Printable paper
2020 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2020 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | D | Algebraic Manipulation — Taking away a negative number is the same as adding it, so the expression becomes 1 + 2 - 3 + 4 - 5 + 6. |
| 2 | E | Solid Geometry — A cube with side 2 has volume 2 times 2 times 2 = 8, not 2 or 4. So the total is 5 times 1 plus 5 times 8. |
| 3 | E | Ratios, Percents & Averages — Give x a handy value that works with every ratio (x = 6 makes z = 1), work out w and y, then divide. |
| 4 | D | Primes — The two acute angles add to 90. Try b = 2, 3, 5, 7 in order and stop at the first one where 90 minus b is also prime. |
| 5 | B | Basic Counting — Place the colors one group at a time: choose 3 of the 7 spots for yellow, then 2 of the remaining 4 for green, then order brown and purple. |
| 6 | B | Bases & Digits — A palindrome is fixed by its first three digits. After 159 comes 160, so the next palindrome is 16061, which is 110 miles later. |
| 7 | A | Number Properties — A square that is a multiple of 6 must be the square of a multiple of 6, so list 6 squared, 12 squared, and so on below 2020. |
| 8 | D | Triangles: Area & Pythagorean — The area forces R to be exactly 3 units from line PQ. Then check each place the right angle could be: at P, at Q, or at R. |
| 9 | D | Diophantine Equations — Rewrite it as x to the 2020th power plus (y - 1) squared equals 1. Two never-negative whole numbers adding to 1 must be 0 and 1. |
| 10 | C | Solid Geometry — When the sector is rolled up, its radius 4 becomes the slanted side of the cone, and its curved edge wraps around the base. That makes the base radius 3. |
| 11 | D | Basic Probability — Whatever five books Harold picks, Betty must take 2 of his 5 books and 3 of the other 5. Count her ways and divide by all her possible choices. |
| 12 | D | Fractions & Decimals — 20 to the 20th power is 1,048,576 followed by 20 zeros, a 27-digit number. So 1 divided by it has 26 zeros before the first nonzero digit. |
| 13 | B | Coordinate Geometry — Group the moves in fours (east, north, west, south). Each group moves Andy 2 left and 2 down, and 2020 moves make exactly 505 groups. |
| 14 | D | Circles — Neighboring semicircles cross halfway between the center and each corner. The shaded part is six small triangles of side 1, each missing a curved bite. |
| 15 | D | Modular Arithmetic — Work with blocks of 30 digits: erasing leaves 20, then 15, then 12. So the final list repeats every 12 digits. |
| 16 | A | Games & Processes — Bela takes the exact middle first, then answers each of Jenn's numbers with its mirror image across the middle. That reply is always allowed, so Jenn gets stuck first. |
| 17 | C | Basic Counting — Split into cases by how many pairs are 'across' pairs. Everyone left must pair with a neighbor, so each leftover stretch of the circle needs an even number of people. |
| 18 | B | Conditional Probability & States — Every order of two red and two blue draws has chance 1/30: denominators are always 2, 3, 4, 5 and numerators 1, 2, 1, 2. There are 6 orders. |
| 19 | A | Divisibility & Factors — The number of hands is divisible by 4 but not 8, and not by 3. The digit tests for 3, 4 and 8 then leave only A = 2. |
| 20 | B | Solid Geometry — The region is the box, plus slabs on the six faces, plus quarter-cylinders along the twelve edges, plus eighth-balls at the eight corners. Each kind gives one power of r. |
| 21 | B | Quadrilaterals & Polygon Areas — Extend CB and CD to meet line EH. The pentagon is a big half-square triangle minus two small half-square corners, each of area FI squared over 2. |
| 22 | D | Algebraic Manipulation — Multiply the divisor by 2^101 - 2^51 + 1 and you get exactly 2^202 + 1. So 2^202 + 202 is a multiple of the divisor plus 201. |
| 23 | C | Basic Counting — Each move sends corner A to a neighboring corner, so after 19 moves the square is exactly one move from home, and exactly one 20th move finishes. |
| 24 | C | Number Properties — Call the common value k. Then n = 70k - 1000, and n must be at least k squared but less than (k+1) squared. Solve both conditions. |
| 25 | A | Distributions & Stars and Bars — 96 is five 2s times a 3. Put the 3 in one factor, give every other factor one 2, then share out the leftover 2s freely among all the factors. |
2020 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.