2004 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.

Take it timed (75 min)Print the paper PDF 205 KB Blank answer sheet Enter answers from paper

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 125 KB — a separate file from the paper, so one can be handed over without the other.

#	Answer	Topic	Key insight
1	C	Basic Counting	Rows 12 through 22 means both end rows count, so there are 11 rows, not 10. Then multiply 11 rows by 33 seats.
2	B	Basic Counting	It is easier to count the two-digit numbers with no 7 at all (8 choices for the tens digit, 9 for the ones digit) and take them away from 90.
3	A	Sequences & Series	From practice 1 to practice 5 there are only four doublings. Undo them by halving 48 four times.
4	B	Divisibility & Factors	Only the bottom face is hidden, so P is 720 divided by that face. Check every possible hidden face and see which factors always survive.
5	D	Exponents, Logarithms & Radicals	Put the 0 where it does no damage: as the number being subtracted. Then try the six ways to place 1, 2, 3; one times 3 squared gives 9.
6	C	Number Properties	Write the bigger factorial as the smaller one times the extra numbers. For example 100! is 100 times 99!, so 99! times 100! is (99!) squared times 100, a square.
7	A	Ratios, Percents & Averages	Every 7 U.S. dollars becomes 10 Canadian, a gain of 3. She spent exactly her gain, 60, so she changed 20 groups of 7 dollars: d = 140.
8	A	Triangles: Area & Pythagorean	Seen from the airport, one city is northeast and the other northwest. That is a right angle, so use the Pythagorean theorem with legs 8 and 10.
9	B	Circles	The circle is centered at a corner of the square, so exactly a quarter of the circle overlaps the square. Add the square and the other three quarters.
10	D	Sequences & Series	The rows hold 1, 3, 5, 7, ... cans, and adding up the first few odd numbers always gives a perfect square. So 100 cans means 10 rows.
11	C	Basic Probability	Count the rolls that fail: any roll with a 1, plus 2 and 2 (a tie). That is 16 of the 64 rolls.
12	A	Circles	A radius meets a tangent line at a right angle, so triangle OZX is a right triangle. The Pythagorean theorem gives b squared minus c squared equals a squared.
13	B	Modular Arithmetic	In hundredths of a millimeter, every coin is 15 more than a multiple of 20. So the coin count must be a multiple of 4: 8.
14	C	Ratios, Percents & Averages	Only the last step matters. One fifth blue means 1 blue per 4 others; doubling blue gives 2 per 4 others, which is one third.
15	A	Linear Equations & Word Problems	Each nickel swapped gains 5 cents and each dime swapped loses 5. A 70-cent gain means 14 more nickels than dimes: 17 and 3.
16	D	Circles	Join the three small centers: they form an equilateral triangle with side 2. The big circle's radius is the distance from the triangle's center to a corner, plus 1.
17	B	Bases & Digits	Write the ages as 10a + b and 10b + a. The five-years rule becomes 8a = 19b + 5, and only a = 3, b = 1 works.
18	E	Triangles: Area & Pythagorean	Find the middle triangle by cutting off the three corner triangles. Each corner triangle turns out to have area 18, and the big triangle has area 96.
19	C	Sequences & Series	List the first ten terms. The even-numbered terms drop by 2 each time: 2002, 2000, 1998, ... After 1001 drops, term 2004 is 0.
20	D	Triangle Centers & Cevians	Put weights on the corners so the triangle balances at T along both lines. A weighs 5, B weighs 4, C weighs 11, and then CD/BD = 4/11.
21	A	Sequences & Series	Shared numbers start at 16 and go up by 21, but only up to 6010, where the first list ends. That gives 286 to subtract.
22	D	Triangle Centers & Cevians	It is a right triangle. The outer center is the middle of the longest side, (6, 2.5); the inner center is (2, 2).
23	B	Basic Probability	It works when the less common color is on no faces, one face, or two opposite faces (top and bottom). That is 20 of 64.
24	B	Circles	Equal angles at A cut off equal arcs, so BD = CD. Ptolemy's theorem then gives 9 times AD = 15 times CD.
25	B	Circles	Find the overlap of the two big circles, then remove the small circle. The overlap is two 120-degree slices, each minus a triangle.

2004 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.