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.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | D | Algebraic Manipulation | Subtracting a negative is adding, so the expression is 1 + 2 - 3 + 4 - 5 + 6, which pairs up neatly. | |
| 2 | E | Solid Geometry | A cube of side 2 has volume 8, not 2 or 4; the total is 5 times 1 plus 5 times 8. | |
| 3 | E | Ratios, Percents & Averages | Pick a concrete value for x that fits every ratio (x = 6 makes z = 1), then read off w and y. | |
| 4 | D | Primes | The angles sum to 90, so test the primes b = 2, 3, 5, 7 in order until 90 - b is also prime. | |
| 5 | B | Basic Counting | Arrange 7 tiles as if distinct (7!), then divide by 2! and 3! for the identical green and yellow tiles. | |
| 6 | B | Bases & Digits | The middle digit 9 cannot go up, so the next palindrome is 16061; the trip is 110 miles in 2 hours. | |
| 7 | A | Number Properties | An even multiple of 3 that is a square must be divisible by 36, so count k with 36k^2 < 2020. | |
| 8 | D | Triangles: Area & Pythagorean | Area 12 with base 8 forces R to lie 3 units from line PQ; split by where the right angle is and count intersections. | |
| 9 | D | Diophantine Equations | Complete the square: x^2020 + (y-1)^2 = 1, a sum of two nonnegative integers equal to 1, so one is 0 and the other 1. | |
| 10 | C | Solid Geometry | The sector's arc (three quarters of 8 pi) becomes the base circumference, so the base radius is 3 and the slant height is 4. | |
| 11 | D | Basic Probability | Fix Harold's five books; Betty must choose 2 of those 5 and 3 of the other 5 out of C(10,5) equally likely selections. | |
| 12 | D | Fractions & Decimals | Write 20^20 = 2^20 * 10^20; since 2^20 is about 1.05 million, 1/2^20 is about 9.5 * 10^-7, then shift 20 more places. | |
| 13 | B | Coordinate Geometry | Every block of four moves (east, north, west, south) shifts Andy by (-2, -2); 2020 turns is exactly 505 blocks. | |
| 14 | D | Circles | Adjacent semicircles overlap, so do not subtract 3 pi; instead the shaded region is a central hexagon of side 1 minus six 60-degree circular segments. | |
| 15 | D | Modular Arithmetic | Each erasure keeps the list periodic: 30 original digits become 20, then 15, then 12, so the final list repeats every 12 digits. | |
| 16 | A | Games & Processes | Bela takes the midpoint n/2 first, then mirrors every Jenn move across it; the mirror move is always legal, so Jenn runs out first. | |
| 17 | C | Basic Counting | Split by how many pairs are 'across' pairs; the leftover people must pair with neighbors along the circle, which forces even-length arcs. | |
| 18 | B | Conditional Probability & States | Every sequence of two red and two blue draws has the same probability 1/30, because the denominators are always 2,3,4,5 and the numerators are 1,2 for each color. | |
| 19 | A | Divisibility & Factors | Legendre's formula shows C(52,10) is divisible by 4 but not 8 and not by 3; digit rules for 3, 4, 8 leave only A = 2. | |
| 20 | B | Solid Geometry | The r-neighborhood of a box is the box, plus slabs on the faces, quarter-cylinders along the edges, and eighth-spheres at the corners; each contributes one power of r. | |
| 21 | B | Quadrilaterals & Polygon Areas | Extend CB, CD to line EH: the pentagon is a right isosceles triangle of altitude 2 sqrt(2) - 1 minus two 45-degree corners of leg FI. | |
| 22 | D | Algebraic Manipulation | The divisor times 2^101 - 2^51 + 1 equals 2^202 + 1 by difference of squares, so 2^202 is -1 modulo the divisor. | |
| 23 | C | Basic Counting | Each move sends vertex A to an adjacent corner, so after 19 moves the square is one move from home, and exactly one 20th move finishes: 4^19. | |
| 24 | C | Number Properties | Set the common value to k: then n = 70k - 1000 and k^2 <= 70k - 1000 < (k+1)^2, two quadratic inequalities in k. | |
| 25 | A | Distributions & Stars and Bars | 96 = 2^5 * 3; with k factors, place the 3 in one slot, then give every other slot a 2 and distribute the rest: k * C(5, k-1). |
Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic tags, difficulty ratings and key insights on this page are original to this site. No problem statements are reproduced here — each links to its own page.