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2002 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2002 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | E | Exponents, Logarithms & Radicals — Six is 2 times 3, so write the bottom as powers of 2 and 3. Then cancel twos against twos and threes against threes. |
| 2 | C | Functions — The rule D just says: multiply the three numbers, add the three numbers, and divide the product by the sum. |
| 3 | A | Bases & Digits — Every number made of nines is 9 times a number made of ones. So the mean is 1 + 11 + 111 + ... + 111111111, which is 123456789. |
| 4 | D | Algebraic Manipulation — The first two parts both contain (3x - 2). Pull it out and the rest cancels, leaving 3x - 1, which is 11 when x = 4. |
| 5 | E | Circles — The two small circles sit side by side across the big circle, so the big circle's diameter is 6 + 4 = 10 and its radius is 5. |
| 6 | B | Primes — The expression factors as (n - 1)(n - 2), two whole numbers next to each other, so it is always even. The only even prime is 2. |
| 7 | E | Fractions & Decimals — The known fractions add to 41/42, just under 1. Since 1/n is at most 1, the sum stays below 2, so it must equal 1 and n = 42. |
| 8 | D | Clocks, Calendars & Time — A 31-day month has three weekdays that appear five times: those of the 1st, 2nd and 3rd. Try each possible start for July and see which August day always appears. |
| 9 | D | Basic Counting — Count the words that come before USAMO in blocks: first all words starting with A, M, O or S, then the words starting UA, UM or UO. USAMO is next. |
| 10 | C | Quadratics — With solutions a and b, the quadratic is (x-a)(x-b). Matching terms gives a+b = -a and ab = b, so a = 1 and b = -2. |
| 11 | B | Algebraic Manipulation — Call the middle number n. The sum is 3n and the product is (n-1)n(n+1), so the condition becomes (n-1)(n+1) = 24, giving n = 5. |
| 12 | E | Linear Equations & Word Problems — Cross-multiply. The x squared terms cancel, leaving (k-5)x = 2k-6. At k = 5 the left side is always 0 but the right side is 4. |
| 13 | D | Algebraic Manipulation — Write it as (8x-12)y + (2x-3). For it to be zero for every y, the part multiplying y and the leftover part must both be zero. |
| 14 | B | Exponents, Logarithms & Radicals — N is 5 to the 64th times 2 to the 75th. Pair each 5 with a 2 to make 64 tens; 2048 is left over. |
| 15 | E | Primes — A-B and A+B are both odd primes, so B must be the even prime 2. Then A-2, A, A+2 are all prime, which only 3, 5, 7 manage. |
| 16 | D | Diophantine Equations — A square is never negative, so n is from 0 to 19 and the fraction is at most 19. Only the squares 0, 1, 4, 9 give whole-number n. |
| 17 | C | Quadrilaterals & Polygon Areas — See the octagon as a square with corners cut off. DG spans that square, length 2 + 2 root 2, and A is 2 + root 2 below DG. |
| 18 | D | Circles — Two circles cross in at most 2 points. Four circles make 6 pairs, so at most 12 points, and slightly shifted copies of one circle reach 12. |
| 19 | C | Sequences & Series — Each term in the second block is 100 steps after its partner in the first, so it is 100d bigger. The sums differ by 10000d = 100. |
| 20 | B | Algebraic Manipulation — Move b to the right, square both equations and add. The mixed terms cancel, leaving 65(a^2 + c^2) = 65(1 + b^2). |
| 21 | B | Ratios, Percents & Averages — Only comparisons are given, so pick easy numbers: lawns 6, 3, 2 and mowing speeds 3, 2, 1. Then times are 2, 1.5 and 2, so Beth finishes first. |
| 22 | B | Triangles: Area & Pythagorean — Call OM = x and ON = y. The two Pythagorean equations are mirror images; adding them gives x^2 + y^2 = 169, and XY is twice 13. |
| 23 | D | Sequences & Series — Put n = 1: each term is the previous one plus the next counting number. That gives 1, 3, 6, 10, ..., the triangular numbers, so a_12 = 78. |
| 24 | D | Circles — Ten feet up is halfway from the bottom to the center, so bottom, center and rider form an equilateral triangle: a 60 degree turn, one sixth of a minute. |
| 25 | A | Linear Equations & Word Problems — Call the count n and the mean m, so the total is mn. Each mean change gives one equation: m + 2n = 13 and 2m + n = 14. |
2002 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.