2019 AMC 10A
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 126 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Exponents, Logarithms & Radicals | Work each tower from the top down. The first becomes 0 to the power 1, which is 0, then 2 to the power 0, which is 1. | |
| 2 | A | Divisibility & Factors | 15! already contains three 5s and plenty of 2s, so it is a multiple of 1000. So is 20!, and so is their difference. | |
| 3 | D | Linear Equations & Word Problems | With Bonita b and Ana b squared, last year's condition is b squared minus 1 = 5(b minus 1). Cancel the shared factor b minus 1. | |
| 4 | B | Sets, Estimation & Miscellaneous | Build the worst draw with no 15 of a color: 14 of each big color plus every small-color ball, 75 in all. One more forces it. | |
| 5 | D | Number Properties | The integers may be negative. A run from -44 up to 44 adds to 0, so adding 45 on the end gives sum 45 with 90 numbers in the run. | |
| 6 | C | Circles | A point equally far from all four corners is the center of a circle through them. That needs opposite angles adding to 180 degrees. | |
| 7 | C | Coordinate Geometry | Write the two lines through (2,2), find where each meets x + y = 10, then use the side on that line as the base. | |
| 8 | C | Transformations & Symmetry | Watch the little tails. A mirror flip changes which way a tail slants; a half-turn does not. The lower row looks exactly like the upper row turned halfway around. | |
| 9 | B | Divisibility & Factors | Dividing n! by n(n+1)/2 leaves 2 times (n-1)! over n+1. This fails to be whole only when n+1 is a prime, and the largest prime below 1000 is 997. | |
| 10 | C | Paths & Grids | Each grid line crossed means a new tile. The bug crosses 16 lines one way and 9 the other, never two at once, so it visits 26 tiles. | |
| 11 | C | Divisibility & Factors | 201 = 3 times 67, so each divisor is a power of 3 times a power of 67. Count squares (25) and cubes (16), then remove the 4 counted twice. | |
| 12 | E | Statistics & Data | The modes are 1 to 28, so d = 14.5. The 183rd value is 16, so M = 16. Missing large dates pull the mean just below 16. | |
| 13 | D | Circles | Any point on a circle sees the diameter at a right angle, so BD and CE are both heights of the triangle. Then triangle BFC has angles 50 and 20. | |
| 14 | D | Angles & Polygons | Four lines give at most 6 crossing points. Parallel lines and three lines through one point reduce that. Every count from 0 to 6 can happen except 2. | |
| 15 | E | Sequences & Series | Flip every term upside down. The flipped terms go up by the same amount each time: 1, 7/3, 11/3, 15/3, and so on. | |
| 16 | A | Circles | Join the centers of touching circles; each join is 2 long. The top center is 2 times root 3 above the middle, so the big radius is that plus 1. | |
| 17 | D | Basic Counting | Stack all 9 cubes and then take off the top one. Every 8-cube tower comes from exactly one 9-cube stack, so just count 9-cube stacks. | |
| 18 | D | Bases & Digits | In base ten 0.2323... is 23/99; in base k it is (2k + 3) over (k squared minus 1). So 51 must divide k squared minus 1. | |
| 19 | B | Algebraic Manipulation | Pair outside and inside factors: both give x squared plus 5x, plus a number. Call that u; the product is (u + 4)(u + 6), never below minus 1. | |
| 20 | B | Basic Probability | Only odd or even matters. The five odd numbers must fill one whole row and one whole column, a cross shape. There are 9 crosses out of 126 odd-even patterns. | |
| 21 | D | Solid Geometry | The sphere cuts the triangle's plane in a circle that touches all three sides, which is the inscribed circle, radius 4. Then use a right triangle with hypotenuse 6. | |
| 22 | B | Conditional Probability & States | Each number is 0 (chance 1/4), 1 (chance 1/4) or a random point in between (chance 1/2). Split into cases by type and add: 1/8 + 1/4 + 1/16. | |
| 23 | C | Sequences & Series | Tadd speaks in every third turn, saying 1, 4, 7, ... numbers. Add these to find which turn holds his 2019th number, then find where that turn starts. | |
| 24 | B | Polynomials | Clear the fractions and plug in each root: 1/A is the product of p minus each other root. Adding the three needs only the sum and pair-products of the roots. | |
| 25 | D | Divisibility & Factors | Counting ways to split n squared people into n teams shows the fraction is a whole number times (n-1)!/n. Only primes and n = 4 fail. |
2019 AMC 10A. 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.