2020 AMC 10B
All 25 problems, 75 minutes, scored 6 / 1.5 / 0. The answer key and the idea behind each problem are on this page too, folded away until you ask for them.
The problems
Show the answer key, topics and key insights
This gives away all 25 answers and the idea behind each one. Sit the paper first if you mean to.
The same key as a PDF 124 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 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. 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.