2000 AMC 10
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.
The same key as a PDF 125 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 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. 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.