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2015 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2015 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Exponents, Logarithms & Radicals — Any nonzero number to the power 0 is 1, so the parentheses hold 25. A power of negative one means flip the number over, giving 1/25, then times 5. |
| 2 | D | Linear Equations & Word Problems — Pretend all 25 tiles are triangles: that is 75 edges. Each triangle swapped for a square adds one edge, so the 9 extra edges mean 9 squares. |
| 3 | D | Basic Counting — Count the flat toothpicks row by row, then use the fact that there are just as many standing ones. The 3-step staircase has 18, the 5-step one has 40. |
| 4 | B | Ratios, Percents & Averages — Only the ratio matters, so give Mia 1 egg, Sofia 2 and Pablo 6. Everyone should end with 3, so Sofia gets 1 of Pablo's 6 eggs. |
| 5 | E | Ratios, Percents & Averages — Turn each average into a total: 14 tests averaging 80 add to 1120, and 15 tests averaging 81 add to 1215. The difference, 95, is Payton's score. |
| 6 | B | Linear Equations & Word Problems — The sum equals 5 times the difference. Gathering terms gives 4 times the larger equals 6 times the smaller, so the ratio is 3 to 2. |
| 7 | B | Sequences & Series — From 13 to 73 is a climb of 60, which is 20 jumps of 3. Twenty jumps connect 21 numbers, because you also count the starting number. |
| 8 | B | Linear Equations & Word Problems — Name the ages from two years ago so Pete is 3 times Claire. The four-years-ago fact then gives Claire 6, so today they are 20 and 8. |
| 9 | D | Ratios, Percents & Averages — Volume uses the radius squared. A radius 1.1 times as big gives 1.21 times the base area, so the first cylinder must be 1.21 times as tall. |
| 10 | C | Arrangements with Restrictions — The letter b may only sit next to d, and c only next to a. With one allowed neighbor each, both must sit at the ends. |
| 11 | C | Triangles: Area & Pythagorean — Try the 3-by-4 rectangle: its diagonal is 5 and its area is 12. Area divided by diagonal squared is 12/25, and that stays the same at every size. |
| 12 | C | Algebraic Manipulation — The equation rearranges to (y minus x squared) squared equals 1. So y is x squared plus 1 or minus 1, and the two heights always differ by 2. |
| 13 | C | Basic Counting — Count in nickels, with a dime worth 2. With at least one nickel, every amount up to the total can be made, so the total is 17 nickels. |
| 14 | C | Circles — The disk turns from rolling and also from traveling around the clock: 3 turns per lap. So the arrow is up again after one third of a lap. |
| 15 | B | Diophantine Equations — Cross-multiply and rearrange to get y times (10 minus x) equals 11x. So x is at most 9; checking x from 1 to 9 leaves only 5/11 in lowest terms. |
| 16 | B | Systems of Equations — Subtract the equations: the difference of squares lets you cancel x minus y, leaving x + y = 3. Adding the equations then gives 15. |
| 17 | D | Coordinate Geometry — The given slope means a 30-degree tilt. The third line tilts 30 degrees downward, so the triangle's vertical side runs between the two lines' heights at x = 1. |
| 18 | E | Bases & Digits — In base 16, 1000 is 3E8. Every three-digit string of digits 0 to 9 starting with 0 to 3 is below it: 400 strings, minus zero. |
| 19 | D | Triangles: Area & Pythagorean — Cut along the mirror line, the height from C. Each half is a right triangle with a 15-degree angle, and tan 15 degrees is 2 minus root 3. |
| 20 | B | Algebraic Manipulation — Area plus perimeter plus 4 always factors as (width + 2) times (length + 2), both at least 3. But 106 is only 2 times 53. |
| 21 | C | Solid Geometry — Faces ABC and ABD are matching 3-4-5 right triangles on side AB. Their heights to AB meet at one point, and CD shows they are perpendicular. |
| 22 | A | Recursive Counting — Rows with no two neighbors standing follow 2, 3, 5, 8, 13, 21, 34. Open the circle at one person: 34 ways if they sit, 13 if they stand. |
| 23 | C | Quadratics — The zeros multiply to twice their sum. That rearranges to (r minus 2)(s minus 2) = 4; use every factor pair of 4, negatives included. |
| 24 | B | Diophantine Equations — A right triangle inside shows BC squared is 4 times (AD minus 1). So BC = 2k, AD = k squared plus 1, with k from 1 to 31. |
| 25 | A | Geometric Probability — Split by where the two points land: same side (chance 1/4), next-door sides (1/2), or opposite sides (1/4). Find the success chance in each case and weight it. |
2015 AMC 10A — answer key. 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.