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2023 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2023 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | E | Linear Equations & Word Problems — Each hour the two riders get 18 + 12 = 30 miles closer, so they meet after 1.5 hours. Then find how far Alicia rode. |
| 2 | A | Linear Equations & Word Problems — Take away what the two sides share. What is left says 5/12 of a pizza weighs the same as 3 cups of orange slices, which is 3/4 pound. |
| 3 | A | Number Properties — A square is a multiple of 5 only when the number squared is a multiple of 5. Count 5, 10, 15, and so on while the square stays below 2023. |
| 4 | D | Geometric Optimization — The longest side must be shorter than the other three sides added together. With a perimeter of 26 that means it is less than 13, and 12 works. |
| 5 | E | Exponents, Logarithms & Radicals — Break every number into primes. Pair each 2 with a 5 to make 10s. What is left is 243 followed by fifteen zeros. |
| 6 | D | Basic Counting — Follow one vertex's number: 3 edges use it and each edge sits on 2 faces, so it is counted 6 times. The cube's value is 6 times 21. |
| 7 | B | Basic Probability — Each roll adds at least 1, so the total can reach 3 only in the first three rolls. List the orders adding to 3: 3; 1,2; 2,1; 1,1,1. |
| 8 | D | Linear Equations & Word Problems — 200 is 90 of the 240 degrees from 110 to 350, which is 3/8 of the way. On the new scale that is 3/8 of 100, or 37.5. |
| 9 | E | Bases & Digits — The year 2023 already has one 0 and one 3. So the month and day together must use one 0, one 3, and two copies of some other digit. |
| 10 | D | Ratios, Percents & Averages — Turn each mean into a total: n quizzes with mean m add up to n times m. Each story then gives a simple equation. |
| 11 | C | Triangles: Area & Pythagorean — The legs add to the big square's side and their squares add to 2. That gives their product, and sum plus product gives the ratio. |
| 12 | B | Divisibility & Factors — Reversing N moves its first digit to the end. A multiple of 5 ends in 0 or 5, and N cannot start with 0, so N is in the 500s. |
| 13 | C | Circles — All the spots where Bharat sees Abdul and Chiang at a 60 degree angle lie on one circle through them. Abdul-to-Bharat is longest when it is a diameter. |
| 14 | B | Conditional Probability & States — Only the 9 multiples of 11 up to 99 can work. For each, exactly half its divisors are multiples of 11, because each divisor d pairs with 11 times d. |
| 15 | E | Sequences & Series — Each shaded ring is the difference of two circles, and those areas are 3, 7, 11, ... times pi. With n rings the total is n(2n + 1) times pi. |
| 16 | B | Games & Processes — Every game has exactly one winner, so total wins equal total games. Left wins are 1.4 times right wins, which makes the total a multiple of 12. |
| 17 | A | Triangles: Area & Pythagorean — Go around the rectangle one right triangle at a time. Leg 30 forces 16-30-34; then leg 12 allows 5, 9 or 16, and only 9 makes the last triangle work. |
| 18 | D | Solid Geometry — Count edges from the faces (12 times 4, halved), then vertices from Euler's formula. Counting edge ends two ways gives the split into 3-edge and 4-edge vertices. |
| 19 | E | Coordinate Geometry — A rotation keeps each point the same distance from the center. So the center is on the perpendicular bisector of A and A', and of B and B'. |
| 20 | D | Paths & Grids — The center square is in all four 2 by 2 blocks, so pick it first. The four edge-middle squares then use the other three colors, and each corner is forced. |
| 21 | D | Polynomials — Each clue is a value of P: P(1) is 1, and P is 0 at 0, 4 and 9. Those roots alone make P(1) equal 24, so add another root. |
| 22 | D | Circles — C3 sits at the middle point between the two centers with radius 3/4. By symmetry C4's center is straight above that point, which gives one right triangle. |
| 23 | C | Diophantine Equations — N equals a(a + 20) and b(b + 23). If a were b + 2 or more, the first product would be too big, so a is b + 1. |
| 24 | C | Quadrilaterals & Polygon Areas — Only the frame's side matters. Neighboring blocks share a side line, which forces the two gaps on each frame edge to add up to 2, so the side is 3. |
| 25 | A | Basic Probability — Stand at R and sort the other 11 vertices by distance: 5 at distance 1, 5 at distance 2, 1 at distance 3. Then count pairs. |
2023 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.