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2004 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2004 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Basic Counting — Rows 12 through 22 means both end rows count, so there are 11 rows, not 10. Then multiply 11 rows by 33 seats. |
| 2 | B | Basic Counting — It is easier to count the two-digit numbers with no 7 at all (8 choices for the tens digit, 9 for the ones digit) and take them away from 90. |
| 3 | A | Sequences & Series — From practice 1 to practice 5 there are only four doublings. Undo them by halving 48 four times. |
| 4 | B | Divisibility & Factors — Only the bottom face is hidden, so P is 720 divided by that face. Check every possible hidden face and see which factors always survive. |
| 5 | D | Exponents, Logarithms & Radicals — Put the 0 where it does no damage: as the number being subtracted. Then try the six ways to place 1, 2, 3; one times 3 squared gives 9. |
| 6 | C | Number Properties — Write the bigger factorial as the smaller one times the extra numbers. For example 100! is 100 times 99!, so 99! times 100! is (99!) squared times 100, a square. |
| 7 | A | Ratios, Percents & Averages — Every 7 U.S. dollars becomes 10 Canadian, a gain of 3. She spent exactly her gain, 60, so she changed 20 groups of 7 dollars: d = 140. |
| 8 | A | Triangles: Area & Pythagorean — Seen from the airport, one city is northeast and the other northwest. That is a right angle, so use the Pythagorean theorem with legs 8 and 10. |
| 9 | B | Circles — The circle is centered at a corner of the square, so exactly a quarter of the circle overlaps the square. Add the square and the other three quarters. |
| 10 | D | Sequences & Series — The rows hold 1, 3, 5, 7, ... cans, and adding up the first few odd numbers always gives a perfect square. So 100 cans means 10 rows. |
| 11 | C | Basic Probability — Count the rolls that fail: any roll with a 1, plus 2 and 2 (a tie). That is 16 of the 64 rolls. |
| 12 | A | Circles — A radius meets a tangent line at a right angle, so triangle OZX is a right triangle. The Pythagorean theorem gives b squared minus c squared equals a squared. |
| 13 | B | Modular Arithmetic — In hundredths of a millimeter, every coin is 15 more than a multiple of 20. So the coin count must be a multiple of 4: 8. |
| 14 | C | Ratios, Percents & Averages — Only the last step matters. One fifth blue means 1 blue per 4 others; doubling blue gives 2 per 4 others, which is one third. |
| 15 | A | Linear Equations & Word Problems — Each nickel swapped gains 5 cents and each dime swapped loses 5. A 70-cent gain means 14 more nickels than dimes: 17 and 3. |
| 16 | D | Circles — Join the three small centers: they form an equilateral triangle with side 2. The big circle's radius is the distance from the triangle's center to a corner, plus 1. |
| 17 | B | Bases & Digits — Write the ages as 10a + b and 10b + a. The five-years rule becomes 8a = 19b + 5, and only a = 3, b = 1 works. |
| 18 | E | Triangles: Area & Pythagorean — Find the middle triangle by cutting off the three corner triangles. Each corner triangle turns out to have area 18, and the big triangle has area 96. |
| 19 | C | Sequences & Series — List the first ten terms. The even-numbered terms drop by 2 each time: 2002, 2000, 1998, ... After 1001 drops, term 2004 is 0. |
| 20 | D | Triangle Centers & Cevians — Put weights on the corners so the triangle balances at T along both lines. A weighs 5, B weighs 4, C weighs 11, and then CD/BD = 4/11. |
| 21 | A | Sequences & Series — Shared numbers start at 16 and go up by 21, but only up to 6010, where the first list ends. That gives 286 to subtract. |
| 22 | D | Triangle Centers & Cevians — It is a right triangle. The outer center is the middle of the longest side, (6, 2.5); the inner center is (2, 2). |
| 23 | B | Basic Probability — It works when the less common color is on no faces, one face, or two opposite faces (top and bottom). That is 20 of 64. |
| 24 | B | Circles — Equal angles at A cut off equal arcs, so BD = CD. Ptolemy's theorem then gives 9 times AD = 15 times CD. |
| 25 | B | Circles — Find the overlap of the two big circles, then remove the small circle. The overlap is two 120-degree slices, each minus a triangle. |
2004 AMC 10B — 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.