Difference between revisions of "2010 AMC 12A Problems"
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<math>\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 22 \qquad \textbf{(D)}\ 23 \qquad \textbf{(E)}\ 24</math> | <math>\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 22 \qquad \textbf{(D)}\ 23 \qquad \textbf{(E)}\ 24</math> | ||
− | [[2010 AMC 12A Problems/Problem 6|Solution] | + | [[2010 AMC 12A Problems/Problem 6|Solution]] |
== Problem 7 == | == Problem 7 == | ||
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== Problem 8 == | == Problem 8 == | ||
− | + | Triangle <math>ABC</math> has <math>AB=2 \cdot AC</math>. Let <math>D</math> and <math>E</math> be on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, such that <math>\angle BAE = \angle ACD</math>. Let <math>F</math> be the intersection of segments <math>AE</math> and <math>CD</math>, and suppose that <math>\triangle CFE</math> is equilateral. What is <math>\angle ACB</math>? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 60^\circ \qquad \textbf{(B)}\ 75^\circ \qquad \textbf{(C)}\ 90^\circ \qquad \textbf{(D)}\ 105^\circ \qquad \textbf{(E)}\ 120^\circ</math> |
− | [[2010 AMC 12A Problems/Problem 8|Solution] | + | [[2010 AMC 12A Problems/Problem 8|Solution]] |
+ | |||
+ | == Problem 9 == | ||
+ | A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? | ||
+ | |||
+ | <math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15</math> | ||
+ | |||
+ | [[2010 AMC 12A Problems/Problem 9|Solution] | ||
+ | |||
+ | == Problem 10 == | ||
+ | The first four terms of an arithmetic sequence are <math>p</math>, <math>9</math>, <math>3p-q</math>, and <math>3p+q</math>. What is the <math>2010^\text{th}</math> term of this sequence? | ||
+ | |||
+ | <math>\textbf{(A)}\ 8041 \qquad \textbf{(B)}\ 8043 \qquad \textbf{(C)}\ 8045 \qquad \textbf{(D)}\ 8047 \qquad \textbf{(E)}\ 8049</math> | ||
+ | |||
+ | [[2010 AMC 12A Problems/Problem 10|Solution] | ||
+ | |||
+ | == Problem 11 == | ||
+ | The solution of the equation <math>7^{x+7} = 8^x</math> can be expressed in the form <math>x = \log_b 7^7</math>. What is <math>b</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{7}{15} \qquad \textbf{(B)}\ \frac{7}{8} \qquad \textbf{(C)}\ \frac{8}{7} \qquad \textbf{(D)}\ \frac{15}{8} \qquad \textbf{(E)}\ \frac{15}{7}</math> | ||
+ | |||
+ | [[2010 AMC 12A Problems/Problem 11|Solution] |
Revision as of 13:05, 10 February 2010
Contents
Problem 1
What is ?
Problem 2
A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?
Problem 3
Rectangle , pictured below, shares of its area with square . Square shares of its area with rectangle . What is ?
Problem 4
If , then which of the following must be positive?
Problem 5
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next shots are bullseyes she will be guaranteed victory. What is the minimum value for ?
Problem 6
A $\texti{palindrome}$ (Error compiling LaTeX. Unknown error_msg), such as 83438, is a number that remains the same when its digits are reversed. The numbers and are three-digit and four-digit palindromes, respectively. What is the sum of the digits of ?
Problem 7
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?
Problem 8
Triangle has . Let and be on and , respectively, such that . Let be the intersection of segments and , and suppose that is equilateral. What is ?
Problem 9
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
[[2010 AMC 12A Problems/Problem 9|Solution]
Problem 10
The first four terms of an arithmetic sequence are , , , and . What is the term of this sequence?
[[2010 AMC 12A Problems/Problem 10|Solution]
Problem 11
The solution of the equation can be expressed in the form . What is ?
[[2010 AMC 12A Problems/Problem 11|Solution]