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2023 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2023 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Ratios, Percents & Averages — Find the level every glass must end at first: the total juice split four ways. Then each full glass gives up the difference between full and that level. |
| 2 | B | Ratios, Percents & Averages — A 20% discount means times 0.8 and a 7.5% tax means times 1.075. Together Carlos pays 0.86 of the original price, so divide 43 by 0.86. |
| 3 | D | Circles — When a right triangle sits inside a circle with all three corners on it, its longest side is a diameter. So the diameters are 5 and 13. |
| 4 | C | Sets, Estimation & Miscellaneous — Change both lengths into centimeters before multiplying: 6.5 mm is 0.65 cm and 25 m is 2500 cm. Then the area comes out directly in square centimeters. |
| 5 | A | Linear Equations & Word Problems — Tripling every number triples the total, so the original total is 15. Adding 3 to every number adds 3 once for each number, and that extra is 30. |
| 6 | E | Modular Arithmetic — Look only at odd or even. The terms go odd, odd, even, and that pattern repeats, so a term is even exactly when its position is a multiple of 3. |
| 7 | B | Transformations & Symmetry — All eight corners lie on one circle around the center. Angle EAB is half of arc EB, and arc EB is 90 minus 20 = 70 degrees. |
| 8 | A | Modular Arithmetic — Only the last digit of each base matters. Last digits of powers of 2 and of 3 repeat every 4 steps, so use the remainders of 2023 and 2022. |
| 9 | B | Number Properties — Neighboring squares n squared and (n+1) squared always differ by 2n + 1. So count the values of n that make 2n + 1 at most 2023. |
| 10 | C | Paths & Grids — Every possible rectangle covers one of the four middle-of-edge squares, so 4 guesses always work. Three guesses can touch at most 10 of the 12 rectangle positions. |
| 11 | B | Diophantine Equations — Everything except the fifty-dollar bills comes in multiples of 20 or 100, so the fifties must come in pairs. Then list the choices by the number of hundreds. |
| 12 | C | Polynomials — To the right of 10 every factor is positive. Moving left, P changes sign only when you pass a root with an odd power: 9, 7, 5, 3, 1. |
| 13 | B | Absolute Value & Inequalities — The region looks the same in all four quadrants. In one quadrant it is a tilted square around (1, 1) with area 2, so the total is 4 times 2. |
| 14 | C | Diophantine Equations — Add mn to both sides to get (m+n) squared on the left. Then (mn) squared plus mn must be a perfect square, so mn is 0 or -1. |
| 15 | C | Number Properties — Pair each factorial with the next, like 3! times 4! = (3!) squared times 4. What remains is a square times 8!, and 8! needs 2 times 5 times 7. |
| 16 | E | Basic Counting — Picking the digits decides the number, so count sets of digits. The digit 0 can end a downno but can never start an upno. |
| 17 | D | Algebraic Manipulation — The diagonal squared is a squared plus b squared plus c squared. That equals (a+b+c) squared minus 2(ab+bc+ca), and the edge total and face total give both parts. |
| 18 | E | GCD & LCM — Multiply by 210 to get c = 15a + 14b. Then 2 and 7 divide c exactly when they divide a, and 3 and 5 exactly when they divide b. |
| 19 | B | Geometric Probability — All four directions are alike, so assume she hops north. She leaves when her height plus the hop passes 6: a small triangle in a 1 by 6 rectangle. |
| 20 | A | Solid Geometry — The four corners where the semicircles meet are evenly spaced around the sphere's equator. Neighboring corners are 2 times root 2 apart, so each semicircle has radius root 2. |
| 21 | E | Conditional Probability & States — Only odd or even matters for each bin. Track one question, 'do all three bins match?', ball by ball. The chance quickly settles at 1/4. |
| 22 | B | Number Properties — Call the floor k. Then x = (k squared + 2)/3, and this x must really have floor k. Only k = 0, 1, 2, 3 pass that test. |
| 23 | B | Sequences & Series — The real sum is 221 or 223. The number of terms divides twice the sum and is at most 14, since steps are at least 2. Only 13 fits. |
| 24 | E | Coordinate Geometry — For each fixed w the points form a 2 by 1 rectangle. As w goes from 0 to 1, the rectangle slides along (-3, 4) and sweeps out a hexagon. |
| 25 | B | Angles & Polygons — Folding a corner to the center creases halfway between them. So the new center-to-side distance is half the old center-to-corner distance: a golden-ratio shrink. |
2023 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.