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2016 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2016 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | D | Exponents, Logarithms & Radicals — A negative exponent means flip the number over, so with a = 1/2 the flipped value is 2. Put in numbers first and the problem becomes simple arithmetic. |
| 2 | B | Functions — Follow the rule exactly: cube the first number and square the second. Then 2 heart 4 is 128 and 4 heart 2 is 256. |
| 3 | D | Absolute Value & Inequalities — Since x is negative, both |x| and -x equal 2016. Replace them with 2016 and work from the innermost bars outward. |
| 4 | B | Clocks, Calendars & Time — Book 15 is finished on day 1 + 2 + ... + 15 = 120. Day 120 is 119 days after day 1, and 119 is exactly 17 weeks. |
| 5 | D | Statistics & Data — The mean makes the four ages add to 32; the median makes the middle two add to 10. The youngest and oldest are the rest: 22. |
| 6 | B | Bases & Digits — The hundreds digits differ and are not zero, so S is at least 300. Digit sums 1 to 3 fail, and 143 + 257 = 400 gives 4. |
| 7 | C | Linear Equations & Word Problems — Call the angles 5x and 4x. The smaller angle has the bigger complement, so 90 - 4x = 2(90 - 5x), which gives x = 15. |
| 8 | A | Modular Arithmetic — Only the last two digits matter. 2015 squared ends in 25, higher powers keep ending in 25, and 25 - 17 = 8, so the number ends in 08. |
| 9 | C | Coordinate Geometry — The parabola is symmetric about the y-axis, so B and C are (-b, b^2) and (b, b^2). The area is (1/2)(2b)(b^2) = b^3 = 64. |
| 10 | D | Similar & Congruent Triangles — Same wood and same thickness means weight goes with area. Sides grow 5/3 times, so area grows (5/3)^2 = 25/9 times: 12 * 25/9 is about 33.3. |
| 11 | B | Linear Equations & Word Problems — Corners are shared, so n posts per short side and 2n per long side give 6n - 4 = 20. Each side is (posts - 1) gaps of 4 yards. |
| 12 | D | Basic Probability — A product is odd only when both numbers are odd. Only 3 of the 10 pairs are both odd, so the other 7 pairs give an even product. |
| 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 — The line y = pi x slopes up, so a square can only cross it at its top-left corner. Check that corner, then count squares by side length. |
| 15 | C | Paths & Grids — Color the grid like a checkerboard. Consecutive numbers alternate colors, so the five odd numbers fill the four corners and the center. The center is 25 - 18 = 7. |
| 16 | E | Sequences & Series — With second term 1 and ratio r, the sum is S = 1/(r(1 - r)). The bottom is largest, 1/4, at r = 1/2, so S is at least 4. |
| 17 | D | Algebraic Manipulation — Each corner uses one face from each opposite pair, so the eight products add to (a+b)(c+d)(e+f). The pair sums total 27, so the best is 9 x 9 x 9. |
| 18 | E | Divisibility & Factors — A run of k consecutive numbers starting at a adds to k(2a + k - 1)/2. So k(2a + k - 1) = 690: count factor pairs of 690. |
| 19 | D | Similar & Congruent Triangles — Put D at the origin. P and Q both lie on line EF, so PQ/EF is just the ratio of the horizontal distances: (40/13 - 20/7) divided by 2. |
| 20 | C | Transformations & Symmetry — The radii give scale factor 3/2. Use A and its image to find the center (-4, -6); then the origin moves to (2, 3), a distance of sqrt(13). |
| 21 | B | Circles — The shape is the same in all four quadrants. In the first quadrant the curve is a circle through (0,0), (1,0), (0,1); the region is a triangle plus a half-circle. |
| 22 | A | Games & Processes — There are 21 teams. A trio is not a cycle exactly when one team beat both others; each team does that in 45 trios, so 1330 - 945 = 385. |
| 23 | C | Quadrilaterals & Polygon Areas — The three strips have equal height, but the hexagon is wider in the middle. The top and bottom strips are trapezoids worth 8/27 of the hexagon each, leaving 11/27. |
| 24 | D | Bases & Digits — The digit b is the ones digit of ab and the tens digit of bc. Adding column by column shows the common difference is 11p or 11p + 1. |
| 25 | A | Number Properties — Each term depends only on the part of x after the decimal point. As it grows, f steps up once at each fraction with bottom at most 10. |
2016 AMC 10B — 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.