2007 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	E	Solid Geometry	The four walls unfold into one strip 44 feet long and 8 feet tall. Take away the doors and windows, then multiply by three rooms.
2	E	Functions	The new symbol is just a recipe: add the two numbers, then multiply by the second one. Swapping the order changes which number you multiply by.
3	B	Ratios, Percents & Averages	Average mileage is all the miles divided by all the gallons: 240 miles on 4 + 6 = 10 gallons.
4	D	Circles	The three center angles add to 360, so angle AOC is 100. The angle at B looks at the same arc, so it is half of that.
5	D	Logic Puzzles	The facts stack up like circles inside circles: Crups inside Dramps inside Arogs inside Brafs. So every Crup is also an Arog and a Braf.
6	D	Linear Equations & Word Problems	The three blank problems give 4.5 points for free, so the right answers must bring at least 95.5 more. Fifteen right is 90, too few; sixteen right is 96, enough.
7	E	Angles & Polygons	Draw the line from C to E. It cuts the pentagon into a square and an equilateral triangle, so the angle at E is 90 plus 60.
8	D	Basic Counting	Once a and c are picked, b is their average, so it is fixed. It is a whole number exactly when a and c are both even or both odd.
9	D	Modular Arithmetic	The last s is the 12th s, so it moves 1 + 2 + ... + 12 = 78 places, exactly three trips around the alphabet.
10	A	Triangles: Area & Pythagorean	The base BC is fixed, so area 1 fixes the height. Points at that distance from line BC form two parallel lines, one on each side.
11	C	Circles	The center lies on the triangle's line of symmetry. Setting its distance to the top corner equal to its distance to a bottom corner gives one equation.
12	D	Linear Equations & Word Problems	Going back N years, Tom loses N years but his three children together lose 3N. Writing 'Tom then = twice the children then' gives T = 5N.
13	D	Circles	Both circles pass through (0,0) and (2,2). The overlap is split by the line between those points into two equal pieces, each a quarter circle with a right triangle removed.
14	C	Ratios, Percents & Averages	The group size never changes, so the drop from 40 percent to 30 percent girls is the 2 girls who left: 10 percent of the group is 2.
15	D	Angles & Polygons	Write each angle as a piece of A: B is half of A, C a third, D a quarter. The four angles add to 360.
16	C	Ratios, Percents & Averages	Pretend the class has 10 students: 1 junior and 9 seniors. The class total is 840 and the seniors earned 747, so the junior scored the other 93.
17	D	Triangles: Area & Pythagorean	Join P to the three corners. The three small triangles fill the big one, and adding their areas shows the triangle's height equals 1 + 2 + 3 = 6.
18	B	Circles	Join the middle center to two neighboring outer centers: a right triangle with legs 1 + r and long side 2r.
19	C	Basic Probability	In every column the shaded squares are rows 1 and 3, or rows 2 and 4, and each of those row pairs comes up exactly half the time.
20	C	Basic Counting	Choose which 3 rows (10 ways) and which 3 columns (10 ways) the blocks use, then decide which row goes with which column (6 ways): 10 times 10 times 6.
21	B	Similar & Congruent Triangles	The small triangle above the square has the same shape as the big one, so compare width to height: s over 5 equals (height minus s) over height.
22	B	Expected Value	Of the 16 equal rolls, your number shows on both dice in 1, on exactly one die in 6, and on neither in 9.
23	E	Solid Geometry	The top piece is a shrunken copy of the pyramid. Half the area means every length, including the height, shrinks by a factor of the square root of 2.
24	C	Bases & Digits	A multiple of 4 must end in 44; a multiple of 9 needs the number of 4's to be a multiple of 9. Use nine 4's and one 9.
25	A	Diophantine Equations	Over the common bottom 9ab: since a and b share no factor, b divides 9 and a divides 14, and the 9 forces b to be a multiple of 3.

2007 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.