Difference between revisions of "2010 AMC 12A Problems Raw"

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== Problem 1 ==
What is <math>\left(20-\left(2010-201\right)\right)+\left(2010-\left(201-20\right)\right)</math>?
<math>\textbf{(A)}\ -4020 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 401 \qquad \textbf{(E)}\ 4020</math>
[[2010 AMC 12A Problems/Problem 1|Solution]]
== 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?
<math>\textbf{(A)}\ 585 \qquad \textbf{(B)}\ 594 \qquad \textbf{(C)}\ 672 \qquad \textbf{(D)}\ 679 \qquad \textbf{(E)}\ 694</math>
[[2010 AMC 12A Problems/Problem 2|Solution]]
== Problem 3 ==
Rectangle <math>ABCD</math>, pictured below, shares <math>50\%</math> of its area with square <math>EFGH</math>. Square <math>EFGH</math> shares <math>20\%</math> of its area with rectangle <math>ABCD</math>. What is <math>\frac{AB}{AD}</math>?
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10</math>
[[2010 AMC 12A Problems/Problem 3|Solution]]
== Problem 4 ==
If <math>x<0</math>, then which of the following must be positive?
<math>\textbf{(A)}\ \frac{x}{\left|x\right|} \qquad \textbf{(B)}\ -x^2 \qquad \textbf{(C)}\ -2^x \qquad \textbf{(D)}\ -x^{-1} \qquad \textbf{(E)}\ \sqrt[3]{x}</math>
[[2010 AMC 12A Problems/Problem 4|Solution]]
== 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 <math>n</math> shots are bullseyes she will be guaranteed victory. What is the minimum value for <math>n</math>?
<math>\textbf{(A)}\ 38 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 42 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 46</math>
[[2010 AMC 12A Problems/Problem 5|Solution]]
== Problem 6 ==
A <math>\textit{palindrome}</math>, such as 83438, is a number that remains the same when its digits are reversed. The numbers <math>x</math> and <math>x+32</math> are three-digit and four-digit palindromes, respectively. What is the sum of the digits of <math>x</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]]
== 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?
<math>\textbf{(A)}\ 0.04 \qquad \textbf{(B)}\ \frac{0.4}{\pi} \qquad \textbf{(C)}\ 0.4 \qquad \textbf{(D)}\ \frac{4}{\pi} \qquad \textbf{(E)}\ 4</math>
[[2010 AMC 12A Problems/Problem 7|Solution]]
== 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)}\ 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]]
== Problem 9 ==
A solid cube has side length <math>3</math> inches. A <math>2</math>-inch by <math>2</math>-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]]
== Problem 12 ==
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs?
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math>
[[2010 AMC 12A Problems/Problem 12|Solution]]
== Problem 13 ==
For how many integer values of <math>k</math> do the graphs of <math>x^2+y^2=k^2</math> and <math>xy = k</math> not intersect?
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8</math>
[[2010 AMC 12A Problems/Problem 13|Solution]]
== Problem 14 ==
Nondegenerate <math>\triangle ABC</math> has integer side lengths, <math>\overline{BD}</math> is an angle bisector, <math>AD = 3</math>, and <math>DC=8</math>. What is the smallest possible value of the perimeter?
<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 37</math>
[[2010 AMC 12A Problems/Problem 14|Solution]]
== Problem 15 ==
A coin is altered so that the probability that it lands on heads is less than <math>\frac{1}{2}</math> and when the coin is flipped four times, the probability of an equal number of heads and tails is <math>\frac{1}{6}</math>. What is the probability that the coin lands on heads?
<math>\textbf{(A)}\ \frac{\sqrt{15}-3}{6} \qquad \textbf{(B)}\ \frac{6-\sqrt{6\sqrt{6}+2}}{12} \qquad \textbf{(C)}\ \frac{\sqrt{2}-1}{2} \qquad \textbf{(D)}\ \frac{3-\sqrt{3}}{6} \qquad \textbf{(E)}\ \frac{\sqrt{3}-1}{2}</math>
[[2010 AMC 12A Problems/Problem 15|Solution]]
== Problem 16 ==
Bernardo randomly picks 3 distinct numbers from the set <math>\{1,2,3,...,7,8,9\}</math> and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set <math>\{1,2,3,...,6,7,8\}</math> and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
<math>\textbf{(A)}\ \frac{47}{72} \qquad \textbf{(B)}\ \frac{37}{56} \qquad \textbf{(C)}\ \frac{2}{3} \qquad \textbf{(D)}\ \frac{49}{72} \qquad \textbf{(E)}\ \frac{39}{56}</math>
[[2010 AMC 12A Problems/Problem 16|Solution]]
== Problem 17 ==
Equiangular hexagon <math>ABCDEF</math> has side lengths <math>AB=CD=EF=1</math> and <math>BC=DE=FA=r</math>. The area of <math>\triangle ACE</math> is <math>70\%</math> of the area of the hexagon. What is the sum of all possible values of <math>r</math>?
<math>\textbf{(A)}\ \frac{4\sqrt{3}}{3} \qquad \textbf{(B)} \frac{10}{3} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac{17}{4} \qquad \textbf{(E)}\ 6</math>
[[2010 AMC 12A Problems/Problem 17|Solution]]
== Problem 18 ==
A 16-step path is to go from <math>(-4,-4)</math> to <math>(4,4)</math> with each step increasing either the <math>x</math>-coordinate or the <math>y</math>-coordinate by 1. How many such paths stay outside or on the boundary of the square <math>-2 \le x \le 2</math>, <math>-2 \le y \le 2</math> at each step?
<math>\textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,800</math>
[[2010 AMC 12A Problems/Problem 18|Solution]]
== Problem 19 ==
Each of 2010 boxes in a line contains a single red marble, and for <math>1 \le k \le 2010</math>, the box in the <math>k\text{th}</math> position also contains <math>k</math> white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let <math>P(n)</math> be the probability that Isabella stops after drawing exactly <math>n</math> marbles. What is the smallest value of <math>n</math> for which <math>P(n) < \frac{1}{2010}</math>?
<math>\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005</math>
[[2010 AMC 12A Problems/Problem 19|Solution]]
== Problem 20 ==
Arithmetic sequences <math>\left(a_n\right)</math> and <math>\left(b_n\right)</math> have integer terms with <math>a_1=b_1=1<a_2 \le b_2</math> and <math>a_n b_n = 2010</math> for some <math>n</math>. What is the largest possible value of <math>n</math>?
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 288 \qquad \textbf{(E)}\ 2009</math>
[[2010 AMC 12A Problems/Problem 20|Solution]]
== Problem 21 ==
The graph of <math>y=x^6-10x^5+29x^4-4x^3+ax^2</math> lies above the line <math>y=bx+c</math> except at three values of <math>x</math>, where the graph and the line intersect. What is the largest of these values?
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8</math>
[[2010 AMC 12A Problems/Problem 21|Solution]]
== Problem 22 ==
What is the minimum value of <math>\left|x-1\right| + \left|2x-1\right| + \left|3x-1\right| + \cdots + \left|119x - 1 \right|</math>?
<math>\textbf{(A)}\ 49 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 53</math>
[[2010 AMC 12A Problems/Problem 22|Solution]]
== Problem 23 ==
The number obtained from the last two nonzero digits of <math>90!</math> is equal to <math>n</math>. What is <math>n</math>?
<math>\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68</math>
[[2010 AMC 12A Problems/Problem 23|Solution]]
== Problem 24 ==
Let <math>f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)</math>. The intersection of the domain of <math>f(x)</math> with the interval <math>[0,1]</math> is a union of <math>n</math> disjoint open intervals. What is <math>n</math>?
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 36</math>
[[2010 AMC 12A Problems/Problem 24|Solution]]
== Problem 25 ==
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 32?
<math>\textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad \textbf{(E)}\ 2255</math>
[[2010 AMC 12A Problems/Problem 25|Solution]]

Latest revision as of 11:24, 23 August 2020

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