Printable paper
2001 AMC 10 — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2001 AMC 10 — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | E | Statistics & Data — The list is already in order, so the median is the middle (fifth) entry. Every entry is n plus a fixed number, so find n first, then average. |
| 2 | C | Linear Equations & Word Problems — A number's reciprocal times its opposite is always -1, whatever the number is. So the number equals -1 + 2 = 1. |
| 3 | E | Linear Equations & Word Problems — Adding 3 to each of two numbers adds 6 to the total, and doubling both numbers doubles the total. So the new total is 2 times (S + 6). |
| 4 | E | Circles — Each side of the triangle can cross the circle at most twice, so three sides give at most 6 points, and a slightly enlarged incircle reaches 6. |
| 5 | D | Transformations & Symmetry — Go through the twelve shapes one by one and look for a fold line, including slanted (diagonal) ones. Six shapes have one. |
| 6 | E | Bases & Digits — Write the number as 10 times the tens digit plus the units digit. The units digit cancels, and what is left forces it to be 9. |
| 7 | C | Fractions & Decimals — Moving the decimal point four places to the right is the same as multiplying by 10000. Write that as an equation with the reciprocal and solve. |
| 8 | B | GCD & LCM — The four tutors are all together again on the first day that is a multiple of 3, 4, 6 and 7 at once. That is their least common multiple, 84. |
| 9 | B | Ratios, Percents & Averages — See the tax as p% on everything plus an extra 2% above $28000. The p% parts cancel, so that extra 2% equals 0.25% of her income. |
| 10 | D | Systems of Equations — Divide one equation by another so a letter cancels: xz = 48 and xy = 24 show z is twice y, and then yz = 72 gives y = 6. |
| 11 | C | Sequences & Series — Split a ring into its four sides. The nth ring has two rows of 2n+1 squares and two columns of 2n-1 squares, which adds up to 8n. |
| 12 | D | Divisibility & Factors — Three numbers in a row always include an even number and a multiple of 3, so the product is a multiple of 42, but not always of 4. |
| 13 | E | Logic Puzzles — Ten letters and ten digits means every digit 0 to 9 is used once. Pick the odd run and the even run; A, B, C are the leftovers. |
| 14 | A | Diophantine Equations — Double every price so there are no halves. Then full price times (full-price tickets + 140) equals 4002, so look for a factor of 4002 between 140 and 280. |
| 15 | C | Quadrilaterals & Polygon Areas — The crosswalk is a parallelogram. Its area is 15 times 40 using the curb as the base, and 50 times the unknown distance using a stripe as the base. |
| 16 | D | Statistics & Data — Call the mean m. Then the numbers are m - 10, 5 and m + 15, and three numbers with mean m always add up to 3m. |
| 17 | C | Solid Geometry — Rolled up, the sector's radius of 10 becomes the cone's slant side, and its curved edge, of length 14 pi, wraps around the base circle. |
| 18 | D | Quadrilaterals & Polygon Areas — The pattern repeats in 3 by 3 blocks. The four corner squares of a block cover 4 of its 9 units of area, so pentagons cover 5 out of 9. |
| 19 | D | Distributions & Stars and Bars — Only how many of each type Pat picks matters, not the order. List the ways to split 4 donuts among 3 types, or picture 4 dots split by 2 dividers. |
| 20 | B | Angles & Polygons — Each side of the square is made of two triangle legs and one octagon side. Each leg is the octagon side divided by the square root of 2. |
| 21 | B | Solid Geometry — Slice straight down through the center: the cone becomes a triangle, the cylinder a square, and the small triangle above the square has the same shape as the big one. |
| 22 | D | Systems of Equations — Compare two lines that share an unknown number; that unknown cancels. The left column against the top row gives w = 19, and against the diagonal gives x = 22. |
| 23 | D | Basic Probability — Imagine drawing all five chips. The whites run out first exactly when the last chip in the box is red, and each chip is equally likely to be last. |
| 24 | B | Triangles: Area & Pythagorean — Draw a line from B straight across to CD. That right triangle has legs 7 and CD - AB, long side AB + CD, giving 4(AB)(CD) = 49. |
| 25 | B | Inclusion-Exclusion — Count multiples of 3 or 4 first. Then remove only those that are also multiples of 5, which are the multiples of 15 or 20. |
2001 AMC 10 — 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.