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.
The same key as a PDF 128 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 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. 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.