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.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Exponents, Logarithms & Radicals | A negative exponent means reciprocal, and squaring kills the sign: (-2)^(-2) = 1/4, so the value is 2 - 1/4. | |
| 2 | B | Clocks, Calendars & Time | From 1:00 to 2:40 is 100 minutes for two tasks, so each task is 50 minutes and the third ends 50 minutes after 2:40. | |
| 3 | A | Linear Equations & Word Problems | If 28 were written twice the other number would be 44/3, not an integer; so 28 is written three times and the other is 8. | |
| 4 | C | Fractions & Decimals | Put everything over 60: Alex 12, Beth 20, Cyril 15, so Dan gets 60 - 47 = 13, which slots between Cyril's 15 and Alex's 12. | |
| 5 | B | Logic Puzzles | Marta's 6th place anchors a chain: Jack 5th, Todd 3rd, Rand 2nd, then Hikmet is 6 behind Rand in 8th. | |
| 6 | E | Logic Puzzles | Three non-consecutive running days among Tue, Thu, Fri, Sat, Sun force Tuesday plus two of Thu/Sat, Thu/Sun, Fri/Sun; only the first leaves a legal tennis day. | |
| 7 | A | Functions | Evaluate inside out: (1 diamond 2) diamond 3 = 1/2 - 1/3 = 1/6, and 1 diamond (2 diamond 3) = 1 - 3/5 = 2/5; the difference is -7/30. | |
| 8 | E | Transformations & Symmetry | Track a general point: (x,y) goes to (y,-x), then (-y,-x), then (y,x), so the three moves together are one reflection across the line y = x. | |
| 9 | B | Circles | The falcata is a quarter disk of radius 3 with a half disk of radius 3/2 removed: 9pi/4 - 9pi/8 = 9pi/8. | |
| 10 | C | Modular Arithmetic | There are 1007 factors, an odd count, so the product is negative; it contains the factor -5 and only odd factors, so it ends in 5. | |
| 11 | B | Primes | The prime digits are 2, 3, 5, 7, so there are only 4 + 16 = 20 candidates; list them and test the sixteen two-digit ones for primality. | |
| 12 | A | Coordinate Geometry | Plug (x, -x) into the circle inequality; the cross terms cancel and it collapses to x^2 <= 25, so x runs from -5 to 5. | |
| 13 | E | Triangles: Area & Pythagorean | The intercepts give a 5-12-13 right triangle; two altitudes are the legs, and the third is (leg times leg)/hypotenuse = 60/13. | |
| 14 | D | Quadratics | Factor out (x - b): the roots are b and (a + c)/2, so make b the largest digit 9 and give a, c the next two, 8 and 7. | |
| 15 | B | Diophantine Equations | With h horses and c cows the total is 13h + 5c; test each choice by subtracting multiples of 13 and checking for a multiple of 5. | |
| 16 | C | Basic Probability | Distinct numbers in a divisor chain strictly increase, so Cal's number is at most 2; listing the chains c | b | a gives 9 of 720 outcomes. | |
| 17 | B | Solid Geometry | The four side-face centers form a rhombus in the middle horizontal slice with area half of 5 x 4; the octahedron is two pyramids on it of height 3/2 each. | |
| 18 | D | Expected Value | A coin stays tails only after three tails in a row, probability 1/8; so each coin is heads with probability 7/8, and 64 * 7/8 = 56. | |
| 19 | C | Circles | The center lies on the perpendicular bisectors of XY and ZW, i.e. of AB and AC, so it is the midpoint of AB; equal radii force AC = BC. | |
| 20 | A | Paths & Grids | Cube corners alternate color, so after 7 moves Erin is at odd distance; not adjacent means the opposite corner, reached by 2 routes per first move. | |
| 21 | D | Number Properties | Jumps are ceilings, and ceil(n/2) - ceil(n/5) stays within 1 of 3n/10, so only n = 62 to 66 need checking. | |
| 22 | D | Angles & Polygons | Every small triangle is a golden triangle, so lengths are powers of phi: FG = 1/phi, JH = 1, CD = phi, summing to 2 phi. | |
| 23 | B | Divisibility & Factors | Trailing zeros count factors of 5; tripling needs (2n)! to gain an extra factor from 25 that n! lacks, first at n = 8, 9, 13, 14. | |
| 24 | D | Sequences & Series | At perfect-square indices the spiral sits on a diagonal: p_{(2k)^2} = (-k, k); from p_1936 = (-22, 22) walk 44 south, then 35 east. | |
| 25 | B | Diophantine Equations | Divide by abc to get 1/a + 1/b + 1/c = 1/2; ordering forces 3 <= a <= 6, then bound b likewise and solve for c. |
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.