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2013 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2013 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Linear Equations & Word Problems — The 1.50 is paid once, but the 0.25 is paid again for every mile, so add 1.50 to 5 times 0.25. |
| 2 | B | Fractions & Decimals — Count how many quarter cups fit into two and a half cups: four in each whole cup, two in the half cup. |
| 3 | E | Triangles: Area & Pythagorean — The corner of the square at B is a right angle, so AB and BE are the base and height of the triangle; half of 10 times BE equals 40. |
| 4 | C | Linear Equations & Word Problems — A win means the team scored double the other team, so its score must be even. That makes the even scores the wins and the odd scores the one-run losses. |
| 5 | B | Linear Equations & Word Problems — Everyone should pay the same amount, one third of 405, which is 135. Each person gives Sammy the difference between 135 and what they already paid. |
| 6 | D | Logic Puzzles — The 5-year-old is home, so the movie pair is 3 and 13 or 7 and 9. Only 3 and 13 leaves two boys under 10 for baseball. |
| 7 | C | Basic Counting — English is always in, so pick 3 more from the other 5 courses. Of those 10 choices, only History, Art, Latin has no math, so 9 are allowed. |
| 8 | C | Exponents, Logarithms & Radicals — Write 2 to the 2014 as 4 times 2 to the 2012, pull 2 to the 2012 out of top and bottom, and 5 over 3 is left. |
| 9 | B | Ratios, Percents & Averages — A three-point try averages 0.2 times 3 = 0.6 points and a two-point try averages 0.3 times 2 = 0.6, so 30 tries give 18. |
| 10 | E | Ratios, Percents & Averages — Pretend there are 100 flowers: 60 pink (40 of them carnations) and 40 red (30 of them carnations). That is 70 carnations. |
| 11 | A | Basic Counting — Ten ways to pick 2 people means the council has 5 members. Then count the ways to pick 3 of those 5, which is also 10. |
| 12 | C | Similar & Congruent Triangles — Parallel lines copy the equal base angles, so both small corner triangles are isosceles. Then the perimeter equals AB + AC. |
| 13 | B | Bases & Digits — The number is aba with a not 5. Small a allows every middle digit b; for a from 6 to 9, b must be below 20 - 2a. |
| 14 | D | Solid Geometry — Each of the 12 old edges keeps its middle piece, and each of the 8 corner holes adds 9 new edges: 12 + 72 = 84. |
| 15 | D | Triangles: Area & Pythagorean — Every altitude equals twice the area divided by its side. Putting that into the averaging rule makes the area cancel and leaves an equation for the third side. |
| 16 | E | Coordinate Geometry — After reflecting, (9, 1) lies on the segment from (8, -3) to (10, 5), so the union is a big triangle minus a small top triangle. |
| 17 | B | Inclusion-Exclusion — Two friends meet on multiples of 12, 15 or 20. All three meet on multiples of 60, and those days must be taken out of each pair's count. |
| 18 | B | Coordinate Geometry — The piece next to D is a triangle with base AD = 4 on the x-axis. Half the area fixes the point's height; line CD gives x. |
| 19 | C | Bases & Digits — The last digit in base b is the remainder after dividing by b, so b divides 2010 and must be bigger than 3. |
| 20 | C | Transformations & Symmetry — Where a corner travels, the region is a pie slice of the corner circle; elsewhere it is only what the first or last square covers. |
| 21 | D | Divisibility & Factors — Work backwards: each step multiplies by 12 and divides by 1 to 11. Twelves supply every 2 and 3, so the last share needs 5, 5, 7, 11. |
| 22 | B | Solid Geometry — The big sphere has radius 3. The eighth center sits straight above the hexagon's center, giving a right triangle with legs 2 and 3 - r, hypotenuse r + 1. |
| 23 | D | Circles — Drop a perpendicular from A to BC. Pythagoras twice gives CX times CB = 97 squared minus 86 squared = 2013, and the triangle rule leaves only CB = 61. |
| 24 | E | Games & Processes — Each round is one of 6 matchings, in two groups of 3. Either all six are used once, or one group is used twice. |
| 25 | A | Basic Counting — Any 4 corners give one crossing, so 70 in all, but the center and eight other points have extra diagonals through them and are overcounted. |
2013 AMC 10A — 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.