2007 AMC 10A Problems
Contents
[hide]Problem 1
One ticket to a show costs 20$at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?$ (Error compiling LaTeX. Unknown error_msg)\mathrm{(A)}\ 2\qquad \mathrm{(B)}\ 5\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 15\qquad \mathrm{(E)}\ 20$[[2007 AMC 10A Problems/Problem 1|Solution]]
== Problem 2 ==
Define$ (Error compiling LaTeX. Unknown error_msg)a@b = ab - b^{2}a\#b = a + b - ab^{2}
\frac {6@2}{6\#2}
\text{(A)}\ - \frac {1}{2}\qquad \text{(B)}\ - \frac {1}{4}\qquad \text{(C)}\ \frac {1}{8}\qquad \text{(D)}\ \frac {1}{4}\qquad \text{(E)}\ \frac {1}{2}$[[2007 AMC 10A Problems/Problem 2|Solution]]
== Problem 3 == An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?$ (Error compiling LaTeX. Unknown error_msg)\text{(A)}\ 0.5 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 1.5 \qquad \text{(D)}\ 2 \qquad \text{(E)}\ 2.5$[[2007 AMC 10A Problems/Problem 3|Solution]]
== Problem 4 == The larger of two consecutive odd integers is three times the smaller. What is their sum?$ (Error compiling LaTeX. Unknown error_msg)\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20$[[2007 AMC 10A Problems/Problem 4|Solution]]
== Problem 5 ==
A school store sells 7 pencils and 8 notebooks for$ (Error compiling LaTeX. Unknown error_msg)$ . It also sells 5 pencils and 3 notebooks for
1.77
\text{(A)}\ $
5.84 \qquad \text{(C)}\ $
6.16 \qquad \text{(E)}\ $
Problem 6
At Euclid High School, the number of students taking the AMC 10 was in 2002,
in 2003,
in 2004,
in 2005,
and 2006, and is
in 2007. Between what two consecutive years was there the largest percentage increase?
Problem 7
Last year Mr. Jon Q. Public received an inheritance. He paid in federal taxes on the inheritance, and paid
of what he had left in state taxes. He paid a total of
10500
(\mathrm {A})\ 30000 \qquad (\mathrm {B})\ 32500 \qquad(\mathrm {C})\ 35000 \qquad(\mathrm {D})\ 37500 \qquad(\mathrm {E})\ 40000$[[2007 AMC 10A Problems/Problem 7|Solution]]
== Problem 8 ==
Triangles$ (Error compiling LaTeX. Unknown error_msg)ABCADC
AB=BC
AD=DC
D
ABC
ABC
ADC
BAD
\mathrm{(A)}\ 20\qquad \mathrm{(B)}\ 30\qquad \mathrm{(C)}\ 40\qquad \mathrm{(D)}\ 50\qquad \mathrm{(E)}\ 60$[[2007 AMC 10A Problems/Problem 8|Solution]]
== Problem 9 ==
Real numbers$ (Error compiling LaTeX. Unknown error_msg)ab
3^{a} = 81^{b + 2}
125^{b} = 5^{a - 3}
ab
\text{(A)}\ -60 \qquad \text{(B)}\ -17 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 60$[[2007 AMC 10A Problems/Problem 9|Solution]]
== Problem 10 ==
The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is$ (Error compiling LaTeX. Unknown error_msg)2048
16
\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$[[2007 AMC 10A Problems/Problem 10|Solution]]
== Problem 11 ==
The numbers from$ (Error compiling LaTeX. Unknown error_msg)18
\text{(A)}\ 14 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 24$[[2007 AMC 10A Problems/Problem 11|Solution]]
== Problem 12 == Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?$ (Error compiling LaTeX. Unknown error_msg)\text{(A)}\ 56 \qquad \text{(B)}\ 58 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 62 \qquad \text{(E)}\ 64$[[2007 AMC 10A Problems/Problem 12|Solution]]
== Problem 13 == Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the [[ratio]] of Yan's distance from his home to his distance from the stadium?$ (Error compiling LaTeX. Unknown error_msg)\mathrm{(A)}\ \frac 23\qquad \mathrm{(B)}\ \frac 34\qquad \mathrm{(C)}\ \frac 45\qquad \mathrm{(D)}\ \frac 56\qquad \mathrm{(E)}\ \frac 78$[[2007 AMC 10A Problems/Problem 13|Solution]]
== Problem 14 ==
A triangle with side lengths in the ratio$ (Error compiling LaTeX. Unknown error_msg)3 : 4 : 53
\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18$[[2007 AMC 10A Problems/Problem 14|Solution]]
== Problem 15 ==
Four circles of radius$ (Error compiling LaTeX. Unknown error_msg)12
\text{(A)}\ 32 \qquad \text{(B)}\ 22 + 12\sqrt {2}\qquad \text{(C)}\ 16 + 16\sqrt {3}\qquad \text{(D)}\ 48 \qquad \text{(E)}\ 36 + 16\sqrt {2}$[[2007 AMC 10A Problems/Problem 15|Solution]]
== Problem 16 ==
Integers$ (Error compiling LaTeX. Unknown error_msg)a, b, c,d
ad-bc
\mathrm{(A)}\ \frac 38\qquad \mathrm{(B)}\ \frac 7{16}\qquad \mathrm{(C)}\ \frac 12\qquad \mathrm{(D)}\ \frac 9{16}\qquad \mathrm{(E)}\ \frac 58$[[2007 AMC 10A Problems/Problem 16|Solution]]
== Problem 17 ==
Suppose that$ (Error compiling LaTeX. Unknown error_msg)mn
75m = n^{3}
m + n
\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 60 \qquad \text{(E)}\ 5700$[[2007 AMC 10A Problems/Problem 17|Solution]]
== Problem 18 ==
Consider the$ (Error compiling LaTeX. Unknown error_msg)12ABCDEFGHIJKL
4
\overline{AG}
\overline{CH}
M
ABCM
\text{(A)}\ \frac {44}{3}\qquad \text{(B)}\ 16 \qquad \text{(C)}\ \frac {88}{5}\qquad \text{(D)}\ 20 \qquad \text{(E)}\ \frac {62}{3}$[[2007 AMC 10A Problems/Problem 18|Solution]]
== Problem 19 == A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width? <center> [[Image:2007 AMC 10A Problems-Problem 19 Picture.png|120]] </center>$ (Error compiling LaTeX. Unknown error_msg)\text{(A)}\ 2\sqrt {2} + 1 \qquad \text{(B)}\ 3\sqrt {2}\qquad \text{(C)}\ 2\sqrt {2} + 2 \qquad \text{(D)}\ 3\sqrt {2} + 1 \qquad \text{(E)}\ 3\sqrt {2} + 2$[[2007 AMC 10A Problems/Problem 19|Solution]]
== Problem 20 ==
Suppose that the number$ (Error compiling LaTeX. Unknown error_msg)a4 = a + a^{ - 1}
a^{4} + a^{ - 4}
\text{(A)}\ 164 \qquad \text{(B)}\ 172 \qquad \text{(C)}\ 192 \qquad \text{(D)}\ 194 \qquad \text{(E)}\ 212$[[2007 AMC 10A Problems/Problem 20|Solution]]
== Problem 21 ==
A sphere is inscribed in a cube that has a surface area of$ (Error compiling LaTeX. Unknown error_msg)24\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 12$[[2007 AMC 10A Problems/Problem 21|Solution]]
== Problem 22 ==
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let$ (Error compiling LaTeX. Unknown error_msg)SS
\mathrm{(A)}\ 3\qquad \mathrm{(B)}\ 7\qquad \mathrm{(C)}\ 13\qquad \mathrm{(D)}\ 37\qquad \mathrm{(E)}\ 43$[[2007 AMC 10A Problems/Problem 22|Solution]]
== Problem 23 ==
How many ordered pairs$ (Error compiling LaTeX. Unknown error_msg)(m,n)m \ge n
96
\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 12$[[2007 AMC 10A Problems/Problem 23|Solution]]
== Problem 24 ==
Circles centered at$ (Error compiling LaTeX. Unknown error_msg)AB
2
O
\overline{AB}
OA = 2\sqrt {2}
OC
OD
A
B
EF
ECODF
\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qquad \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}$[[2007 AMC 10A Problems/Problem 24|Solution]]
== Problem 25 ==
For each positive integer$ (Error compiling LaTeX. Unknown error_msg)nS(n)
n.
n
n + S(n) + S(S(n)) = 2007?$$ (Error compiling LaTeX. Unknown error_msg)\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$
See also
- AMC 10
- AMC 10 Problems and Solutions
- 2007 AMC 10A
- 2007 AMC A Math Jam Transcript
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.