2017 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	B	Bases & Digits	Undo the steps from the end. Switched back, the result was 17, 27, 37, 47 or 57, and only 47 = 3 times 12 plus 11 works.
2	C	Linear Equations & Word Problems	Find the time for each part of a lap: 100 meters takes 25 seconds and 300 meters takes 60 seconds, so one lap is 85 seconds.
3	E	Absolute Value & Inequalities	Add the two facts: y is more than -1 and z is more than 1, so y + z is more than 0. Every other choice can be negative.
4	D	Algebraic Manipulation	Clear the fraction in the given equation: it simplifies to x = y. Then the fraction asked for is 4x over 2x, which is 2.
5	D	Linear Equations & Word Problems	Call the cherry count c, so blueberry is 2c. After eating 10 of each, 2c - 10 = 3(c - 10), so c = 20 and blueberry is 40.
6	B	Solid Geometry	Volume allows at most 4 blocks, since 5 blocks need 20 cubic inches and the box has 18. Three flat blocks plus one on its side fit.
7	C	Linear Equations & Word Problems	Both halves have the same length d, and the walking distance is exactly d. Biking time d/17 plus walking time d/5 equals 44 minutes, written in hours.
8	C	Coordinate Geometry	Because AB = AC, the altitude from A lands at the middle of BC. So D is the midpoint, and C is the step from B to D taken twice.
9	D	Basic Probability	There are 27 equally likely ways to answer. Exactly two right happens in 3 times 2 = 6 ways, all three right in 1 way: 7 out of 27.
10	E	Coordinate Geometry	The slopes are a/2 and -2/b, and perpendicular slopes multiply to -1, so a = b. Putting (1, -5) into both equations then gives c.
11	D	Ratios, Percents & Averages	Picture 100 students. 12 like dancing but say they dislike it, and 36 dislike it and say so. Of the 48 who say dislike, 12 actually like it.
12	A	Ratios, Percents & Averages	Try real numbers: a trip needing 3 liters in the old car needs 2 in the new one. At 1.20 dollars instead of 1, cost drops from 3.00 to 2.40.
13	C	Inclusion-Exclusion	Sort students by how many classes they take. 20 students in all, 9 take two or more, so 11 take one; the class lists add to 32 sign-ups.
14	D	Modular Arithmetic	The fourth powers of 1, 2, 3, 4 all leave remainder 1 when divided by 5. So N to the 16th leaves 1 unless N is a multiple of 5.
15	E	Similar & Congruent Triangles	Similar right triangles give AE = 9/5, which is 9/25 of the diagonal AC = 5. Triangle AED is then 9/25 of triangle ACD, whose area is 6.
16	A	Bases & Digits	Count the numbers with no 0, where every digit is 1 to 9: 9 + 81 + 729 + 729 = 1548. The other 469 contain a 0.
17	B	Basic Counting	Choosing a set of different digits gives exactly one increasing number and one decreasing number. Count digit sets: 511 increasing, 1022 decreasing, minus the 9 one-digit numbers counted twice.
18	D	Arrangements with Restrictions	Put the single green disk at the top corner or the bottom middle. Only a left-right flip is left, and the 10 red placements give 6 paintings each.
19	E	Triangles: Area & Pythagorean	Cut the big triangle into ABC plus three outside triangles. Each has sides 4s and 3s with a 120-degree angle, so area 12 times ABC.
20	B	Divisibility & Factors	21! has eighteen factors of 2. Divisors split into 19 equal groups by how many 2s they contain, and the odd ones are one group.
21	D	Triangle Centers & Cevians	The right angle at A makes AD = 5. Each half-triangle has area 12, and each radius is its area divided by half its perimeter.
22	D	Circles	AB is a diameter, so the angle at C is a right angle. Then triangle ACB has the same shape as right triangle ADE, which has area 35/2.
23	C	Modular Arithmetic	Split 45 into 5 and 9. N ends in 4, so its remainder by 5 is 4; its digit sum matches 1 through 44 added up, 990.
24	C	Coordinate Geometry	The centroid (1, 1) is also the center of the triangle's circle. Mirror symmetry across y = x forces one corner to be (-1, -1).
25	E	Modular Arithmetic	Subtract 90 from each score. The 7-test total is a multiple of 7 from 28 to 49, and minus 5 it must be a multiple of 6: only 35.

2017 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.