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2003 AMC 10A — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2003 AMC 10A — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | D | Sequences & Series — Match the first even number with the first odd number, the second with the second, and so on. Each even number is exactly 1 more than its odd partner. |
| 2 | B | Linear Equations & Word Problems — Work out what one member spends: two pairs of socks and two shirts, which is 26 dollars. Then divide the total bill by 26. |
| 3 | D | Solid Geometry — A box has 8 corners, not 4. So 8 small cubes of volume 27 are cut out of a box of volume 1200. |
| 4 | A | Linear Equations & Word Problems — Average speed is total distance divided by total time: 2 km in 40 minutes, which is 3 km per hour. |
| 5 | B | Quadratics — Putting x = 1 into the equation gives zero, so 1 is a solution. One factor is then 1 minus 1, and the whole product is zero. |
| 6 | C | Absolute Value & Inequalities — x heart 0 is the absolute value of x, which is never negative. So for a negative x, like -3, it cannot equal x. |
| 7 | B | Triangles: Area & Pythagorean — The longest side must be shorter than the other two together, so it is less than half of 7. That makes it 3, and the others add to 4. |
| 8 | E | Divisibility & Factors — 60 has 12 factors, and every whole number from 1 to 6 divides 60, so 6 of the 12 factors are less than 7. |
| 9 | A | Exponents, Logarithms & Radicals — Start at the innermost part: the cube root of x times the square root of x is just the square root of x. Every layer then repeats the same step. |
| 10 | E | Solid Geometry — Fold the four solid squares first. They make a bottom and three walls, so a fifth square works exactly when it folds onto one of the two faces still missing. |
| 11 | E | Bases & Digits — Both numbers start with the same three digits AMC. Written as place values, the sum is twice AMC hundreds, plus 10 plus 12, which gives AMC = 617. |
| 12 | A | Geometric Probability — Probability equals area here. The points with x less than y fill only a small triangle of area one half, inside a rectangle of area 4. |
| 13 | A | Systems of Equations — Treat the second and third numbers together as one amount. The first is 4 times that amount, so the total 20 is 5 of those amounts. |
| 14 | A | Primes — d and e must be among 2, 3, 5, 7, and a two-digit prime cannot end in 2 or 5. That leaves only a few cases to list and compare. |
| 15 | C | Divisibility & Factors — Count the 50 even numbers, then take away the even numbers that are also multiples of 3, which are the 16 multiples of 6. That leaves 34. |
| 16 | C | Modular Arithmetic — Only the last digit matters, so look at powers of 3. Their last digits repeat 3, 9, 7, 1, and 2003 is 3 past a multiple of 4. |
| 17 | B | Circles — In an equilateral triangle drawn inside a circle, each side equals the radius times the square root of 3. Set 3 times that side equal to pi times radius squared. |
| 18 | B | Quadratics — Replace x by 1 over y. The new equation, y squared plus y plus 2003/2004 = 0, has the reciprocals as its solutions, and they add to -1. |
| 19 | C | Circles — The lune is the small half-circle minus the piece of the big circle above its diameter. That piece is a 60 degree slice minus an equilateral triangle of side 1. |
| 20 | E | Bases & Digits — Three digits in base 9 means 81 to 728; in base 11, 121 to 1330. Both hold from 121 to 728: 608 of the 900 numbers. |
| 21 | D | Distributions & Stars and Bars — Cookies of the same kind are identical, so an assortment is just how many of each kind. Count the ways three whole numbers can add to 6. |
| 22 | B | Similar & Congruent Triangles — G sits above two parallel sides. The triangles from G to each side have the same shape, with bases 3 and 5, and heights differing by 8. |
| 23 | C | Basic Counting — Every toothpick is a side of exactly one upward-pointing small triangle. Count those, 1 + 2 + ... + 1002, and multiply by 3. |
| 24 | E | Logic Puzzles — Red 5 fits beside only blue 5, and red 4 beside only blue 4. So those reds go at the two ends, and every other card is forced. |
| 25 | B | Modular Arithmetic — Write n as 99 times q plus the sum q + r. Since 11 divides 99, q + r is a multiple of 11 exactly when n is. |
2003 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.