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2000 AMC 10 — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2000 AMC 10 — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | E | Divisibility & Factors — Break 2001 into primes: 3, 23 and 29. To make the sum big, make two of the numbers as small as possible, 1 and 3. |
| 2 | A | Exponents, Logarithms & Radicals — 2000 is the same as 2000 to the first power. Multiplying one more 2000 onto 2000 copies of 2000 gives 2001 copies. |
| 3 | B | Ratios, Percents & Averages — Eating 20% leaves 80%, which is four fifths. Two days multiply the start by four fifths twice, so 32 is 16/25 of the start. |
| 4 | D | Linear Equations & Word Problems — January used exactly one extra December's worth of connect time, so the difference between the two bills is the cost of December's time. |
| 5 | B | Triangles: Area & Pythagorean — P moves along a line parallel to AB, so its height above AB never changes. Since MN is always half of AB, only the perimeter can change. |
| 6 | C | Modular Arithmetic — Only the last digits matter. Build the list of last digits by adding the previous two and keeping the ones digit, and check off each digit when it first appears. |
| 7 | B | Triangles: Area & Pythagorean — Cutting the right angle at D into three parts gives 30 degree angles, so every triangle here is a 30-60-90 triangle and AD = 1 fixes all the other lengths. |
| 8 | D | Ratios, Percents & Averages — Two fifths of the freshmen equals four fifths of the sophomores. The group with the smaller fraction taking part must be the bigger group, twice as big. |
| 9 | C | Absolute Value & Inequalities — Since x < 2, the number inside the bars is negative, so p = 2 - x. Then x - p = 2 - 2p. |
| 10 | D | Triangles: Area & Pythagorean — The third side must be strictly between 2 and 10. Two numbers in that range can differ by anything less than 8, but never by exactly 8. |
| 11 | C | Primes — Both primes are odd, so the product is odd and the sum is even, and the result is odd. The largest result is 191, which leaves only 119. |
| 12 | C | Sequences & Series — Figure n has n squared plus (n+1) squared unit squares, so figure 100 has 100 squared plus 101 squared, which is 20201. |
| 13 | B | Arrangements with Restrictions — Five yellow pegs need five different rows and five different columns, and the board has exactly five of each. That forces their places, and every other color's too. |
| 14 | C | Modular Arithmetic — A whole-number average means the first k scores add to a multiple of k. Work backward from the total 400, using multiples of 4 and then multiples of 3. |
| 15 | E | Algebraic Manipulation — Write a/b + b/a as one fraction, (a squared + b squared)/(ab). Then use the given ab = a - b to rewrite the top, and everything cancels to 2. |
| 16 | B | Coordinate Geometry — Put the grid on coordinates and find where lines AB and CD cross. E is 5/9 of the way from A to B, so AE is 5/9 of AB. |
| 17 | D | Modular Arithmetic — Trading a quarter or a nickel keeps the money the same. Trading a penny adds exactly 124 cents. So any total is 1 cent more than a multiple of 124. |
| 18 | C | Quadrilaterals & Polygon Areas — She sees everything within 1 km of the square's edge: a frame inside the square, four 5 by 1 strips outside, and four quarter-circles at the corners. |
| 19 | D | Similar & Congruent Triangles — Make the square's side 1. Both small slanted sides lie on the same line, so they are equally steep: legs 2m and 1 match legs 1 and 1/(2m). |
| 20 | C | Algebraic Manipulation — Adding A + M + C + 1 = 11 turns the expression into (A+1)(M+1)(C+1). Three numbers adding to 13 have the biggest product when nearly equal: 4, 4, 5. |
| 21 | B | Logic Puzzles — Only statement II must be true: a creepy crawler that is an alligator is also ferocious. Statements I and III each fail in a small made-up example. |
| 22 | C | Linear Equations & Word Problems — Angela drank one person's share, 1/p of everything, made of a quarter of the milk and a sixth of the coffee. So 1/p is between 1/6 and 1/4. |
| 23 | E | Statistics & Data — The mode is always 2 and the mean is (25 + x)/7. The median is 2, x or 4, so check each case for three evenly spaced values. |
| 24 | B | Functions — First find f of a plain number t by writing x = 3t. Then f(3z) = 7 is a quadratic in z, and its solutions add to -b/a. |
| 25 | A | Clocks, Calendars & Time — Two Tuesdays are a whole number of weeks apart. The gap is 265 days, or 266 if year N is a leap year, and only 266 divides by 7. |
2000 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.