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.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | D | Statistics & Data | Add the five widths, which pair up nicely (6 + 10 = 16, 2.5 + 0.5 + 1 = 4), then divide by 5 to get 20/5 = 4. | |
| 2 | B | Quadrilaterals & Polygon Areas | Four small squares of side s span the big square's side, so the rectangle is 4s wide and 4s - s = 3s tall. | |
| 3 | D | Linear Equations & Word Problems | The total 108 marbles is fixed, so Tyrone ends with 72 and Eric with 36; Tyrone gave away 97 - 72 = 25. | |
| 4 | B | Linear Equations & Word Problems | 412/56 is a bit more than 7, so 8 discs are needed, and then each disc holds 412/8 = 51.5 minutes. | |
| 5 | E | Circles | Circumference 2 pi r = 24 pi gives r = 12, so the area is pi r^2 = 144 pi. | |
| 6 | C | Functions | Evaluate the inner operation first: 2 spade 2 = 3/2, then 2 spade (3/2) = 2 - 2/3 = 4/3. | |
| 7 | C | Triangles: Area & Pythagorean | The northeast and southeast miles are perpendicular legs of a right isosceles triangle, so together they move her sqrt(2) east and 0 north; the return leg is sqrt(1 + 2). | |
| 8 | D | Linear Equations & Word Problems | Tony earns his age in dollars per day, and 630/50 = 12.6 is not an integer, so he turned 13 during the period. | |
| 9 | E | Bases & Digits | x + 32 has four digits, so x is at least 968; only four three-digit palindromes remain, and only 969 + 32 = 1001 is a palindrome. | |
| 10 | E | Clocks, Calendars & Time | May 27 shifts one weekday per year, two when a February 29 is crossed; tally the shifts from Tuesday until the total reaches 4 mod 7. | |
| 11 | D | Absolute Value & Inequalities | Solving for x squeezes the interval by a factor of 2: the x-interval has length (b - a)/2, so b - a = 20. | |
| 12 | C | Solid Geometry | Volume scales as the cube of length: a volume ratio of one to a million means a length ratio of one to 100, so 40/100 = 0.4. | |
| 13 | A | Linear Equations & Word Problems | Driving time is 3 hours minus the 20-minute stop, i.e. 8/3 hours; distance is 80 times the first stretch plus 100 times the rest. | |
| 14 | C | Angles & Polygons | Angle chasing around the equilateral triangle forces angle BAC = 60 degrees; together with AB = 2AC that makes ABC a 30-60-90 triangle, so angle C is 90. | |
| 15 | D | Logic Puzzles | Chris and LeRoy accuse each other, so exactly one is a toad; a toad Mike makes Brian's statement impossible, so Mike and Brian are frogs. | |
| 16 | B | Triangle Centers & Cevians | The angle bisector theorem forces AB : BC = 3 : 8, so sides are 3m, 8m, 11; m = 1 is degenerate, m = 2 gives 6, 16, 11. | |
| 17 | A | Solid Geometry | Each 2-by-2 hole leaves a border of width 1/2 on its face, so what survives is eight half-inch corner cubes plus twelve 1/2-by-1/2-by-2 edge bars. | |
| 18 | B | Basic Probability | If Bernardo draws a 9 (probability 1/3) he wins outright; otherwise both draw from the same set and he wins with probability (1 - 1/56)/2 by symmetry. | |
| 19 | E | Quadrilaterals & Polygon Areas | The hexagon is triangle ACE plus three congruent 120-degree triangles with sides 1 and r; the 70% condition gives r^2 - 6r + 1 = 0, root sum 6. | |
| 20 | D | Geometric Optimization | Each corner has one space diagonal and the tour uses two legs per corner, so at most 4 legs are space diagonals; the rest can be face diagonals. | |
| 21 | A | Polynomials | By Vieta the roots multiply to 2010 = 2 * 3 * 5 * 67 and a is their sum; keep 67 alone and split 30 as 5 and 6. | |
| 22 | A | Basic Counting | An interior triangle needs three chords that pairwise cross inside the circle, which uses six distinct endpoints, and any six points on the circle produce exactly one such triangle. | |
| 23 | A | Conditional Probability & States | Box k holds k + 1 marbles, so drawing white at boxes 1 through n-1 and red at box n has probability (1/2)(2/3)...((n-1)/n) * 1/(n+1), which telescopes to 1/(n(n+1)). | |
| 24 | A | Modular Arithmetic | Strip 5^21 and 2^21; mod 25 each block of four non-multiples of 5 has product -1, so 90!/5^21 is -1, and dividing by 2^21 = 2 gives 12. | |
| 25 | B | Number Properties | If n = m^2 + r with r at most 2m, one step sends n to r; the minimal number needing k+1 steps is s_k plus the smallest legal square. |
Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic tags, difficulty ratings and key insights on this page are original to this site. No problem statements are reproduced here — each links to its own page.