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2018 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2018 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | A | Basic Counting — Count how many 2-inch pieces fit along each side of the pan, 10 along one side and 9 along the other, then multiply. |
| 2 | D | Linear Equations & Word Problems — Each 30 minutes is half an hour. Find the miles driven in the first two half hours, subtract from 96, and double what is left. |
| 3 | B | Basic Counting — The value only depends on which two digits get multiplied together. There are just three ways to split 1, 2, 3, 4 into two pairs. |
| 4 | B | Systems of Equations — Opposite faces of a box match, so the three different areas 24, 48, 72 are the three products of two edges. Find three edges that make them. |
| 5 | D | Basic Counting — Count all subsets, then take away the ones with no prime at all, which are the subsets made only from 4, 6, 8, 9. |
| 6 | D | Basic Probability — Three draws are needed exactly when the first two chips add to 4 or less, which only happens for the pairs 1 and 2 or 1 and 3. |
| 7 | D | Circles — If the big radius is N times the small one, the big semicircle holds N squared small areas, so the leftover is N squared minus N. |
| 8 | C | Sequences & Series — Half the toothpicks are horizontal. Column by column the horizontal ones number 2, 3, 4, and so on, so find how many columns make 90. |
| 9 | D | Basic Probability — Turn every die upside down: a face k becomes 7 minus k. A roll with sum 10 becomes a roll with sum 49 minus 10, which is 39. |
| 10 | E | Solid Geometry — The base is a slanted 1-by-square-root-of-13 rectangle. The height is the distance from M to that slanted rectangle, which you can measure from F in the front face. |
| 11 | C | Primes — The square of any prime other than 3 is one more than a multiple of 3, so adding 26 always gives a multiple of 3. The prime 3 gives 35. |
| 12 | C | Triangle Centers & Cevians — The centroid sits one third of the way from the circle's center to C. So it moves on a circle of radius 4, with area 16 pi, about 50. |
| 13 | C | Modular Arithmetic — The terms are 1 followed by zeros and a 1. Every fourth term, starting with 101 itself, is a multiple of 101, so count 1st, 5th, 9th, and so on. |
| 14 | D | Statistics & Data — The mode is the only value that appears 10 times, so every other value appears at most 9 times. Fill the other 2008 spots using as few values as possible. |
| 15 | A | Solid Geometry — Follow the paper from its center to a corner: half the width across the bottom, up the side, half the width on top. That is w plus h. |
| 16 | E | Modular Arithmetic — A number and its cube always leave the same remainder when divided by 6, so the sum of cubes acts like the plain sum. |
| 17 | B | Triangles: Area & Pythagorean — The octagon is the rectangle with four matching right triangles cut off its corners. Write both legs using the side length, then use the Pythagorean theorem. |
| 18 | D | Arrangements with Restrictions — Siblings can never share a row, so each row has one child per family. Choose and order the front row; then only 2 back rows work. |
| 19 | E | Divisibility & Factors — Chloe's age is a multiple of Zoe's exactly when Zoe's age divides their age gap. So the gap has 9 divisors, which makes it 36, Chloe 37 and Joey 38. |
| 20 | B | Sequences & Series — Work out the first dozen values. Every 6 steps the value goes up by exactly 6, so jump from f(2) to f(2018) in steps of 6. |
| 21 | C | Divisibility & Factors — The next divisor must share the factor 17 or 19 with 323; otherwise n would be a multiple of their product, far too big. |
| 22 | C | Geometric Probability — Side 1 is longest. A triangle needs x + y > 1 and an obtuse one needs x squared + y squared < 1: a quarter circle minus a triangle. |
| 23 | B | GCD & LCM — Write a = gx and b = gy, with g the gcd and x, y sharing no factor. Then 63 equals g times a whole number, so g divides 63. |
| 24 | C | Quadrilaterals & Polygon Areas — Each side of XYZ cuts a corner off ACE through the middle of one side and a quarter of the way along another, removing 1/8 each time. |
| 25 | C | Number Properties — Split x into whole part n and decimal part. Each interval from n to n + 1 has one solution when (n + 1) squared is below 10,000. |
2018 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.