2007 AMC 10A

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	C	Ratios, Percents & Averages	Turn each discount into a price per ticket first: 25% off 20 dollars is 15 dollars, and 30% off is 14 dollars. Then multiply by the number of tickets.
2	A	Functions	The strange symbols are just recipes. Put 6 in place of a and 2 in place of b in each recipe, work out the two numbers, then divide.
3	D	Solid Geometry	The brick sits fully under water and pushes up exactly its own volume of water. Spread that volume over the tank's floor to get the rise.
4	A	Linear Equations & Word Problems	Two odd numbers in a row are 2 apart. If the bigger one is three times the smaller, the difference of 2 must equal twice the smaller.
5	B	Systems of Equations	Work in cents. The two purchases give two equations for the pencil price and the notebook price; solve them, then price the new order.
6	A	Ratios, Percents & Averages	A percent increase compares the gain to the starting number. From 60 to 66 is exactly 10%, and every other year's gain is less than 10% of its start.
7	D	Ratios, Percents & Averages	The state tax is 10% of what is left, not of the whole. So the taxes are 20% plus 8%, or 28% of the inheritance.
8	D	Angles & Polygons	Both isosceles triangles have AC as the base. Find the base angle at A in each one (70 and 20 degrees); angle BAD is the gap between them.
9	E	Exponents, Logarithms & Radicals	Rewrite 81 as 3 to the 4th and 125 as 5 to the 3rd. Then each equation says two exponents are equal, giving two simple equations in a and b.
10	E	Ratios, Percents & Averages	Turn averages into totals. The family totals 20 times its size, the mother and children 16 times theirs; the gap is the father's 48.
11	C	Sets, Estimation & Miscellaneous	A face and its opposite face use each of the 8 corners once. Their sums add to 36, the total of 1 through 8, so each face sums to 18.
12	D	Basic Counting	Each tourist has 2 choices, giving 64 ways with no rule. Only 2 of them leave a guide with nobody, so the answer is 62.
13	B	Linear Equations & Word Problems	Set the walking speed to 1 and biking to 7. Walking straight takes as long as walking home plus biking the whole way from home to the stadium.
14	A	Circles	A 3-4-5 triangle has a right angle, so its longest side is a diameter of the circle, which is 6. Scale the 3-4-5 triangle up by 6/5.
15	B	Circles	Walk along a diagonal from the center to a corner: 2 + 1 = 3 to a small circle's center, then the diagonal of a 1-by-1 square to the corner.
16	E	Basic Probability	Only odd and even matter. A product of two random numbers is odd just 1/4 of the time, and ad - bc is even when both products match.
17	D	Number Properties	75 is 3 times 5 times 5. To make a perfect cube, m must add two more 3s and one more 5, so m = 45 and n = 15.
18	C	Quadrilaterals & Polygon Areas	Put the shape on a grid. Then ABCM is triangle ABG with triangle CMG cut off, and M's position comes from where the two lines cross.
19	C	Quadrilaterals & Polygon Areas	Measure the unpainted half: four equal right triangles against the sides. What remains of each side is two short painted pieces set by the brush width.
20	D	Algebraic Manipulation	Square a + 1/a. Since a times 1/a is 1, the middle term is just 2. Square twice: 16 - 2 = 14, then 196 - 2 = 194.
21	C	Solid Geometry	One length is passed along: the outer cube's edge equals the sphere's diameter, which equals the long diagonal through the inner cube, corner to opposite corner.
22	D	Bases & Digits	The digits wrap around, so each digit appears once as a hundreds, once as a tens and once as a ones digit. The sum is 111 times the digit total.
23	B	Diophantine Equations	Write the difference of squares as (m + n) times (m - n) = 96. These two factors must both be even, which leaves only four factor pairs.
24	B	Circles	Radii to E and F make a rectangle. The shaded part is that rectangle minus two right triangles with legs 2 and two 45-degree circle slices.
25	D	Bases & Digits	Digit sums are small, so n must be between 1969 and 2007. A remainder argument with 9 shows n is a multiple of 3, leaving 13 numbers to test.

2007 AMC 10A. 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.