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2010 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2010 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 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 — 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.