2005 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	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. 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.