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2006 AMC 10B — answer key
The answer to each problem, with the topic it tests and the idea it turns on.
2006 AMC 10B — answer key
amcstep.com| # | Answer | Topic and the idea it turns on |
|---|---|---|
| 1 | C | Sequences & Series — The terms go -1, 1, -1, 1, and so on. Each neighboring pair adds to 0, and 2006 terms make 1003 whole pairs. |
| 2 | A | Functions — The spade rule is just the first number squared minus the second number squared. Work out the inside part first, then use its answer, sign included. |
| 3 | A | Linear Equations & Word Problems — Take away the 14-point winning margin from the total 34. What is left, 20, is split equally between the two teams, so the losing team has 10. |
| 4 | D | Circles — The big circle is 3 times as wide, so its area is 9 times the red circle. The blue ring is that minus the red: 8 times. |
| 5 | B | Geometric Optimization — The rectangles cover 18 square units, more than a 4 by 4 square holds. Side by side they make a 5 by 4 block, which fits. |
| 6 | D | Circles — Each square side is the diameter of one half-circle. Half of pi times 2/pi is exactly 1, so the four arcs total 4. |
| 7 | A | Exponents, Logarithms & Radicals — The messy fraction simplifies to the square root of x squared. A square root is never negative, and x is negative, so the answer is -x. |
| 8 | B | Circles — A line from the center to a top corner is a radius: the long side of a right triangle with legs equal to the side and half the side. |
| 9 | B | Ratios, Percents & Averages — The batch weighs 600 grams and has 411 calories. A 200-gram glass is one third of the batch, so it has 411 divided by 3 = 137. |
| 10 | A | Triangles: Area & Pythagorean — With sides x, 3x and 15, the two shorter sides must add to more than 3x. So x + 15 > 3x, and x is at most 7. |
| 11 | C | Modular Arithmetic — Every factorial from 10! onward ends in 00, so it cannot change the tens digit. Only 7! + 8! + 9! matters. |
| 12 | E | Coordinate Geometry — The meeting point is on both lines, so putting x = 1 and y = 2 into each equation gives a and b right away. No system needs solving. |
| 13 | E | Ratios, Percents & Averages — Joe adds cream after drinking, so he keeps all 2 ounces. JoAnn drinks 2 of her 14 stirred ounces, so she loses one seventh of her cream. |
| 14 | D | Quadratics — The number q is the product of the two new roots. Multiplying them out leaves only ab, which equals 2, so m never matters. |
| 15 | C | Quadrilaterals & Polygon Areas — BD is the short diagonal of the big rhombus but the long diagonal of the small one, so lengths shrink by the square root of 3 and area by 3. |
| 16 | E | Clocks, Calendars & Time — A normal year moves the weekday 1 forward; a year with a leap day moves it 2. That is 12 + 8 = 20, which is 6 past Sunday. |
| 17 | D | Basic Probability — Whatever color Alice gives, Bob then has six balls with two of that color. The bags match only if Bob sends back one of those two. |
| 18 | E | Sequences & Series — Write out terms until 2 and 3 appear again. The sequence repeats every 6 terms, and 2006 leaves remainder 2, so use the second term. |
| 19 | A | Circles — Draw OD and OE to make a 30 degree slice of the circle. The slice is the shaded piece plus two thin triangles, OBD and OBE. |
| 20 | E | Coordinate Geometry — From A, B is 2000 right and 200 up. AD turns a right angle and goes 2 right, so it goes 20 down. AB is 100 times as long. |
| 21 | C | Basic Probability — The weights 1 through 6 add to 21, so face k comes up with chance k/21. Then add up the chances of the six ordered ways to make 7. |
| 22 | D | Diophantine Equations — In cents, sandwiches times cost per sandwich is 253 = 11 times 23. Only 11 sandwiches at 23 cents each works: 2 globs, 3 blobs. |
| 23 | D | Triangle Centers & Cevians — Draw a line from the top corner to where the two lines cross. Triangles with the same height have areas in the ratio of their bases. |
| 24 | B | Circles — Radii meet a tangent line at right angles, so OADP is a trapezoid with parallel sides 2 and 4. The Pythagorean theorem gives its height. |
| 25 | B | Divisibility & Factors — If 5 were an age, the even plate number would end in 0 and look like d0d0. The rule for 9 forces 9090, which 4 does not divide. |
2006 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.