometric Probability — The board is a tilted square of area 128. The target is a ring of area 14 pi that fits exactly inside it. |
| 15 | C | Statistics & Data — The mean makes x + y + z equal 11.2 or 20.2, and the median must be x, y or z. Then split by the smallest and largest numbers. |
| 16 | A | Games & Processes — Each move removes three numbers, so the count always leaves remainder 2 when divided by 3. The odd numbers never grow past 1012. Only 1010 fits both. |
| 17 | D | Arrangements with Restrictions — Split into cases by how many snails tie. Choose the tied snails, treat them as one block, then line up the block and the other snails. |
| 18 | B | Modular Arithmetic — Multiples of 5 give remainder 0. Any other number to the 100th power is 1 more than a multiple of 125, built up one factor of 5 at a time. |
| 19 | C | Coordinate Geometry — A line through two lattice points has a fraction slope. A fraction slope through one lattice point hits infinitely many. So each row has exactly two Possible entries. |
| 20 | A | Arrangements with Restrictions — A left shoe next to a right shoe must be its own mate. First find the allowed left-right patterns, then see which pairs are forced into place. |
| 21 | C | Circles — Two touching circles of radii r and s sit 2 times root(rs) apart along the floor. Measure along the floor for each spot the third pipe could take. |
| 22 | A | Basic Counting — Line up the 16 people, cut the line into four groups of 4, and divide out orders that do not matter. Then count the 3s on top and bottom. |
| 23 | B | Sequences & Series — The ten fractions come out to whole numbers 1, 3, 4, 7, 11, ... and each one is the sum of the two before it, just like Fibonacci numbers. |
| 24 | E | Modular Arithmetic — Put everything over 8. For even m each top term is a multiple of 8. For odd m the top becomes 4(m + 1), a multiple of 8. |
| 25 | E | Systems of Equations — A new side cannot grow from the same brick edge as before. That forces 7a = 3c + 1, 2b = 3a + 1, 2c = 3b + 1. |
2024 AMC 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.