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2005 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2005 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | A | Ratios, Percents & Averages — Find the price of one bar when buying (40 cents) and when selling (50 cents). Each bar earns 10 cents, and there are 1000 bars. |
| 2 | D | Ratios, Percents & Averages — The phrase 'x percent of x' means x divided by 100, then times x. Setting that equal to 4 gives x times x = 400, so x = 20. |
| 3 | D | Ratios, Percents & Averages — Using one third of the paint leaves two thirds. This happens twice, so what remains is two thirds of two thirds, which is 4/9. |
| 4 | D | Functions — A negative number squared is positive, so both inner parts equal 13. Then 13 diamond 13 is 13 times the square root of 2. |
| 5 | C | Ratios, Percents & Averages — If one third of the CDs costs one fifth of her money, then all the CDs cost three times as much: three fifths. Two fifths is left. |
| 6 | B | Ratios, Percents & Averages — Her goal is 40 A's out of all 50 quizzes. She has 22, so she needs 18 of the last 20 quizzes to be A's. Only 2 can be lower. |
| 7 | B | Circles — A circle in a square has diameter equal to the side; a square in a circle has diagonal equal to the diameter. Start with side 2. |
| 8 | A | Circles — On one tile, the four white quarter circles fit together into one whole circle of radius 1/2. So each tile has 1 minus pi/4 shaded, and there are 80 tiles. |
| 9 | D | Basic Probability — The sum is odd only when one die is odd and the other even. The first die is odd 2/3 of the time, the second only 1/3. |
| 10 | A | Triangles: Area & Pythagorean — Drop a perpendicular from C; it lands at the middle of AB. Its length squared is 48, so its foot is 4 from D, and BD = 4 - 1. |
| 11 | E | Sequences & Series — Write out terms until one repeats. From term 2 on, the sequence cycles 133, 55, 250 every 3 terms, and term 2005 lands on 250, like term 4. |
| 12 | E | Basic Probability — A prime product needs eleven dice showing 1 and one die showing 2, 3 or 5: 12 times 3 = 36 good outcomes out of 6 to the 12th. |
| 13 | C | Inclusion-Exclusion — Multiples of 12 appear in both the list of multiples of 3 and the list of multiples of 4. To remove them completely, subtract their count twice. |
| 14 | C | Triangles: Area & Pythagorean — Use CD = 2 as the base. M is halfway up side AC, so its height above line BD is half the triangle's height, or half of root 3. |
| 15 | D | Basic Probability — Only two kinds of pairs reach 20 dollars: any pair with a twenty, or the two tens. That is 14 of the 28 possible pairs. |
| 16 | D | Quadratics — Doubling both roots doubles their sum and multiplies their product by 4. Comparing coefficients gives m = 2p and n = 4m, so n = 8p. |
| 17 | B | Exponents, Logarithms & Radicals — Plug each equation into the next to get 8 = 4 raised to the power abcd. Since 8 is 2 cubed and 4 is 2 squared, abcd = 3/2. |
| 18 | D | Basic Counting — Seven different digits in increasing order can be arranged only one way. So a phone number is fixed by which of the eight digits 2 through 9 is left out. |
| 19 | B | Statistics & Data — Pretend there are 20 students. The total of all scores divided by 20 gives the mean, 86. The middle students score 85, the median. |
| 20 | C | Basic Counting — Each of the five digits sits in each place equally often, so every place averages (1+3+5+7+8)/5 = 4.8. The average number is 4.8 times 11111. |
| 21 | A | Basic Probability — Both probabilities have the same bottom, so q/p is just the count of two-pair draws (1620) divided by the count of four-of-a-kind draws (10). |
| 22 | C | Divisibility & Factors — After simplifying, the question is whether n + 1 divides 2 times (n - 1) factorial. This fails only when n + 1 is an odd prime: 8 times. |
| 23 | C | Quadrilaterals & Polygon Areas — EF joins the midpoints of the legs, so its length is the average of the two bases, and both smaller trapezoids have half the height. |
| 24 | E | Diophantine Equations — With digits a and b, x squared minus y squared equals 99(a + b)(a - b). So a + b must be 11 and a - b a perfect square. |
| 25 | C | Sets, Estimation & Miscellaneous — Numbers 1 to 24 have no partner making 125, so keep them all. Numbers 25 to 100 form 38 pairs; keep one from each: 62. |
2005 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.