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Difference between revisions of "2007 AMC 12B Problems"

m (→‎Problem 3)
Line 3: Line 3:
  
 
<math>\mathrm {(A)} 678</math>  <math>\mathrm {(B)} 768</math>  <math>\mathrm {(C)} 786</math>  <math>\mathrm {(D)} 867</math>  <math>\mathrm {(E)} 876</math>
 
<math>\mathrm {(A)} 678</math>  <math>\mathrm {(B)} 768</math>  <math>\mathrm {(C)} 786</math>  <math>\mathrm {(D)} 867</math>  <math>\mathrm {(E)} 876</math>
−
 
  
 
[[2007 AMC 12B Problems/Problem 1 | Solution]]
 
[[2007 AMC 12B Problems/Problem 1 | Solution]]
  
 
==Problem 2==
 
==Problem 2==
−
 
 
A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
 
A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
  
 
<math>\mathrm {(A)} 22</math>  <math>\mathrm {(B)} 24</math>  <math>\mathrm {(C)} 25</math>  <math>\mathrm {(D)} 26</math>  <math>\mathrm {(E)} 28</math>
 
<math>\mathrm {(A)} 22</math>  <math>\mathrm {(B)} 24</math>  <math>\mathrm {(C)} 25</math>  <math>\mathrm {(D)} 26</math>  <math>\mathrm {(E)} 28</math>
−
 
  
 
[[2007 AMC 12B Problems/Problem 2 | Solution]]
 
[[2007 AMC 12B Problems/Problem 2 | Solution]]
  
 
==Problem 3==
 
==Problem 3==
−
 
−
 
 
The point <math>O</math> is the center of the circle circumscribed about triangle <math>ABC</math>, with <math>\angle BOC = 120^{\circ}</math> and <math>\angle AOB = 140^{\circ}</math>, as shown. What is the degree measure of <math>\angle ABC</math>?
 
The point <math>O</math> is the center of the circle circumscribed about triangle <math>ABC</math>, with <math>\angle BOC = 120^{\circ}</math> and <math>\angle AOB = 140^{\circ}</math>, as shown. What is the degree measure of <math>\angle ABC</math>?
  
Line 24: Line 19:
  
 
<math>\mathrm {(A)} 35</math>  <math>\mathrm {(B)} 40</math>  <math>\mathrm {(C)} 45</math>  <math>\mathrm {(D)} 50</math>  <math>\mathrm {(E)} 60</math>
 
<math>\mathrm {(A)} 35</math>  <math>\mathrm {(B)} 40</math>  <math>\mathrm {(C)} 45</math>  <math>\mathrm {(D)} 50</math>  <math>\mathrm {(E)} 60</math>
−
 
  
 
[[2007 AMC 12B Problems/Problem 3 | Solution]]
 
[[2007 AMC 12B Problems/Problem 3 | Solution]]
  
 
==Problem 4==
 
==Problem 4==
−
 
−
 
 
At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?  
 
At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?  
  
 
<math>\mathrm {(A)} 6</math>  <math>\mathrm {(B)} 8</math>  <math>\mathrm {(C)} 9</math>  <math>\mathrm {(D)} 12</math>  <math>\mathrm {(E)} 18</math>
 
<math>\mathrm {(A)} 6</math>  <math>\mathrm {(B)} 8</math>  <math>\mathrm {(C)} 9</math>  <math>\mathrm {(D)} 12</math>  <math>\mathrm {(E)} 18</math>
−
 
  
 
[[2007 AMC 12B Problems/Problem 4 | Solution]]
 
[[2007 AMC 12B Problems/Problem 4 | Solution]]
Line 42: Line 33:
  
 
<math>\mathrm {(A)} 13</math>  <math>\mathrm {(B)} 14</math>  <math>\mathrm {(C)} 15</math>  <math>\mathrm {(D)} 16</math>  <math>\mathrm {(E)} 17</math>
 
<math>\mathrm {(A)} 13</math>  <math>\mathrm {(B)} 14</math>  <math>\mathrm {(C)} 15</math>  <math>\mathrm {(D)} 16</math>  <math>\mathrm {(E)} 17</math>
−
 
−
 
  
 
[[2007 AMC 12B Problems/Problem 5 | Solution]]
 
[[2007 AMC 12B Problems/Problem 5 | Solution]]
  
 
==Problem 6==
 
==Problem 6==
−
 
 
Triangle <math>ABC</math> has side lengths <math>AB = 5</math>, <math>BC = 6</math>, and <math>AC = 7</math>. Two bugs start simultaneously from <math>A</math> and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point <math>D</math>. What is <math>BD</math>?  
 
Triangle <math>ABC</math> has side lengths <math>AB = 5</math>, <math>BC = 6</math>, and <math>AC = 7</math>. Two bugs start simultaneously from <math>A</math> and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point <math>D</math>. What is <math>BD</math>?  
  
Line 59: Line 47:
  
 
<math>\mathrm {(A)} 90</math>  <math>\mathrm {(B)} 108</math>  <math>\mathrm {(C)} 120</math>  <math>\mathrm {(D)} 144</math>  <math>\mathrm {(E)} 150</math>
 
<math>\mathrm {(A)} 90</math>  <math>\mathrm {(B)} 108</math>  <math>\mathrm {(C)} 120</math>  <math>\mathrm {(D)} 144</math>  <math>\mathrm {(E)} 150</math>
−
 
  
 
[[2007 AMC 12B Problems/Problem 7 | Solution]]
 
[[2007 AMC 12B Problems/Problem 7 | Solution]]
  
 
==Problem 8==
 
==Problem 8==
−
 
 
Tom's age is <math>T</math> years, which is also the sum of the ages of his three children. His age <math>N</math> years ago was twice the sum of their ages then. What is <math>T/N</math> ?  
 
Tom's age is <math>T</math> years, which is also the sum of the ages of his three children. His age <math>N</math> years ago was twice the sum of their ages then. What is <math>T/N</math> ?  
  
 
<math>\mathrm {(A)} 2</math>  <math>\mathrm {(B)} 3</math>  <math>\mathrm {(C)} 4</math>  <math>\mathrm {(D)} 5</math>  <math>\mathrm {(E)} 6</math>
 
<math>\mathrm {(A)} 2</math>  <math>\mathrm {(B)} 3</math>  <math>\mathrm {(C)} 4</math>  <math>\mathrm {(D)} 5</math>  <math>\mathrm {(E)} 6</math>
−
 
  
 
[[2007 AMC 12B Problems/Problem 8 | Solution]]
 
[[2007 AMC 12B Problems/Problem 8 | Solution]]
  
 
==Problem 9==
 
==Problem 9==
 +
A function <math>f</math> has the property that <math>f(3x-1)=x^2+x+1</math> for all real numbers <math>x</math>. What is <math>f(5)</math>?
  
−
 
+
<math>\mathrm {(A)} 7</math>  <math>\mathrm {(B)} 13</math>  <math>\mathrm {(C)} 31</math>  <math>\mathrm {(D)} 111</math>  <math>\mathrm {(E)} 211</math>
−
 
 
  
 
[[2007 AMC 12B Problems/Problem 9 | Solution]]
 
[[2007 AMC 12B Problems/Problem 9 | Solution]]
  
 
==Problem 10==
 
==Problem 10==
 +
Some boys and girls are having a car wash to raise money for a class trip to China. Initially <math>40%</math> of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then <math>30%</math> of the group are girls. How many girls were initially in the group?
 +
 +
<math>\mathrm {(A)} 4</math>  <math>\mathrm {(B)} 6</math>  <math>\mathrm {(C)} 8</math>  <math>\mathrm {(D)} 10</math>  <math>\mathrm {(E)} 12</math>
  
 
[[2007 AMC 12B Problems/Problem 10 | Solution]]
 
[[2007 AMC 12B Problems/Problem 10 | Solution]]
  
 
==Problem 11==
 
==Problem 11==
 +
The angles of quadrilateral <math>ABCD</math> satisfy <math>\angle A=2\angle B=3\angle C=4\angle D</math>. What is the degree measure of <math>\angle A</math>, rounded to the nearest whole number?
  
 +
<math>\mathrm {(A)} 125</math>  <math>\mathrm {(B)} 144</math>  <math>\mathrm {(C)} 153</math>  <math>\mathrm {(D)} 173</math>  <math>\mathrm {(E)} 180</math>
  
 
[[2007 AMC 12B Problems/Problem 11 | Solution]]
 
[[2007 AMC 12B Problems/Problem 11 | Solution]]
  
 
==Problem 12==
 
==Problem 12==
 +
A teacher gave a test to a class in which <math>10%</math> of the students are juniors and <math>90%</math> are seniors. The average score on the test was <math>84</math>. The juniors all received the same score, and the average score of the seniors was <math>83</math>. What score did each of the juniors receive on the test?
  
−
 
+
<math>\mathrm {(A)} 85</math>  <math>\mathrm {(B)} 88</math>  <math>\mathrm {(C)} 93</math>  <math>\mathrm {(D)} 94</math>  <math>\mathrm {(E)} 98</math>
  
 
[[2007 AMC 12B Problems/Problem 12 | Solution]]
 
[[2007 AMC 12B Problems/Problem 12 | Solution]]
  
 
==Problem 13==
 
==Problem 13==
 +
A traffic light runs repeatedly through the following cycle: green for <math>30</math> seconds, then yellow for <math>3</math> seconds, and then red for <math>30</math> seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?
  
−
 
+
<math>\mathrm {(A)} \frac{1}{63}</math>  <math>\mathrm {(B)} \frac{1}{21}</math>  <math>\mathrm {(C)} \frac{1}{10}</math>  <math>\mathrm {(D)} \frac{1}{7}</math>  <math>\mathrm {(E)} \frac{1}{3}</math>
  
 
[[2007 AMC 12B Problems/Problem 13 | Solution]]
 
[[2007 AMC 12B Problems/Problem 13 | Solution]]
  
 
==Problem 14==
 
==Problem 14==
 +
Point <math>P</math> is inside equilateral <math>\triangle ABC</math>. Points <math>Q</math>, <math>R</math>, and <math>S</math> are the feet of the perpendiculars from <math>P</math> to <math>\overline{AB}</math>, <math>\overline{BC}</math>, and <math>\overline{CA}</math>, respectively. Given that <math>PQ=1</math>, <math>PR=2</math>, and <math>PS=3</math>, what is <math>AB</math>?
  
 +
<math>\mathrm {(A)} 4</math>  <math>\mathrm {(B)} 3\sqrt{3}</math>  <math>\mathrm {(C)} 6</math>  <math>\mathrm {(D)} 4\sqrt{3}</math>  <math>\mathrm {(E)} 9</math>
  
 
[[2007 AMC 12B Problems/Problem 14 | Solution]]
 
[[2007 AMC 12B Problems/Problem 14 | Solution]]
  
 
==Problem 15==
 
==Problem 15==
 +
The geometric series <math>a+ar+ar^2\cdots</math> has a sum of <math>7</math>, and the terms involving odd powers of <math>r</math> have a sum of <math>3</math>. What is <math>a+r</math>?
  
 +
<math>\mathrm {(A)} \frac{4}{3}</math>  <math>\mathrm {(B)} \frac{12}{7}</math>  <math>\mathrm {(C)} \frac{3}{2}</math>  <math>\mathrm {(D)} \frac{7}{3}</math>  <math>\mathrm {(E)} \frac{5}{2}</math>
  
 
[[2007 AMC 12B Problems/Problem 15 | Solution]]
 
[[2007 AMC 12B Problems/Problem 15 | Solution]]
  
 
==Problem 16==
 
==Problem 16==
 +
Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?
  
 +
<math>\mathrm {(A)} 15</math>  <math>\mathrm {(B)} 18</math>  <math>\mathrm {(C)} 27</math>  <math>\mathrm {(D)} 54</math>  <math>\mathrm {(E)} 81</math>
  
 
[[2007 AMC 12B Problems/Problem 16 | Solution]]
 
[[2007 AMC 12B Problems/Problem 16 | Solution]]
  
 
==Problem 17==
 
==Problem 17==
 +
If <math>a</math> is a nonzero integer and <math>b</math> is a positive number such that <math>ab^2=\log_{10}b</math>, what is the median of the set <math>\{0,1,a,b,1/b\}</math>?
  
 +
<math>\mathrm {(A)} 0</math>  <math>\mathrm {(B)} 1</math>  <math>\mathrm {(C)} a</math>  <math>\mathrm {(D)} b</math>  <math>\mathrm {(E)} \frac{1}{b}</math>
  
 
[[2007 AMC 12B Problems/Problem 17 | Solution]]
 
[[2007 AMC 12B Problems/Problem 17 | Solution]]
  
 
==Problem 18==
 
==Problem 18==
 +
Let <math>a</math>, <math>b</math>, and <math>c</math> be digits with <math>a\ne 0</math>. THe three-digit integer <math>abc</math> lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer <math>acb</math> lies two thirds of the way between the same two squares. What is <math>a+b+c</math>?
  
 +
<math>\mathrm {(A)} 10</math>  <math>\mathrm {(B)} 13</math>  <math>\mathrm {(C)} 16</math>  <math>\mathrm {(D)} 18</math>  <math>\mathrm {(E)} 21</math>
  
 
[[2007 AMC 12B Problems/Problem 18 | Solution]]
 
[[2007 AMC 12B Problems/Problem 18 | Solution]]
  
 
==Problem 19==
 
==Problem 19==
 +
Rhombus <math>ABCD</math>, with side length <math>6</math>, is rolled to form a cylinder of volume <math>6</math> by taping <math>\overline{AB}</math> to <math>\overline{DC}</math>. What is <math>\sin(\angle ABC)</math>?
  
 +
<math>\mathrm {(A)} \frac{\pi}{9}</math>  <math>\mathrm {(B)} \frac{1}{2}</math>  <math>\mathrm {(C)} \frac{\pi}{6}</math>  <math>\mathrm {(D)} \frac{\pi}{4}</math>  <math>\mathrm {(E)} \frac{\sqrt{3}}{2}</math>
  
 
[[2007 AMC 12B Problems/Problem 19 | Solution]]
 
[[2007 AMC 12B Problems/Problem 19 | Solution]]
  
 
==Problem 20==
 
==Problem 20==
 +
The parallelogram bounded by the lines <math>y=ax+c</math>, <math>y=ax+d</math>, <math>y=bx+c</math>, and <math>y=bx+d</math> has area <math>18</math>. The parallelogram bounded by the lines <math>y=ax+c</math>, <math>y=ax-d</math>, <math>y=bx+c</math>, and <math>y=bx-d</math> has area <math>72</math>. Given that <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are positive integers, what is the smallest possible value of <math>a+b+c+d</math>?
  
 +
<math>\mathrm {(A)} 13</math>  <math>\mathrm {(B)} 14</math>  <math>\mathrm {(C)} 15</math>  <math>\mathrm {(D)} 16</math>  <math>\mathrm {(E)} 17</math>
  
 
[[2007 AMC 12B Problems/Problem 20 | Solution]]
 
[[2007 AMC 12B Problems/Problem 20 | Solution]]
  
 
==Problem 21==
 
==Problem 21==
 +
The first <math>2007</math> positive integers are each written in base <math>3</math>. How many of these base-<math>3</math> representations are palindromes? (A palindrome is a number that reads the same forward and backward.)
  
−
 
+
<math>\mathrm {(A)} 100</math>  <math>\mathrm {(B)} 101</math>  <math>\mathrm {(C)} 102</math>  <math>\mathrm {(D)} 103</math>  <math>\mathrm {(E)} 104</math>
  
 
[[2007 AMC 12B Problems/Problem 21 | Solution]]
 
[[2007 AMC 12B Problems/Problem 21 | Solution]]
  
 
==Problem 22==
 
==Problem 22==
 +
Two particles move along the edges of equilateral <math>\triangle ABC</math> in the direction <cmath>A\Rightarrow B\Rightarrow C\Rightarrow A,</cmath> starting simultaneously and moving at the same speed. One starts at <math>A</math>, and the other starts at the midpoint of <math>\overline{BC}</math>. The midpoint of the line segment joining the two particles traces out a path that encloses a region <math>R</math>. What is the ratio of the area of <math>R</math> to the area of <math>\triangle ABC</math>?
  
 +
<math>\mathrm {(A)} \frac{1}{16}</math>  <math>\mathrm {(B)} \frac{1}{12}</math>  <math>\mathrm {(C)} \frac{1}{9}</math>  <math>\mathrm {(D)} \frac{1}{6}</math>  <math>\mathrm {(E)} \frac{1}{4}</math>
  
 
[[2007 AMC 12B Problems/Problem 22 | Solution]]
 
[[2007 AMC 12B Problems/Problem 22 | Solution]]
  
 
==Problem 23==
 
==Problem 23==
 +
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to <math>3</math> times their perimeters?
  
 +
<math>\mathrm {(A)} 6</math>  <math>\mathrm {(B)} 7</math>  <math>\mathrm {(C)} 8</math>  <math>\mathrm {(D)} 10</math>  <math>\mathrm {(E)} 12</math>
  
 
[[2007 AMC 12B Problems/Problem 23 | Solution]]
 
[[2007 AMC 12B Problems/Problem 23 | Solution]]
  
 
==Problem 24==
 
==Problem 24==
 +
How many pairs of positive integers <math>(a,b)</math> are there such that <math>gcd(a,b)=1</math> and <cmath>\frac{a}{b}+\frac{14b}{9a}</cmath> is an integer?
 +
 +
<math>\mathrm {(A)} 4</math>  <math>\mathrm {(B)} 6</math>  <math>\mathrm {(C)} 9</math>  <math>\mathrm {(D)} 12</math>  <math>\mathrm {(E)} \text{infinitely many}</math>
  
 
[[2007 AMC 12B Problems/Problem 24 | Solution]]
 
[[2007 AMC 12B Problems/Problem 24 | Solution]]
  
 
==Problem 25==
 
==Problem 25==
 +
Points <math>A,B,C,D</math> and <math>E</math> are located in 3-dimensional space with <math>AB=BC=CD=DE=EA=2</math> and <math>\angle ABC=\angle CDE=\angle DEA=90^o</math>. The plane of <math>\triangle ABC</math> is parallel to <math>\overline{DE}</math>. What is the area of <math>\triangle BDE</math>?
  
 +
<math>\mathrm {(A)} \sqrt{2}</math>  <math>\mathrm {(B)} \sqrt{3}</math>  <math>\mathrm {(C)} 2</math>  <math>\mathrm {(D)} \sqrt{5}</math>  <math>\mathrm {(E)} \sqrt{6}</math>
  
 
[[2007 AMC 12B Problems/Problem 25 | Solution]]
 
[[2007 AMC 12B Problems/Problem 25 | Solution]]

Revision as of 19:26, 9 November 2007

Problem 1

Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted?

$\mathrm {(A)} 678$ $\mathrm {(B)} 768$ $\mathrm {(C)} 786$ $\mathrm {(D)} 867$ $\mathrm {(E)} 876$

Solution

Problem 2

A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?

$\mathrm {(A)} 22$ $\mathrm {(B)} 24$ $\mathrm {(C)} 25$ $\mathrm {(D)} 26$ $\mathrm {(E)} 28$

Solution

Problem 3

The point $O$ is the center of the circle circumscribed about triangle $ABC$, with $\angle BOC = 120^{\circ}$ and $\angle AOB = 140^{\circ}$, as shown. What is the degree measure of $\angle ABC$?

2007 12B AMC-3.png

$\mathrm {(A)} 35$ $\mathrm {(B)} 40$ $\mathrm {(C)} 45$ $\mathrm {(D)} 50$ $\mathrm {(E)} 60$

Solution

Problem 4

At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?

$\mathrm {(A)} 6$ $\mathrm {(B)} 8$ $\mathrm {(C)} 9$ $\mathrm {(D)} 12$ $\mathrm {(E)} 18$

Solution

Problem 5

The 2007 AMC 12 contests will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points?

$\mathrm {(A)} 13$ $\mathrm {(B)} 14$ $\mathrm {(C)} 15$ $\mathrm {(D)} 16$ $\mathrm {(E)} 17$

Solution

Problem 6

Triangle $ABC$ has side lengths $AB = 5$, $BC = 6$, and $AC = 7$. Two bugs start simultaneously from $A$ and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point $D$. What is $BD$?

$\mathrm {(A)} 1$ $\mathrm {(B)} 2$ $\mathrm {(C)} 3$ $\mathrm {(D)} 4$ $\mathrm {(E)} 5$

Solution

Problem 7

All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = \angle B = 90^{\circ}$. What is the degree measure of $\angle E$?

$\mathrm {(A)} 90$ $\mathrm {(B)} 108$ $\mathrm {(C)} 120$ $\mathrm {(D)} 144$ $\mathrm {(E)} 150$

Solution

Problem 8

Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T/N$ ?

$\mathrm {(A)} 2$ $\mathrm {(B)} 3$ $\mathrm {(C)} 4$ $\mathrm {(D)} 5$ $\mathrm {(E)} 6$

Solution

Problem 9

A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$?

$\mathrm {(A)} 7$ $\mathrm {(B)} 13$ $\mathrm {(C)} 31$ $\mathrm {(D)} 111$ $\mathrm {(E)} 211$

Solution

Problem 10

Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40%$ (Error compiling LaTeX. Unknown error_msg) of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30%$ (Error compiling LaTeX. Unknown error_msg) of the group are girls. How many girls were initially in the group?

$\mathrm {(A)} 4$ $\mathrm {(B)} 6$ $\mathrm {(C)} 8$ $\mathrm {(D)} 10$ $\mathrm {(E)} 12$

Solution

Problem 11

The angles of quadrilateral $ABCD$ satisfy $\angle A=2\angle B=3\angle C=4\angle D$. What is the degree measure of $\angle A$, rounded to the nearest whole number?

$\mathrm {(A)} 125$ $\mathrm {(B)} 144$ $\mathrm {(C)} 153$ $\mathrm {(D)} 173$ $\mathrm {(E)} 180$

Solution

Problem 12

A teacher gave a test to a class in which $10%$ (Error compiling LaTeX. Unknown error_msg) of the students are juniors and $90%$ (Error compiling LaTeX. Unknown error_msg) are seniors. The average score on the test was $84$. The juniors all received the same score, and the average score of the seniors was $83$. What score did each of the juniors receive on the test?

$\mathrm {(A)} 85$ $\mathrm {(B)} 88$ $\mathrm {(C)} 93$ $\mathrm {(D)} 94$ $\mathrm {(E)} 98$

Solution

Problem 13

A traffic light runs repeatedly through the following cycle: green for $30$ seconds, then yellow for $3$ seconds, and then red for $30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?

$\mathrm {(A)} \frac{1}{63}$ $\mathrm {(B)} \frac{1}{21}$ $\mathrm {(C)} \frac{1}{10}$ $\mathrm {(D)} \frac{1}{7}$ $\mathrm {(E)} \frac{1}{3}$

Solution

Problem 14

Point $P$ is inside equilateral $\triangle ABC$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. Given that $PQ=1$, $PR=2$, and $PS=3$, what is $AB$?

$\mathrm {(A)} 4$ $\mathrm {(B)} 3\sqrt{3}$ $\mathrm {(C)} 6$ $\mathrm {(D)} 4\sqrt{3}$ $\mathrm {(E)} 9$

Solution

Problem 15

The geometric series $a+ar+ar^2\cdots$ has a sum of $7$, and the terms involving odd powers of $r$ have a sum of $3$. What is $a+r$?

$\mathrm {(A)} \frac{4}{3}$ $\mathrm {(B)} \frac{12}{7}$ $\mathrm {(C)} \frac{3}{2}$ $\mathrm {(D)} \frac{7}{3}$ $\mathrm {(E)} \frac{5}{2}$

Solution

Problem 16

Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?

$\mathrm {(A)} 15$ $\mathrm {(B)} 18$ $\mathrm {(C)} 27$ $\mathrm {(D)} 54$ $\mathrm {(E)} 81$

Solution

Problem 17

If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\log_{10}b$, what is the median of the set $\{0,1,a,b,1/b\}$?

$\mathrm {(A)} 0$ $\mathrm {(B)} 1$ $\mathrm {(C)} a$ $\mathrm {(D)} b$ $\mathrm {(E)} \frac{1}{b}$

Solution

Problem 18

Let $a$, $b$, and $c$ be digits with $a\ne 0$. THe three-digit integer $abc$ lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer $acb$ lies two thirds of the way between the same two squares. What is $a+b+c$?

$\mathrm {(A)} 10$ $\mathrm {(B)} 13$ $\mathrm {(C)} 16$ $\mathrm {(D)} 18$ $\mathrm {(E)} 21$

Solution

Problem 19

Rhombus $ABCD$, with side length $6$, is rolled to form a cylinder of volume $6$ by taping $\overline{AB}$ to $\overline{DC}$. What is $\sin(\angle ABC)$?

$\mathrm {(A)} \frac{\pi}{9}$ $\mathrm {(B)} \frac{1}{2}$ $\mathrm {(C)} \frac{\pi}{6}$ $\mathrm {(D)} \frac{\pi}{4}$ $\mathrm {(E)} \frac{\sqrt{3}}{2}$

Solution

Problem 20

The parallelogram bounded by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$, and $y=bx+d$ has area $18$. The parallelogram bounded by the lines $y=ax+c$, $y=ax-d$, $y=bx+c$, and $y=bx-d$ has area $72$. Given that $a$, $b$, $c$, and $d$ are positive integers, what is the smallest possible value of $a+b+c+d$?

$\mathrm {(A)} 13$ $\mathrm {(B)} 14$ $\mathrm {(C)} 15$ $\mathrm {(D)} 16$ $\mathrm {(E)} 17$

Solution

Problem 21

The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)

$\mathrm {(A)} 100$ $\mathrm {(B)} 101$ $\mathrm {(C)} 102$ $\mathrm {(D)} 103$ $\mathrm {(E)} 104$

Solution

Problem 22

Two particles move along the edges of equilateral $\triangle ABC$ in the direction \[A\Rightarrow B\Rightarrow C\Rightarrow A,\] starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of $\triangle ABC$?

$\mathrm {(A)} \frac{1}{16}$ $\mathrm {(B)} \frac{1}{12}$ $\mathrm {(C)} \frac{1}{9}$ $\mathrm {(D)} \frac{1}{6}$ $\mathrm {(E)} \frac{1}{4}$

Solution

Problem 23

How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters?

$\mathrm {(A)} 6$ $\mathrm {(B)} 7$ $\mathrm {(C)} 8$ $\mathrm {(D)} 10$ $\mathrm {(E)} 12$

Solution

Problem 24

How many pairs of positive integers $(a,b)$ are there such that $gcd(a,b)=1$ and \[\frac{a}{b}+\frac{14b}{9a}\] is an integer?

$\mathrm {(A)} 4$ $\mathrm {(B)} 6$ $\mathrm {(C)} 9$ $\mathrm {(D)} 12$ $\mathrm {(E)} \text{infinitely many}$

Solution

Problem 25

Points $A,B,C,D$ and $E$ are located in 3-dimensional space with $AB=BC=CD=DE=EA=2$ and $\angle ABC=\angle CDE=\angle DEA=90^o$. The plane of $\triangle ABC$ is parallel to $\overline{DE}$. What is the area of $\triangle BDE$?

$\mathrm {(A)} \sqrt{2}$ $\mathrm {(B)} \sqrt{3}$ $\mathrm {(C)} 2$ $\mathrm {(D)} \sqrt{5}$ $\mathrm {(E)} \sqrt{6}$

Solution

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