2001 AMC 10

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

#	Answer	Topic	Key insight
1	E	Statistics & Data	The list is already in order, so the median is the middle (fifth) entry. Every entry is n plus a fixed number, so find n first, then average.
2	C	Linear Equations & Word Problems	A number's reciprocal times its opposite is always -1, whatever the number is. So the number equals -1 + 2 = 1.
3	E	Linear Equations & Word Problems	Adding 3 to each of two numbers adds 6 to the total, and doubling both numbers doubles the total. So the new total is 2 times (S + 6).
4	E	Circles	Each side of the triangle can cross the circle at most twice, so three sides give at most 6 points, and a slightly enlarged incircle reaches 6.
5	D	Transformations & Symmetry	Go through the twelve shapes one by one and look for a fold line, including slanted (diagonal) ones. Six shapes have one.
6	E	Bases & Digits	Write the number as 10 times the tens digit plus the units digit. The units digit cancels, and what is left forces it to be 9.
7	C	Fractions & Decimals	Moving the decimal point four places to the right is the same as multiplying by 10000. Write that as an equation with the reciprocal and solve.
8	B	GCD & LCM	The four tutors are all together again on the first day that is a multiple of 3, 4, 6 and 7 at once. That is their least common multiple, 84.
9	B	Ratios, Percents & Averages	See the tax as p% on everything plus an extra 2% above $28000. The p% parts cancel, so that extra 2% equals 0.25% of her income.
10	D	Systems of Equations	Divide one equation by another so a letter cancels: xz = 48 and xy = 24 show z is twice y, and then yz = 72 gives y = 6.
11	C	Sequences & Series	Split a ring into its four sides. The nth ring has two rows of 2n+1 squares and two columns of 2n-1 squares, which adds up to 8n.
12	D	Divisibility & Factors	Three numbers in a row always include an even number and a multiple of 3, so the product is a multiple of 42, but not always of 4.
13	E	Logic Puzzles	Ten letters and ten digits means every digit 0 to 9 is used once. Pick the odd run and the even run; A, B, C are the leftovers.
14	A	Diophantine Equations	Double every price so there are no halves. Then full price times (full-price tickets + 140) equals 4002, so look for a factor of 4002 between 140 and 280.
15	C	Quadrilaterals & Polygon Areas	The crosswalk is a parallelogram. Its area is 15 times 40 using the curb as the base, and 50 times the unknown distance using a stripe as the base.
16	D	Statistics & Data	Call the mean m. Then the numbers are m - 10, 5 and m + 15, and three numbers with mean m always add up to 3m.
17	C	Solid Geometry	Rolled up, the sector's radius of 10 becomes the cone's slant side, and its curved edge, of length 14 pi, wraps around the base circle.
18	D	Quadrilaterals & Polygon Areas	The pattern repeats in 3 by 3 blocks. The four corner squares of a block cover 4 of its 9 units of area, so pentagons cover 5 out of 9.
19	D	Distributions & Stars and Bars	Only how many of each type Pat picks matters, not the order. List the ways to split 4 donuts among 3 types, or picture 4 dots split by 2 dividers.
20	B	Angles & Polygons	Each side of the square is made of two triangle legs and one octagon side. Each leg is the octagon side divided by the square root of 2.
21	B	Solid Geometry	Slice straight down through the center: the cone becomes a triangle, the cylinder a square, and the small triangle above the square has the same shape as the big one.
22	D	Systems of Equations	Compare two lines that share an unknown number; that unknown cancels. The left column against the top row gives w = 19, and against the diagonal gives x = 22.
23	D	Basic Probability	Imagine drawing all five chips. The whites run out first exactly when the last chip in the box is red, and each chip is equally likely to be last.
24	B	Triangles: Area & Pythagorean	Draw a line from B straight across to CD. That right triangle has legs 7 and CD - AB, long side AB + CD, giving 4(AB)(CD) = 49.
25	B	Inclusion-Exclusion	Count multiples of 3 or 4 first. Then remove only those that are also multiples of 5, which are the multiples of 15 or 20.

2001 AMC 10. 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.