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2020 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2020 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | E | Linear Equations & Word Problems — Work out the right side first (it is 1/12), then add 3/4 to both sides. Using twelfths for every fraction keeps the arithmetic small. |
| 2 | C | Ratios, Percents & Averages — Five numbers averaging 15 add up to 75. Take away 3 + 5 + 7 = 15 to get a + b = 60, then halve it. |
| 3 | A | Algebraic Manipulation — Each top is a bottom with its sign flipped, like a - 3 and 3 - a. Matched up, each fraction is -1, so the product is -1. |
| 4 | E | Linear Equations & Word Problems — The answer is per hour, so look at one hour. She drives 60 miles, earns 30 dollars, uses 2 gallons costing 4 dollars, and keeps 26 dollars. |
| 5 | C | Absolute Value & Inequalities — The inside must equal 2 or -2. The first case gives x = 4 and x = 8; the second gives only x = 6. Add the three different numbers. |
| 6 | B | Basic Counting — The last digit must be 0 (even and divisible by 5). The first digit has 4 choices, since it cannot be 0, and each middle digit has 5. |
| 7 | C | Sequences & Series — The five rows together use every number exactly once. So five times the row sum equals the total of all 25 numbers, which is 50. The row sum is 10. |
| 8 | B | Sequences & Series — Cut the sum into 50 groups of four. The groups add to 2, 10, 18, and so on, going up by 8 each time. Add those 50 numbers. |
| 9 | B | Divisibility & Factors — Cut a bench into 77 pieces: an adult uses 11 and a child 7, so each adult-child pair uses 18. So 77 times N must be a multiple of 18. |
| 10 | B | Solid Geometry — Side faces are never covered. From above you see exactly a 7 by 7 square of top faces, and from below another 7 by 7 square. |
| 11 | C | Statistics & Data — With 4040 numbers the median is halfway between the 2020th and 2021st. The squares 1, 4, ..., 1936 (44 of them) squeeze in below, pushing the middle down by 44. |
| 12 | C | Triangle Centers & Cevians — The medians cross two thirds of the way along each, so triangle MPC has legs 8 and 8 and area 32. It is one third of the whole. |
| 13 | B | Conditional Probability & States — The points (1,1), (1,3) and the center sit on diagonals of the square, so from them both kinds of side are equally likely. Look one jump ahead. |
| 14 | D | Algebraic Manipulation — Combine the fractions over (xy) squared, which is 4. Then build the sum of fifth powers from the sum 4 and product -2, never finding x or y. |
| 15 | E | Divisibility & Factors — Write 12! as prime powers. A divisor chooses a power of each prime, and it is a square exactly when every chosen power is even. |
| 16 | B | Geometric Probability — All unit squares look alike. The points within d of the four corners form four quarter-circles, one full circle of area pi d squared. Set that equal to 1/2. |
| 17 | E | Absolute Value & Inequalities — P(n) is zero at the 100 squares. Between squares its sign flips each time, so it is negative in every other gap, starting with the gap from 1 to 4. |
| 18 | C | Number Properties — The difference is odd when exactly one of the two products is odd. A product of two numbers from 0 to 3 is odd for only 4 of 16 pairs. |
| 19 | E | Basic Counting — Every route goes: top face, along the top ring, down, along the bottom ring, bottom face. Each ring walk has 9 options, so 5 times 9 times 2 times 9. |
| 20 | D | Quadrilaterals & Polygon Areas — Diagonal AC splits the shape into two triangles on one base. ACD is easy. For ABC, find the height from B using similar triangles at E and the right angle. |
| 21 | C | Bases & Digits — Let x be 2 to the 17th. The fraction becomes x^16 - x^15 + ... + 1, and each pair like x^16 - x^15 is seventeen ones in binary. |
| 22 | A | Number Properties — The three floors are equal, so their sum is a multiple of 3, unless n divides 999 or 1000. Count those divisors, leaving out n = 1. |
| 23 | A | Transformations & Symmetry — T has no symmetry, so every point must return home. A reflection makes a mirror image, so there must be 0 or 2 reflections. Each case gives 6. |
| 24 | C | GCD & LCM — Each gcd says a number divides and a bigger power does not. The dividing parts give n = 237 plus multiples of 420; test 1077, 1497, 1917. |
| 25 | A | Basic Probability — Keeping one die and rerolling two is best only when every pair of dice adds to 7 or more and the smallest die is 1, 2 or 3. |
2020 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.