2015 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 127 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 | A negative power means one over the positive power, and squaring -2 gives positive 4. So the expression is 2 minus 1/4. | |
| 2 | B | Clocks, Calendars & Time | From 1:00 to 2:40 is 100 minutes, and that covers two tasks, not three. So each task takes 50 minutes. | |
| 3 | A | Linear Equations & Word Problems | If 28 is written twice, the other number is 44/3, not a whole number. So 28 is written three times and the other is 8. | |
| 4 | C | Fractions & Decimals | Write every share out of 60: Alex 12, Beth 20, Cyril 15. Dan gets the rest, 60 - 47 = 13, which is more than Alex's 12. | |
| 5 | B | Logic Puzzles | Only Marta's place is given as an exact number, 6th. Start there and follow the clues one by one: Jack 5th, Todd 3rd, Rand 2nd, Hikmet 8th. | |
| 6 | E | Logic Puzzles | Tuesday must be a running day, since four days in a row hold at most two runs. The tennis rule then rules out all but one schedule. | |
| 7 | A | Functions | Work from the inside parentheses out. The left side is 1/2 - 1/3 = 1/6, the right side is 1 - 3/5 = 2/5, and 1/6 - 2/5 = -7/30. | |
| 8 | E | Transformations & Symmetry | Follow one point through all three moves: (x, y) ends up at (y, x). So the three moves together just flip the F across the line y = x. | |
| 9 | B | Circles | The shaded fin is a quarter of the big circle with half of the small circle cut out: 9pi/4 - 9pi/8 = 9pi/8. | |
| 10 | C | Modular Arithmetic | There are 1007 negative numbers, an odd count, so the product is negative. All the numbers are odd and one of them is -5, so the product ends in 5. | |
| 11 | B | Primes | Only the digits 2, 3, 5, 7 are allowed, so there are just 4 + 16 = 20 numbers to check. Eight of them are prime. | |
| 12 | A | Coordinate Geometry | Put (x, -x) into the distance formula from (5, 5). The x terms cancel and the condition becomes x squared at most 25, so x goes from -5 to 5. | |
| 13 | E | Triangles: Area & Pythagorean | The triangle is a right triangle with sides 5, 12, 13. The two short sides are already altitudes, and the third altitude comes from the area: 60/13. | |
| 14 | D | Quadratics | Pull out the shared factor (x - b). The roots are b and the average of a and c, so put 9 in b and 8, 7 in a, c. | |
| 15 | B | Diophantine Equations | With h horses and c cows the total is 13h + 5c. Take 13s away from each choice and look for a multiple of 5. | |
| 16 | C | Basic Probability | Each number must be a bigger multiple of the next, so Cal gets 1 or 2. There are 9 good outcomes out of 720. | |
| 17 | B | Solid Geometry | The side-face centers make a diamond in the middle layer, half of 5 by 4. The octahedron is two pyramids on that diamond. | |
| 18 | D | Expected Value | A coin ends tails only after three tails in a row, chance 1/8. So each coin ends heads with chance 7/8, and 64 times 7/8 = 56. | |
| 19 | C | Circles | The circle's center is equally far from both ends of every chord, which puts it at the middle of AB. Then equal distances force equal legs. | |
| 20 | A | Paths & Grids | Color corners in two colors so each edge joins different colors. After 7 moves Erin must end on the corner opposite the start. | |
| 21 | D | Number Properties | Jump counts are steps divided by 2 or 5, rounded up. Their difference stays close to 3/10 of the steps, so only 62 to 66 need checking. | |
| 22 | D | Angles & Polygons | Every triangle in the figure has angles 36, 72, 108 degrees, so lengths grow by the golden ratio: FG = 1/phi, JH = 1, CD = phi. | |
| 23 | B | Divisibility & Factors | End zeros come from factors of 5. Doubling n usually just doubles them, so list small n and find the few that triple: 8, 9, 13, 14. | |
| 24 | D | Sequences & Series | At square step numbers the ant sits at a spiral corner. Step 1936 is at (-22, 22); then 44 steps down and 35 steps right. | |
| 25 | B | Diophantine Equations | Dividing by 2abc gives 1/a + 1/b + 1/c = 1/2. Since a is smallest, a is 3 to 6; try each and solve for b and c. |
2015 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.