2002 AMC 10B
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.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | E | Exponents, Logarithms & Radicals | Split 6^2002 into 2^2002 times 3^2002 and compare exponents prime by prime; one spare 3 on top and one spare 2 below remain. | |
| 2 | C | Functions | D is just a recipe: multiply the three inputs, divide by their sum, and 48/12 = 4. | |
| 3 | A | Bases & Digits | Each number is 9 times a repunit, so the mean is 1 + 11 + 111 + ... + 111111111 = 123456789, which contains every digit except 0. | |
| 4 | D | Algebraic Manipulation | The two products share the factor (3x-2); pulling it out leaves (3x-2)(4x+1-4x) + 1 = 3x - 1, which is 11 when x = 4. | |
| 5 | E | Circles | The two small circles lie along a diameter of the big circle, so the big diameter is 6 plus 4, giving radius 5. | |
| 6 | B | Primes | n^2-3n+2 = (n-1)(n-2) is a product of consecutive integers, hence even, so it can only be the prime 2, which happens at n = 3. | |
| 7 | E | Fractions & Decimals | The three given fractions total 41/42, just below 1, and 1/n adds at most 1, so the integer sum must be exactly 1 and n = 42. | |
| 8 | D | Clocks, Calendars & Time | In a 31-day month the weekdays of the 1st through 3rd occur five times; July 1 is Sat, Sun or Mon, so August 1 is Tue, Wed or Thu. | |
| 9 | D | Basic Counting | Count the words before USAMO block by block: 4 choices of an earlier first letter give 4*24, then UA, UM, UO give 3*6, and USAMO is the very next word. | |
| 10 | C | Quadratics | Vieta gives a+b = -a and ab = b; since b is nonzero, a = 1 and then b = -2. | |
| 11 | B | Algebraic Manipulation | Call the middle integer n: the sum is 3n and the product is n(n^2 - 1), so the condition collapses to n^2 - 1 = 24 and n = 5. | |
| 12 | E | Linear Equations & Word Problems | Cross-multiplying cancels the x^2 terms and leaves (k - 5)x = 2k - 6; the x-coefficient vanishes exactly at k = 5, giving the false statement 0 = 4. | |
| 13 | D | Algebraic Manipulation | Viewed as linear in y, (8x-12)y + (2x-3) vanishes for every y only when coefficient and constant are both zero; both give x = 3/2. | |
| 14 | B | Exponents, Logarithms & Radicals | N = 5^64 * 2^75 = 10^64 * 2^11, which is 2048 followed by 64 zeros, so the digit sum is 2 + 0 + 4 + 8 = 14. | |
| 15 | E | Primes | A-B and A+B differ by 2B, so both are odd, forcing B = 2; then A-2, A, A+2 include a multiple of 3, so A = 5. | |
| 16 | D | Diophantine Equations | A square is nonnegative, so n lies between 0 and 19 and the fraction is at most 19; only k = 0, 1, 2, 3 give integer n. | |
| 17 | C | Quadrilaterals & Polygon Areas | Box the octagon in a square of side 2+2sqrt(2); DG spans the full width and A lies 2+sqrt(2) below it, so area is half their product. | |
| 18 | D | Circles | Each of the C(4,2) = 6 pairs of circles meets in at most 2 points, giving the bound 12, and four slightly shifted copies of one circle achieve it. | |
| 19 | C | Sequences & Series | Each second-block term exceeds its matching first-block term by 100d, so the block sums differ by 10000d = 100, giving d = 0.01. | |
| 20 | B | Algebraic Manipulation | Isolate a and c: a+8c = 4+7b and 8a-c = 7-4b; squaring and adding cancels cross terms, leaving 65(a^2+c^2) = 65(1+b^2). | |
| 21 | B | Ratios, Percents & Averages | Pick concrete numbers: lawns of 6, 3, 2 units and mowing rates 3, 2, 1 give times 2, 1.5, 2, so Beth finishes first. | |
| 22 | B | Triangles: Area & Pythagorean | With OM = x, ON = y: 4x^2+y^2 = 361 and x^2+4y^2 = 484; adding gives x^2+y^2 = 169, and XY^2 = 4(x^2+y^2). | |
| 23 | D | Sequences & Series | Setting n = 1 turns the relation into a_{m+1} = a_m + (m + 1), so a_m is the m-th triangular number and a_12 = 12 * 13 / 2. | |
| 24 | D | Circles | A point 10 feet above the bottom is half a radius below the center, so the radius to it makes 60 degrees with vertical: one sixth of a revolution. | |
| 25 | A | Linear Equations & Word Problems | Describe the list by count n and mean m (sum mn); each appended number yields a linear equation: m+2n = 13 and 2m+n = 14. |
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