2017 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 | B | Bases & Digits | Un-switch the digits: the result 3x+11 must be one of 17, 27, 37, 47, 57, and only 47 = 3(12)+11 comes from a two-digit x. | |
| 2 | C | Linear Equations & Word Problems | Time equals distance over speed for each segment: 100/4 = 25 s plus 300/5 = 60 s gives 85 s per lap, times 5 laps. | |
| 3 | E | Absolute Value & Inequalities | Since y > -1 and z > 1, the sum y + z exceeds 0; every other choice can be pushed negative by taking y close to -1. | |
| 4 | D | Algebraic Manipulation | Cross-multiplying the given equation collapses to x = y, after which the target fraction is 4x/2x = 2. | |
| 5 | D | Linear Equations & Word Problems | Let c be the cherry count; then 2c - 10 = 3(c - 10) gives c = 20, and the blueberry count is 2c = 40. | |
| 6 | B | Solid Geometry | Volume caps the count at floor(18/4) = 4, and a stack of three flat blocks plus one block standing on its edge in the leftover slab achieves 4. | |
| 7 | C | Linear Equations & Word Problems | Both halves have the same length d, so d/17 + d/5 = 44/60 hours; solve for d, which is exactly the walking distance. | |
| 8 | C | Coordinate Geometry | In an isosceles triangle the altitude from the apex bisects the base, so D is the midpoint of BC and C = 2D - B. | |
| 9 | D | Basic Probability | Winning means exactly 2 right or all 3 right; count 3 ways for exactly two (each 1/27 times 2) plus 1 way for three, out of 27. | |
| 10 | E | Coordinate Geometry | Perpendicular slopes a/2 and -2/b multiply to -1, so a = b; then (1, -5) on both lines gives a + 10 = c = 5b - 2. | |
| 11 | D | Ratios, Percents & Averages | With 100 students, 12 like dancing but say they dislike it and 36 dislike it and say so; the requested fraction is 12 out of 48. | |
| 12 | A | Ratios, Percents & Averages | Fuel used scales by 1/1.5 = 2/3 and price by 1.2, so cost scales by (2/3)(1.2) = 0.8, a 20 percent saving. | |
| 13 | C | Inclusion-Exclusion | Count by number of classes taken: a + b + c = 20, b + c = 9, and the roster total gives a + 2b + 3c = 32. | |
| 14 | D | Modular Arithmetic | Fourth powers of 1, 2, 3, 4 are all 1 mod 5, so N^16 is 1 mod 5 exactly when 5 does not divide N. | |
| 15 | E | Similar & Congruent Triangles | Similar right triangles give AE = AB^2/AC = 9/5, and triangles AED and ACD share the height from D, so [AED] = (AE/AC)[ACD] = (9/25)(6). | |
| 16 | A | Bases & Digits | Count numbers with no zero digit (9, 81, 729 for 1 to 3 digits, 729 for 1000-1999, none for 2000-2017) and subtract from 2017. | |
| 17 | B | Basic Counting | An increasing number is a nonempty subset of {1,...,9} and a decreasing one a subset of {0,...,9} other than {0}; subtract the 9 one-digit numbers counted twice. | |
| 18 | D | Arrangements with Restrictions | Fix the unique green disk on a corner or a midpoint; the 10 red placements then fall into 6 classes under the one remaining reflection. | |
| 19 | E | Triangles: Area & Pythagorean | The big triangle is ABC plus three triangles with sides 4s and 3s at a 120-degree angle, each 12 times [ABC], so 1 + 36 = 37. | |
| 20 | B | Divisibility & Factors | 21! has 2^18 in its factorization, so the exponent of 2 in a divisor takes 19 equally sized values, and odd divisors are the one value 0. | |
| 21 | D | Triangle Centers & Cevians | The median to the hypotenuse equals half of it, so AD = 5; each half-triangle has area 12, and r = Area/semiperimeter gives 12/8 + 12/9. | |
| 22 | D | Circles | AB is a diameter, so angle ACB is right; then triangle ACB is similar to triangle ADE, and areas scale by (AB/AE)^2 = 16/74. | |
| 23 | C | Modular Arithmetic | Split 45 = 9 times 5: the last digit gives N = 4 mod 5, and the digit sum equals 1 + ... + 44 = 990, divisible by 9. | |
| 24 | C | Coordinate Geometry | The centroid (1,1) is also the circumcenter; symmetry of the hyperbola about y = x forces (-1,-1) to be a vertex, so R = 2 sqrt 2. | |
| 25 | E | Modular Arithmetic | Shift scores down by 90; the total is a multiple of 7 in [28, 49], and removing the seventh score 5 must leave a multiple of 6. |
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.