2013 AMC 10A
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 124 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 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. 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.