2021 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 | Absolute Value & Inequalities | Since 3π ≈ 9.42, the condition is −9 ≤ x ≤ 9, which contains 19 integers including zero. | |
| 2 | D | Exponents, Logarithms & Radicals | √(t²) = |t|, and 2√3 ≈ 3.46 exceeds 3, so the first term is 2√3 − 3, not 3 − 2√3. | |
| 3 | C | Ratios, Percents & Averages | Equal team counts mean J/4 = S/10, so juniors to seniors is 2 : 5; split 28 in that ratio to get 8 juniors. | |
| 4 | B | Linear Equations & Word Problems | The 23 blue–blue pairs use 46 blue students, so the other 11 blues each sit with a yellow, leaving 64 yellows in 32 pairs. | |
| 5 | B | Divisibility & Factors | 30 has only one single-digit factor pair, 5 × 6, so 6 is taken and 24 must be 3 × 8. | |
| 6 | C | Statistics & Data | Take 3 and 4 students: the total 3·84 + 4·70 = 532 over 7 students gives 76; the answer is a weighted mean. | |
| 7 | D | Circles | Same-side circles nest, so only the largest annulus counts per side; put 7 alone and 5, 3, 1 together: 49 + 16 = 65. | |
| 8 | A | Paths & Grids | Odd squares sit at top-right corners: 169 ends the 13×13 block at row 2, so that row holds 157–169, then 170 and 210 at its ends. | |
| 9 | D | Coordinate Geometry | Undo the transformations in reverse order: reflect (−6,3) over y = −x to get (−3,6), then rotate it 90° clockwise about (1,5). | |
| 10 | A | Solid Geometry | Equate volumes: (1/3)π·12²·18 = π·24²·h, so h = 864/576 = 1.5. | |
| 11 | D | Diophantine Equations | Interior (m−2)(n−2) equals half of mn; substituting a = m−2, b = n−2 gives (a−2)(b−2) = 8, so (m,n) is (5,12) or (6,8). | |
| 12 | C | Divisibility & Factors | Every even divisor is 2, 4, or 8 times an odd divisor, so the even sum is (2 + 4 + 8) = 14 times the odd sum. | |
| 13 | B | Bases & Digits | Both base-n numerals start 3n²: 3n² ≤ 263 forces n ≤ 9, while 324_n ≥ 1101_6 = 253 forces n ≥ 9, so n = 9. | |
| 14 | B | Circles | Equal chords are equidistant from the center, so the center lies midway between the 38-chords: r² = 19² + (d/2)² = 17² + (3d/2)². | |
| 15 | B | Algebraic Manipulation | Squaring twice gives x⁴ + 1/x⁴ = 7; multiply by x⁷ and the target expression x¹¹ − 7x⁷ + x³ is exactly zero. | |
| 16 | C | Divisibility & Factors | Uphill numbers cannot end in 0, so the last digit is 5; choose a subset of {1,2,3,4} whose sum with 5 is a multiple of 3. | |
| 17 | C | Logic Puzzles | Start with the most constrained scores: Oscar's 4 forces {1,3}, then Aditi's 7 forces {2,5}, then Ravon's 11 forces {4,7}. | |
| 18 | C | Basic Probability | Ignore repeats: the six faces first appear in a uniformly random order, and we need the first three to be the evens: 1/C(6,3) = 1/20. | |
| 19 | D | Statistics & Data | Both 'remove the greatest' and 'remove the least' leave n − 1 numbers, and their sums differ by 72, so 72/(n − 1) = 40 − 32. | |
| 20 | D | Quadrilaterals & Polygon Areas | Triangles ABF and BCF are equilateral, so ∠ABC = 120° and AC = 2√3; the pentagon is the fan ABC + ACD + ADE from A. | |
| 21 | A | Transformations & Symmetry | Fold preserves lengths, so the small right triangle at D has perimeter DC' + DC = 4/3; triangle AEC' is similar to it with ratio AC'/DF = 3/2. | |
| 22 | D | Inclusion-Exclusion | Fix Ang's placement; box i is monochromatic iff Ben's and Jasmin's permutations both fix i, so count pairs with a common fixed point by inclusion–exclusion. | |
| 23 | C | Geometric Probability | The center lives in a 7×7 square; it hits black iff within 1/2 of a black shape, so expand the diamond by 1/2 and clip each corner triangle. | |
| 24 | B | Games & Processes | Each wall has a nim-value (mex of reachable values): walls 1–6 give 1,2,3,1,4,3; Beth wins exactly when the XOR is 0, only for (6,2,1). | |
| 25 | E | Coordinate Geometry | The area under y = mx for 0 ≤ x ≤ 30 is 450m, so test m = 2/3 (exactly 300 points); the count holds until slope 19/28. |
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.