2016 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 | D | Exponents, Logarithms & Radicals | With a = 1/2 the reciprocal a^{-1} is 2, so the numerator is 4 + 1 = 5, and dividing by 1/2 doubles it to 10. | |
| 2 | B | Functions | Apply the definition literally: 2 heart 4 = 2^3 * 4^2 = 128 and 4 heart 2 = 4^3 * 2^2 = 256, so the ratio is 1/2. | |
| 3 | D | Absolute Value & Inequalities | Because x is negative, |x| = -x = 2016, so every '-x' adds 2016; peel the absolute values from the inside out. | |
| 4 | B | Clocks, Calendars & Time | Book 15 is finished on day 1+2+...+15 = 120; that is 119 = 7*17 days after the Monday of day 1, so also a Monday. | |
| 5 | D | Statistics & Data | Mean 8 makes the four ages total 32; median 5 makes the two middle ages total 10; the youngest and oldest are what is left: 22. | |
| 6 | B | Bases & Digits | Distinct nonzero hundreds digits force S >= 300, ruling out digit sums 1, 2, 3; the example 143 + 257 = 400 achieves digit sum 4. | |
| 7 | C | Linear Equations & Word Problems | Call the angles 5x and 4x; the smaller angle has the larger complement, so 90 - 4x = 2(90 - 5x) gives x = 15 and angles 75 and 60. | |
| 8 | A | Modular Arithmetic | Work mod 100: 2015^2 ends in 25, and powers of a number ending in 25 keep ending in 25, so the result ends in 25 - 17 = 08. | |
| 9 | C | Coordinate Geometry | A horizontal chord of y = x^2 has endpoints (-b, b^2) and (b, b^2), so base 2b, height b^2, and area b^3 = 64 gives b = 4. | |
| 10 | D | Similar & Congruent Triangles | Same wood and thickness means weight is proportional to area, and area of similar triangles scales by the square of the side ratio: 12 * (5/3)^2 = 100/3. | |
| 11 | B | Linear Equations & Word Problems | Corners are shared, so n posts per short side and 2n per long side give 6n - 4 = 20; a side spans (posts - 1) gaps of 4 yards. | |
| 12 | D | Basic Probability | A product is odd only when both factors are odd; 3 of the 10 pairs are odd-odd, so 7 of 10 products are even. | |
| 13 | D | Linear Equations & Word Problems | With q sets of quadruplets there are 4q sets of triplets and 12q sets of twins, so the babies number 4q + 12q + 24q = 40q = 1000. | |
| 14 | D | Coordinate Geometry | Only the top-left corner can cross y = pi x, so a square with lower-left (a, b) and side s fits iff b + s <= floor(pi a). | |
| 15 | C | Paths & Grids | Checkerboard-color the grid: consecutive numbers alternate colors, so the five odd numbers fill the five corner-and-center squares; 25 - 18 = 7. | |
| 16 | E | Sequences & Series | With ar = 1 the sum is S = 1/(r(1-r)); the denominator peaks at 1/4 when r = 1/2, so S is at least 4. | |
| 17 | D | Algebraic Manipulation | Each vertex takes one face from each pair of opposite faces, so the eight products sum to (a+b)(c+d)(e+f); the pair sums total 27, so the max is 9^3. | |
| 18 | E | Divisibility & Factors | Consecutive integers a, ..., a+k-1 sum to k(2a+k-1)/2, so k(2a+k-1) = 690; count factor pairs of 690 whose smaller factor k is at least 2. | |
| 19 | D | Similar & Congruent Triangles | Put D at the origin; P and Q lie on line EF (y = 2x - 4), so PQ/EF is just the ratio of the x-differences: (40/13 - 20/7)/2. | |
| 20 | C | Transformations & Symmetry | The radii give scale factor 3/2; A moves by (3, 4) and a point's displacement is (k - 1)(P - Z), so O moves by (3,4) - (1/2)(2,2) = (2,3). | |
| 21 | B | Circles | In the first quadrant the curve is a circle through (0,0), (1,0), (0,1); the region there is a right triangle plus a semicircle of radius sqrt(2)/2. | |
| 22 | A | Games & Processes | There are 21 teams; a triple is non-cyclic iff one team beat the other two, and each team supplies C(10,2) such triples, so C(21,3) - 21*45 = 385. | |
| 23 | C | Quadrilaterals & Polygon Areas | WCXYFZ is the middle third of the hexagon between two parallel lines; the top and bottom thirds are trapezoids of area 8/27 each, leaving 11/27. | |
| 24 | D | Bases & Digits | The digit b is both the units digit of ab and the tens digit of bc; comparing the two forces the common difference to be 11p or 11p + 1. | |
| 25 | A | Number Properties | Each term equals floor(k{x}), so f depends only on the fractional part t and jumps exactly at fractions j/k with k <= 10; values = 1 + distinct fractions. |
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.