2006 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.

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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.

The same key as a PDF 126 KB — a separate file from the paper, so one can be handed over without the other.

#	Answer	Topic	Key insight
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. 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.