2010 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 125 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | D | Statistics & Data | Average means total divided by count. Add in easy groups: 6 + 10 = 16 and 1/2 + 2.5 + 1 = 4, so 20 / 5 = 4. | |
| 2 | B | Quadrilaterals & Polygon Areas | Give the small square side 1. The big square is then 4 by 4, and the rectangle under the row is 4 long and 3 tall. | |
| 3 | D | Linear Equations & Word Problems | Passing marbles keeps the total at 108. It ends split 2 to 1, so Tyrone keeps 72 and gave away 25. | |
| 4 | B | Linear Equations & Word Problems | First decide how many discs are needed: 7 discs hold only 392 minutes, so 8 are needed. Then share 412 minutes equally: 412 / 8 = 51.5. | |
| 5 | E | Circles | Go through the radius: the circumference 2 times pi times r equals 24 pi, so r = 12, and the area is pi times 12 squared = 144 pi. | |
| 6 | C | Functions | Do the inside first: 2 spade 2 = 3/2. Then 2 minus 1 divided by 3/2 is 2 - 2/3 = 4/3. | |
| 7 | C | Triangles: Area & Pythagorean | The northeast mile and the southeast mile cancel each other's north-south part and together move her the square root of 2 miles east. | |
| 8 | D | Linear Equations & Word Problems | Each day Tony earns exactly his age in dollars. The daily average 630 / 50 = 12.6 is not whole, so he turned 13 during the period. | |
| 9 | E | Bases & Digits | A three-digit number plus 32 has four digits only if the number is at least 968. That leaves 969, 979, 989, 999, and only 969 + 32 = 1001 works. | |
| 10 | E | Clocks, Calendars & Time | Each year the birthday moves 1 weekday later, or 2 days when a February 29 comes between. Count forward from Tuesday until you land on Saturday. | |
| 11 | D | Absolute Value & Inequalities | Subtracting 3 slides the interval without changing its length; dividing by 2 halves it. So the x-interval is half of b - a, and b - a = 20. | |
| 12 | C | Solid Geometry | Volume shrinks by the length factor cubed. The volume ratio is 1 to 1,000,000 = 100 cubed, so every length shrinks by 100: 40 / 100 = 0.4. | |
| 13 | A | Linear Equations & Word Problems | Take the 20-minute stop out of the 3 hours first: she drove for 8/3 hours. Then distance = 80 times the first part plus 100 times the rest. | |
| 14 | C | Angles & Polygons | Chasing angles around the equilateral triangle shows angle A = 60 degrees. A 60-degree angle between sides 1 and 2 makes a 30-60-90 triangle, so angle C = 90. | |
| 15 | D | Logic Puzzles | Chris and LeRoy call each other frogs, so exactly one is a toad. Nobody could say Brian's sentence if Mike were a toad, so Mike is a frog. | |
| 16 | B | Triangle Centers & Cevians | The angle bisector gives AB : BC = 3 : 8. Sides 3, 8, 11 make a flat triangle, so the smallest real one is 6, 16, 11. | |
| 17 | A | Solid Geometry | Slice the cube into lengths 1/2, 2, 1/2 each way. Only the twelve edge bars and eight corner pieces survive the holes: 6 + 1 = 7. | |
| 18 | B | Basic Probability | If Bernardo picks a 9 (chance 1/3), he surely wins. If not, both choose from the same digits, so apart from a tie each is equally likely to win. | |
| 19 | E | Quadrilaterals & Polygon Areas | Split the hexagon into triangle ACE plus three identical corner triangles with sides 1 and r. The 70% rule becomes r squared - 6r + 1 = 0. | |
| 20 | D | Geometric Optimization | Each corner touches only one longest jump, and the tour uses two jumps at every corner, so at most 4 of the 8 jumps are longest ones. | |
| 21 | A | Polynomials | The roots multiply to 2010 and add to a. One root must contain the prime 67; split the leftover 30 as evenly as possible, 5 and 6. | |
| 22 | A | Basic Counting | Each inside triangle comes from three chords that all cross, using six different points; each group of 6 points gives exactly one triangle. | |
| 23 | A | Conditional Probability & States | Stopping at box n means white from every earlier box, then red. The chances 1/2 x 2/3 x ... cancel, leaving 1 / (n(n+1)). | |
| 24 | A | Modular Arithmetic | Remove the 21 fives and 21 twos that make the zeros, then track only the remainder after dividing by 25. Runs of four non-multiples of 5 act like -1. | |
| 25 | B | Number Properties | Work backwards. If m squared is the largest square below n, the leftover is at most 2m, so each smallest start is the previous one plus the smallest allowed square. |
2010 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.