2016 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 | B | Algebraic Manipulation | 11! is just 11 times 10!, so the top is 10 copies of 10!. Then 10! divided by 9! is 10, giving 10 times 10. | |
| 2 | C | Exponents, Logarithms & Radicals | Write 100 and 1000 as powers of 10. The left side becomes 10 to the power 5x and the right side 10 to the power 15, so 5x = 15. | |
| 3 | C | Ratios, Percents & Averages | David spends 75 cents for each dollar Ben spends, so the 12.50 dollar gap is one quarter of Ben's total. Ben spent 50 dollars and David 37.50. | |
| 4 | B | Number Properties | The division 3/8 divided by -2/5 is -15/16, a bit above -1. Rounding down gives -1, not 0, so the answer is 3/8 - 2/5 = -1/40. | |
| 5 | D | Number Properties | Call the sides k, 3k and 4k. The volume is 12 times k cubed, so it must be 12 times a perfect cube. Only 96 = 12 times 8 works. | |
| 6 | D | Bases & Digits | A 2 turned into 1 costs 1 in the ones place and 10 in the tens place. Count the 2s in each place separately. | |
| 7 | D | Statistics & Data | Only the mean gives an equation: 540 + x must equal 7x, so x = 90. Then check that 90 is also the median and mode. | |
| 8 | C | Linear Equations & Word Problems | Work backwards from 0 coins. To undo a crossing, add back the 40 coins and then cut the amount in half. Three undo steps give 20, then 30, then 35. | |
| 9 | D | Sequences & Series | The rows add up to N(N+1)/2 coins, so N times (N+1) is 4032. Since 63 squared is close to 4032, try 63 times 64, which is exactly 4032. | |
| 10 | B | Quadrilaterals & Polygon Areas | With inner length x, the colored areas are x, 2x + 6 and 2x + 14. Equal gaps need x + 6 = 8, so x = 2. | |
| 11 | D | Quadrilaterals & Polygon Areas | Turning the picture half a turn about the rectangle's center leaves it unchanged, so the slanted lines cross at the center and the two shaded pieces are equal. | |
| 12 | A | Basic Probability | All three numbers must be odd. The first pick is odd half the time, each later pick slightly less, so p is below 1/8. | |
| 13 | B | Logic Puzzles | The seat numbers always add to 15, so all moves cancel. Bea +2, Ceci -1, the swap 0: Ada moved one seat left. | |
| 14 | C | Diophantine Equations | The number of threes must be even, so it can be 0, 2, 4, up to 672. That is 337 choices. | |
| 15 | A | Circles | The dough's radius is 3: the center cookie's radius plus an outer cookie's width. The scrap is 9 pi - 7 pi = 2 pi. | |
| 16 | D | Transformations & Symmetry | Follow the corners: every point (x, y) ends at (y, x). That is a reflection over y = x, and repeating it undoes it. | |
| 17 | A | Basic Probability | Only the red ball's spot matters; all N + 1 spots are equally likely. For N = 5m, exactly m - 1 middle spots fail. | |
| 18 | C | Arrangements with Restrictions | Equal face sums force opposite corners to differ by one fixed amount. The numbers 1 to 8 pair up that way only for differences 1, 2 and 4. | |
| 19 | E | Similar & Congruent Triangles | Because BC is parallel to AD, lines from A cut BD in the ratios 1 to 3 and 2 to 3. Use twentieths. | |
| 20 | B | Distributions & Stars and Bars | Each term records how many brackets give a, b, c, d or 1. With every letter used at least once, stars and bars gives N choose 4. | |
| 21 | D | Circles | For two touching circles on a line, the centers are the radius-sum apart and differ in height by the radius difference, so the Pythagorean theorem gives the sideways gap. | |
| 22 | D | Divisibility & Factors | The primes 2, 5 and 11 each add a factor of at least 2 to the divisor count 110 = 2 times 5 times 11, leaving nothing for other primes. | |
| 23 | A | Functions | Put c = b so that b diamond b = 1 appears, then put b = a. The operation turns out to be division. | |
| 24 | E | Circles | Three equal sides make an isosceles trapezoid. Find the diagonal from the center using areas and the Pythagorean theorem, then compare two right triangles to get the fourth side. | |
| 25 | A | GCD & LCM | The lcm takes the larger power of each prime, so treat 2, 3 and 5 separately: 5, 3 and 1 choices, and 15 in all. |
2016 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.