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Difference between revisions of "2009 AMC 10A Problems"

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{{AMC10 Problems|year=2009|ab=A}}
 
== Problem 1 ==
 
== Problem 1 ==
 
One can holds <math>12</math> ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?
 
One can holds <math>12</math> ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?
−
<math>a)\, 7\qquad
+
 
−
b)\, 8\qquad
+
<math>
−
c)\, 9\qquad
+
\mathrm{(A)}\ 7
−
d)\, 10\qquad
+
\qquad
−
e)\, 11</math>
+
\mathrm{(B)}\ 8
 +
\qquad
 +
\mathrm{(C)}\ 9
 +
\qquad
 +
\mathrm{(D)}\ 10
 +
\qquad
 +
\mathrm{(E)}\ 11
 +
</math>
 +
 
 
[[2009 AMC 10A Problems/Problem 1|Solution]]
 
[[2009 AMC 10A Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
Four coins are picked out of a  piggy bank that contains a collection of pennies, nickels, dimes and quarters. Which of the following could ''not'' be the total value of the four coins, in cents?
 +
 +
<math>
 +
\mathrm{(A)}\ 15
 +
\qquad
 +
\mathrm{(B)}\ 25
 +
\qquad
 +
\mathrm{(C)}\ 35
 +
\qquad
 +
\mathrm{(D)}\ 45
 +
\qquad
 +
\mathrm{(E)}\ 55
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 2|Solution]]
 
[[2009 AMC 10A Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
Which of the following is equal to <math>1 + \frac{1}{1+\frac{1}{1+1}}</math>?
 +
 +
<math>
 +
\mathrm{(A)}\ \frac{5}{4}
 +
\qquad
 +
\mathrm{(B)}\ \frac{3}{2}
 +
\qquad
 +
\mathrm{(C)}\ \frac{5}{3}
 +
\qquad
 +
\mathrm{(D)}\ 2
 +
\qquad
 +
\mathrm{(E)}\ 3
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 3|Solution]]
 
[[2009 AMC 10A Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 +
Eric plans to compete in a triathlon. He can average <math>2</math> miles per hour in the <math>\frac{1}{4}</math>-mile swim and <math>6</math> miles per hour in the <math>3</math>-mile run. His goal is to finish the triathlon in <math>2</math> hours. To accomplish his goal what must his average speed in miles per hour, be for the <math>15</math>-mile bicycle ride?
 +
 +
<math>
 +
\mathrm{(A)}\ \frac{120}{11}
 +
\qquad
 +
\mathrm{(B)}\ 11
 +
\qquad
 +
\mathrm{(C)}\ \frac{56}{5}
 +
\qquad
 +
\mathrm{(D)}\ \frac{45}{4}
 +
\qquad
 +
\mathrm{(E)}\ 12
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 4|Solution]]
 
[[2009 AMC 10A Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
What is the sum of the digits of the square of <math>111,111,111</math>?
 +
 +
<math>
 +
\mathrm{(A)}\ 18
 +
\qquad
 +
\mathrm{(B)}\ 27
 +
\qquad
 +
\mathrm{(C)}\ 45
 +
\qquad
 +
\mathrm{(D)}\ 63
 +
\qquad
 +
\mathrm{(E)}\ 81
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 5|Solution]]
 
[[2009 AMC 10A Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
 +
A circle of radius <math>2</math> is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?
 +
 +
<asy>
 +
unitsize(6mm);
 +
defaultpen(linewidth(.8pt)+fontsize(8pt));
 +
dotfactor=4;
 +
 +
filldraw(Arc((0,0),4,0,180)--cycle,gray,black);
 +
filldraw(Circle((0,2),2),white,black);
 +
dot((0,2));
 +
draw((0,2)--((0,2)+2*dir(60)));
 +
label("$2$",midpoint((0,2)--((0,2)+2*dir(60))),SE);
 +
</asy>
 +
 +
<math>
 +
\mathrm{(A)}\ \frac{1}{2}
 +
\qquad
 +
\mathrm{(B)}\ \frac{\pi}{6}
 +
\qquad
 +
\mathrm{(C)}\ \frac{2}{\pi}
 +
\qquad
 +
\mathrm{(D)}\ \frac{2}{3}
 +
\qquad
 +
\mathrm{(E)}\ \frac{3}{\pi}
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 6|Solution]]
 
[[2009 AMC 10A Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
 +
A carton contains milk that is <math>2</math>% fat, an amount that is <math>40</math>% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
 +
 +
<math>
 +
\mathrm{(A)}\ \frac{12}{5}
 +
\qquad
 +
\mathrm{(B)}\ 3
 +
\qquad
 +
\mathrm{(C)}\ \frac{10}{3}
 +
\qquad
 +
\mathrm{(D)}\ 38
 +
\qquad
 +
\mathrm{(E)}\ 42
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 7|Solution]]
 
[[2009 AMC 10A Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a <math>50</math>% discount as children. The two members of the oldest generation receive a <math>25\%</math> discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs <math>\textdollar 6.00</math>, is paying for everyone. How many dollars must he pay?
 +
 +
<math>
 +
\mathrm{(A)}\ 34
 +
\qquad
 +
\mathrm{(B)}\ 36
 +
\qquad
 +
\mathrm{(C)}\ 42
 +
\qquad
 +
\mathrm{(D)}\ 46
 +
\qquad
 +
\mathrm{(E)}\ 48
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 8|Solution]]
 
[[2009 AMC 10A Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
 +
 +
Positive integers <math>a</math>, <math>b</math>, and <math>2009</math>, with <math>a<b<2009</math>, form a geometric sequence with an integer ratio. What is <math>a</math>?
 +
 +
<math>
 +
\mathrm{(A)}\ 7
 +
\qquad
 +
\mathrm{(B)}\ 41
 +
\qquad
 +
\mathrm{(C)}\ 49
 +
\qquad
 +
\mathrm{(D)}\ 289
 +
\qquad
 +
\mathrm{(E)}\ 2009
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 9|Solution]]
 
[[2009 AMC 10A Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
 +
Triangle <math>ABC</math> has a right angle at <math>B</math>. Point <math>D</math> is the foot of the altitude from <math>B</math>, <math>AD=3</math>, and <math>DC=4</math>. What is the area of <math>\triangle ABC</math>?
 +
 +
<asy>
 +
unitsize(5mm);
 +
defaultpen(linewidth(.8pt)+fontsize(8pt));
 +
dotfactor=4;
 +
 +
pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21));
 +
pair D=foot(B,A,C);
 +
pair[] ps={B,C,A,D};
 +
 +
draw(A--B--C--cycle);
 +
draw(B--D);
 +
draw(rightanglemark(B,D,C));
 +
 +
dot(ps);
 +
label("$A$",A,NW);
 +
label("$B$",B,SW);
 +
label("$C$",C,SE);
 +
label("$D$",D,NE);
 +
label("$3$",midpoint(A--D),NE);
 +
label("$4$",midpoint(D--C),NE);
 +
</asy>
 +
 +
<math>
 +
\mathrm{(A)}\ 4\sqrt3
 +
\qquad
 +
\mathrm{(B)}\ 7\sqrt3
 +
\qquad
 +
\mathrm{(C)}\ 21
 +
\qquad
 +
\mathrm{(D)}\ 14\sqrt3
 +
\qquad
 +
\mathrm{(E)}\ 421
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 10|Solution]]
 
[[2009 AMC 10A Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
 +
One dimension of a cube is increased by <math>1</math>, another is decreased by <math>1</math>, and the third is left unchanged. The volume of the new rectangular solid is <math>5</math> less than that of the cube. What was the volume of the cube?
 +
 +
<math>
 +
\mathrm{(A)}\ 8
 +
\qquad
 +
\mathrm{(B)}\ 27
 +
\qquad
 +
\mathrm{(C)}\ 64
 +
\qquad
 +
\mathrm{(D)}\ 125
 +
\qquad
 +
\mathrm{(E)}\ 216
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 11|Solution]]
 
[[2009 AMC 10A Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
In quadrilateral <math>ABCD</math>, <math>AB = 5</math>, <math>BC = 17</math>, <math>CD = 5</math>, <math>DA = 9</math>, and <math>BD</math> is an integer. What is <math>BD</math>?
 +
<center><asy>
 +
unitsize(4mm);
 +
defaultpen(linewidth(.8pt)+fontsize(8pt));
 +
dotfactor=4;
 +
 +
pair C=(0,0), B=(17,0);
 +
pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0];
 +
pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0];
 +
pair[] dotted={A,B,C,D};
 +
 +
draw(D--A--B--C--D--B);
 +
dot(dotted);
 +
label("$D$",D,NW);
 +
label("$C$",C,W);
 +
label("$B$",B,E);
 +
label("$A$",A,NE);
 +
</asy></center>
 +
 +
<math>
 +
\mathrm{(A)}\ 11
 +
\qquad
 +
\mathrm{(B)}\ 12
 +
\qquad
 +
\mathrm{(C)}\ 13
 +
\qquad
 +
\mathrm{(D)}\ 14
 +
\qquad
 +
\mathrm{(E)}\ 15
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 12|Solution]]
 
[[2009 AMC 10A Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
Suppose that <math>P = 2^m</math> and <math>Q = 3^n</math>. Which of the following is equal to <math>12^{mn}</math> for every pair of integers <math>(m,n)</math>?
 +
 +
<math>
 +
\mathrm{(A)}\ P^2Q
 +
\qquad
 +
\mathrm{(B)}\ P^nQ^m
 +
\qquad
 +
\mathrm{(C)}\ P^nQ^{2m}
 +
\qquad
 +
\mathrm{(D)}\ P^{2m}Q^n
 +
\qquad
 +
\mathrm{(E)}\ P^{2n}Q^m
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 13|Solution]]
 
[[2009 AMC 10A Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
 +
Four congruent rectangles are placed as shown. The area of the outer square is <math>4</math> times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
 +
<center><asy>
 +
unitsize(6mm);
 +
defaultpen(linewidth(.8pt));
 +
 +
path p=(1,1)--(-2,1)--(-2,2)--(1,2);
 +
draw(p);
 +
draw(rotate(90)*p);
 +
draw(rotate(180)*p);
 +
draw(rotate(270)*p);
 +
</asy></center>
 +
 +
<math>
 +
\mathrm{(A)}\ 3
 +
\qquad
 +
\mathrm{(B)}\ \sqrt {10}
 +
\qquad
 +
\mathrm{(C)}\ 2 + \sqrt2
 +
\qquad
 +
\mathrm{(D)}\ 2\sqrt3
 +
\qquad
 +
\mathrm{(E)}\ 4
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 14|Solution]]
 
[[2009 AMC 10A Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
 +
The figures <math>F_1</math>, <math>F_2</math>, <math>F_3</math>, and <math>F_4</math> shown are the first in a sequence of figures. For <math>n\ge3</math>, <math>F_n</math> is constructed from <math>F_{n - 1}</math> by surrounding it with a square and placing one more diamond on each side of the new square than <math>F_{n - 1}</math> had on each side of its outside square. For example, figure <math>F_3</math> has <math>13</math> diamonds. How many diamonds are there in figure <math>F_{20}</math>?
 +
<center><asy>
 +
unitsize(3mm);
 +
defaultpen(linewidth(.8pt)+fontsize(8pt));
 +
 +
path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle;
 +
marker m=marker(scale(5)*d,Fill);
 +
path f1=(0,0);
 +
path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1);
 +
path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1);
 +
path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2);
 +
path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2);
 +
path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)--
 +
(3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3);
 +
path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^
 +
(-2,2)--(-3,3);
 +
 +
draw(f1,m);
 +
draw(shift(5,0)*f2,m);
 +
draw(shift(5,0)*g2);
 +
draw(shift(12,0)*f3,m);
 +
draw(shift(12,0)*g3);
 +
draw(shift(21,0)*f4,m);
 +
draw(shift(21,0)*g4);
 +
label("$F_1$",(0,-4));
 +
label("$F_2$",(5,-4));
 +
label("$F_3$",(12,-4));
 +
label("$F_4$",(21,-4));
 +
</asy></center>
 +
 +
<math>
 +
\mathrm{(A)}\ 401
 +
\qquad
 +
\mathrm{(B)}\ 485
 +
\qquad
 +
\mathrm{(C)}\ 585
 +
\qquad
 +
\mathrm{(D)}\ 626
 +
\qquad
 +
\mathrm{(E)}\ 761
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 15|Solution]]
 
[[2009 AMC 10A Problems/Problem 15|Solution]]
  
 
== Problem 16 ==
 
== Problem 16 ==
 +
 +
Let <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> be real numbers with <math>|a-b|=2</math>, <math>|b-c|=3</math>, and <math>|c-d|=4</math>. What is the sum of all possible values of <math>|a-d|</math>?
 +
 +
<math>
 +
\mathrm{(A)}\ 9
 +
\qquad
 +
\mathrm{(B)}\ 12
 +
\qquad
 +
\mathrm{(C)}\ 15
 +
\qquad
 +
\mathrm{(D)}\ 18
 +
\qquad
 +
\mathrm{(E)}\ 24
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 16|Solution]]
 
[[2009 AMC 10A Problems/Problem 16|Solution]]
  
 
== Problem 17 ==
 
== Problem 17 ==
 +
Rectangle <math>ABCD</math> has <math>AB=4</math> and <math>BC=3</math>. Segment <math>EF</math> is constructed through <math>B</math> so that <math>EF</math> is perpendicular to <math>DB</math>, and <math>A</math> and <math>C</math> lie on <math>DE</math> and <math>DF</math>, respectively. What is <math>EF</math>?
 +
 +
<math>
 +
\mathrm{(A)}\ 9
 +
\qquad
 +
\mathrm{(B)}\ 10
 +
\qquad
 +
\mathrm{(C)}\ \frac {125}{12}
 +
\qquad
 +
\mathrm{(D)}\ \frac {103}{9}
 +
\qquad
 +
\mathrm{(E)}\ 12
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 17|Solution]]
 
[[2009 AMC 10A Problems/Problem 17|Solution]]
  
 
== Problem 18 ==
 
== Problem 18 ==
 +
 +
At Jefferson Summer Camp, <math>60\%</math> of the children play soccer, <math>30\%</math> of the children swim, and <math>40\%</math> of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
 +
 +
<math>
 +
\mathrm{(A)}\ 30\%
 +
\qquad
 +
\mathrm{(B)}\ 40\%
 +
\qquad
 +
\mathrm{(C)}\ 49\%
 +
\qquad
 +
\mathrm{(D)}\ 51\%
 +
\qquad
 +
\mathrm{(E)}\ 70\%
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 18|Solution]]
 
[[2009 AMC 10A Problems/Problem 18|Solution]]
  
 
== Problem 19 ==
 
== Problem 19 ==
 +
 +
Circle <math>A</math> has radius <math>100</math>. Circle <math>B</math> has an integer radius <math>r<100</math> and remains internally tangent to circle <math>A</math> as it rolls once around the circumference of circle <math>A</math>. The two circles have the same points of tangency at the beginning and end of circle <math>B</math>'s trip. How many possible values can <math>r</math> have?
 +
 +
<math>
 +
\mathrm{(A)}\ 4
 +
\qquad
 +
\mathrm{(B)}\ 8
 +
\qquad
 +
\mathrm{(C)}\ 9
 +
\qquad
 +
\mathrm{(D)}\ 50
 +
\qquad
 +
\mathrm{(E)}\ 90
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 19|Solution]]
 
[[2009 AMC 10A Problems/Problem 19|Solution]]
  
 
== Problem 20 ==
 
== Problem 20 ==
 +
 +
Andrea and Lauren are <math>20</math> kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of <math>1</math> kilometer per minute. After <math>5</math> minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?
 +
 +
<math>
 +
\mathrm{(A)}\ 20
 +
\qquad
 +
\mathrm{(B)}\ 30
 +
\qquad
 +
\mathrm{(C)}\ 55
 +
\qquad
 +
\mathrm{(D)}\ 65
 +
\qquad
 +
\mathrm{(E)}\ 80
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 20|Solution]]
 
[[2009 AMC 10A Problems/Problem 20|Solution]]
  
 
== Problem 21 ==
 
== Problem 21 ==
 +
 +
Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle. In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?
 +
 +
<asy>
 +
unitsize(6mm);
 +
defaultpen(linewidth(.8pt));
 +
 +
draw(Circle((0,0),1+sqrt(2)));
 +
draw(Circle((sqrt(2),0),1));
 +
draw(Circle((0,sqrt(2)),1));
 +
draw(Circle((-sqrt(2),0),1));
 +
draw(Circle((0,-sqrt(2)),1));
 +
</asy>
 +
 +
<math>
 +
\mathrm{(A)}\ 3-2\sqrt2
 +
\qquad
 +
\mathrm{(B)}\ 2-\sqrt2
 +
\qquad
 +
\mathrm{(C)}\ 4(3-2\sqrt2)
 +
\qquad
 +
\mathrm{(D)}\ \frac12(3-\sqrt2)
 +
\qquad
 +
\mathrm{(E)}\ 2\sqrt2-2
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 21|Solution]]
 
[[2009 AMC 10A Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== Problem 22 ==
 +
 +
Two cubical dice each have removable numbers <math>1</math> through <math>6</math>. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is <math>7</math>?
 +
 +
<math>
 +
\mathrm{(A)}\ \frac{1}{9}
 +
\qquad
 +
\mathrm{(B)}\ \frac{1}{8}
 +
\qquad
 +
\mathrm{(C)}\ \frac{1}{6}
 +
\qquad
 +
\mathrm{(D)}\ \frac{2}{11}
 +
\qquad
 +
\mathrm{(E)}\ \frac{1}{5}
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 22|Solution]]
 
[[2009 AMC 10A Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== Problem 23 ==
 +
 +
Convex quadrilateral <math>ABCD</math> has <math>AB=9</math> and <math>CD=12</math>. Diagonals <math>AC</math> and <math>BD</math> intersect at <math>E</math>, <math>AC=14</math>, and <math>\triangle AED</math> and <math>\triangle BEC</math> have equal areas. What is <math>AE</math>?
 +
 +
<math>
 +
\mathrm{(A)}\ \frac{9}{2}
 +
\qquad
 +
\mathrm{(B)}\ \frac{50}{11}
 +
\qquad
 +
\mathrm{(C)}\ \frac{21}{4}
 +
\qquad
 +
\mathrm{(D)}\ \frac{17}{3}
 +
\qquad
 +
\mathrm{(E)}\ 6
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 23|Solution]]
 
[[2009 AMC 10A Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== Problem 24 ==
 +
 +
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?
 +
 +
<math>
 +
\mathrm{(A)}\ \frac{1}{4}
 +
\qquad
 +
\mathrm{(B)}\ \frac{3}{8}
 +
\qquad
 +
\mathrm{(C)}\ \frac{4}{7}
 +
\qquad
 +
\mathrm{(D)}\ \frac{5}{7}
 +
\qquad
 +
\mathrm{(E)}\ \frac{3}{4}
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 24|Solution]]
 
[[2009 AMC 10A Problems/Problem 24|Solution]]
  
 
== Problem 25 ==
 
== Problem 25 ==
 +
 +
For <math>k > 0</math>, let <math>I_k = 10\ldots 064</math>, where there are <math>k</math> zeros between the <math>1</math> and the <math>6</math>.  Let <math>N(k)</math> be the number of factors of <math>2</math> in the prime factorization of <math>I_k</math>.  What is the maximum value of <math>N(k)</math>?
 +
 +
<math>
 +
\mathrm{(A)}\ 6
 +
\qquad
 +
\mathrm{(B)}\ 7
 +
\qquad
 +
\mathrm{(C)}\ 8
 +
\qquad
 +
\mathrm{(D)}\ 9
 +
\qquad
 +
\mathrm{(E)}\ 10
 +
</math>
  
 
[[2009 AMC 10A Problems/Problem 25|Solution]]
 
[[2009 AMC 10A Problems/Problem 25|Solution]]
 +
 +
== See also ==
 +
{{AMC10 box|year=2009|before=[[2008 AMC 10A|2008 AMC 10A]], [[2008 AMC 10B|B]] |after=[[2009 AMC 10B Problems]]|ab=A}}
 +
 +
* [[AMC 10]]
 +
* [[AMC 10 Problems and Solutions]]
 +
* [[Mathematics competition resources]]

Latest revision as of 16:46, 13 June 2024

2009 AMC 10A (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

One can holds $12$ ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?

$\mathrm{(A)}\ 7 \qquad \mathrm{(B)}\ 8 \qquad \mathrm{(C)}\ 9 \qquad \mathrm{(D)}\ 10 \qquad \mathrm{(E)}\ 11$

Solution

Problem 2

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes and quarters. Which of the following could not be the total value of the four coins, in cents?

$\mathrm{(A)}\ 15 \qquad \mathrm{(B)}\ 25 \qquad \mathrm{(C)}\ 35 \qquad \mathrm{(D)}\ 45 \qquad \mathrm{(E)}\ 55$

Solution

Problem 3

Which of the following is equal to $1 + \frac{1}{1+\frac{1}{1+1}}$?

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

Solution

Problem 4

Eric plans to compete in a triathlon. He can average $2$ miles per hour in the $\frac{1}{4}$-mile swim and $6$ miles per hour in the $3$-mile run. His goal is to finish the triathlon in $2$ hours. To accomplish his goal what must his average speed in miles per hour, be for the $15$-mile bicycle ride?

$\mathrm{(A)}\ \frac{120}{11} \qquad \mathrm{(B)}\ 11 \qquad \mathrm{(C)}\ \frac{56}{5} \qquad \mathrm{(D)}\ \frac{45}{4} \qquad \mathrm{(E)}\ 12$

Solution

Problem 5

What is the sum of the digits of the square of $111,111,111$?

$\mathrm{(A)}\ 18 \qquad \mathrm{(B)}\ 27 \qquad \mathrm{(C)}\ 45 \qquad \mathrm{(D)}\ 63 \qquad \mathrm{(E)}\ 81$

Solution

Problem 6

A circle of radius $2$ is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?

[asy] unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; filldraw(Arc((0,0),4,0,180)--cycle,gray,black); filldraw(Circle((0,2),2),white,black); dot((0,2)); draw((0,2)--((0,2)+2*dir(60))); label("$2$",midpoint((0,2)--((0,2)+2*dir(60))),SE); [/asy]

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

Solution

Problem 7

A carton contains milk that is $2$% fat, an amount that is $40$% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?

$\mathrm{(A)}\ \frac{12}{5} \qquad \mathrm{(B)}\ 3 \qquad \mathrm{(C)}\ \frac{10}{3} \qquad \mathrm{(D)}\ 38 \qquad \mathrm{(E)}\ 42$

Solution

Problem 8

Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50$% discount as children. The two members of the oldest generation receive a $25\%$ discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $\textdollar 6.00$, is paying for everyone. How many dollars must he pay?

$\mathrm{(A)}\ 34 \qquad \mathrm{(B)}\ 36 \qquad \mathrm{(C)}\ 42 \qquad \mathrm{(D)}\ 46 \qquad \mathrm{(E)}\ 48$

Solution

Problem 9

Positive integers $a$, $b$, and $2009$, with $a<b<2009$, form a geometric sequence with an integer ratio. What is $a$?

$\mathrm{(A)}\ 7 \qquad \mathrm{(B)}\ 41 \qquad \mathrm{(C)}\ 49 \qquad \mathrm{(D)}\ 289 \qquad \mathrm{(E)}\ 2009$

Solution

Problem 10

Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=3$, and $DC=4$. What is the area of $\triangle ABC$?

[asy] unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21)); pair D=foot(B,A,C); pair[] ps={B,C,A,D}; draw(A--B--C--cycle); draw(B--D); draw(rightanglemark(B,D,C)); dot(ps); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NE); label("$3$",midpoint(A--D),NE); label("$4$",midpoint(D--C),NE); [/asy]

$\mathrm{(A)}\ 4\sqrt3 \qquad \mathrm{(B)}\ 7\sqrt3 \qquad \mathrm{(C)}\ 21 \qquad \mathrm{(D)}\ 14\sqrt3 \qquad \mathrm{(E)}\ 421$

Solution

Problem 11

One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube?

$\mathrm{(A)}\ 8 \qquad \mathrm{(B)}\ 27 \qquad \mathrm{(C)}\ 64 \qquad \mathrm{(D)}\ 125 \qquad \mathrm{(E)}\ 216$

Solution

Problem 12

In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$?

[asy] unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair C=(0,0), B=(17,0); pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0]; pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0]; pair[] dotted={A,B,C,D}; draw(D--A--B--C--D--B); dot(dotted); label("$D$",D,NW); label("$C$",C,W); label("$B$",B,E); label("$A$",A,NE); [/asy]

$\mathrm{(A)}\ 11 \qquad \mathrm{(B)}\ 12 \qquad \mathrm{(C)}\ 13 \qquad \mathrm{(D)}\ 14 \qquad \mathrm{(E)}\ 15$

Solution

Problem 13

Suppose that $P = 2^m$ and $Q = 3^n$. Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$?

$\mathrm{(A)}\ P^2Q \qquad \mathrm{(B)}\ P^nQ^m \qquad \mathrm{(C)}\ P^nQ^{2m} \qquad \mathrm{(D)}\ P^{2m}Q^n \qquad \mathrm{(E)}\ P^{2n}Q^m$

Solution

Problem 14

Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); path p=(1,1)--(-2,1)--(-2,2)--(1,2); draw(p); draw(rotate(90)*p); draw(rotate(180)*p); draw(rotate(270)*p); [/asy]

$\mathrm{(A)}\ 3 \qquad \mathrm{(B)}\ \sqrt {10} \qquad \mathrm{(C)}\ 2 + \sqrt2 \qquad \mathrm{(D)}\ 2\sqrt3 \qquad \mathrm{(E)}\ 4$

Solution

Problem 15

The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$?

[asy] unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle; marker m=marker(scale(5)*d,Fill); path f1=(0,0); path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1); path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1); path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2); path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2); path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)-- (3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3); path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^ (-2,2)--(-3,3); draw(f1,m); draw(shift(5,0)*f2,m); draw(shift(5,0)*g2); draw(shift(12,0)*f3,m); draw(shift(12,0)*g3); draw(shift(21,0)*f4,m); draw(shift(21,0)*g4); label("$F_1$",(0,-4)); label("$F_2$",(5,-4)); label("$F_3$",(12,-4)); label("$F_4$",(21,-4)); [/asy]

$\mathrm{(A)}\ 401 \qquad \mathrm{(B)}\ 485 \qquad \mathrm{(C)}\ 585 \qquad \mathrm{(D)}\ 626 \qquad \mathrm{(E)}\ 761$

Solution

Problem 16

Let $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. What is the sum of all possible values of $|a-d|$?

$\mathrm{(A)}\ 9 \qquad \mathrm{(B)}\ 12 \qquad \mathrm{(C)}\ 15 \qquad \mathrm{(D)}\ 18 \qquad \mathrm{(E)}\ 24$

Solution

Problem 17

Rectangle $ABCD$ has $AB=4$ and $BC=3$. Segment $EF$ is constructed through $B$ so that $EF$ is perpendicular to $DB$, and $A$ and $C$ lie on $DE$ and $DF$, respectively. What is $EF$?

$\mathrm{(A)}\ 9 \qquad \mathrm{(B)}\ 10 \qquad \mathrm{(C)}\ \frac {125}{12} \qquad \mathrm{(D)}\ \frac {103}{9} \qquad \mathrm{(E)}\ 12$

Solution

Problem 18

At Jefferson Summer Camp, $60\%$ of the children play soccer, $30\%$ of the children swim, and $40\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?

$\mathrm{(A)}\ 30\% \qquad \mathrm{(B)}\ 40\% \qquad \mathrm{(C)}\ 49\% \qquad \mathrm{(D)}\ 51\% \qquad \mathrm{(E)}\ 70\%$

Solution

Problem 19

Circle $A$ has radius $100$. Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many possible values can $r$ have?

$\mathrm{(A)}\ 4 \qquad \mathrm{(B)}\ 8 \qquad \mathrm{(C)}\ 9 \qquad \mathrm{(D)}\ 50 \qquad \mathrm{(E)}\ 90$

Solution

Problem 20

Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?

$\mathrm{(A)}\ 20 \qquad \mathrm{(B)}\ 30 \qquad \mathrm{(C)}\ 55 \qquad \mathrm{(D)}\ 65 \qquad \mathrm{(E)}\ 80$

Solution

Problem 21

Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle. In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?

[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); draw(Circle((0,0),1+sqrt(2))); draw(Circle((sqrt(2),0),1)); draw(Circle((0,sqrt(2)),1)); draw(Circle((-sqrt(2),0),1)); draw(Circle((0,-sqrt(2)),1)); [/asy]

$\mathrm{(A)}\ 3-2\sqrt2 \qquad \mathrm{(B)}\ 2-\sqrt2 \qquad \mathrm{(C)}\ 4(3-2\sqrt2) \qquad \mathrm{(D)}\ \frac12(3-\sqrt2) \qquad \mathrm{(E)}\ 2\sqrt2-2$

Solution

Problem 22

Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $7$?

$\mathrm{(A)}\ \frac{1}{9} \qquad \mathrm{(B)}\ \frac{1}{8} \qquad \mathrm{(C)}\ \frac{1}{6} \qquad \mathrm{(D)}\ \frac{2}{11} \qquad \mathrm{(E)}\ \frac{1}{5}$

Solution

Problem 23

Convex quadrilateral $ABCD$ has $AB=9$ and $CD=12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC=14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$?

$\mathrm{(A)}\ \frac{9}{2} \qquad \mathrm{(B)}\ \frac{50}{11} \qquad \mathrm{(C)}\ \frac{21}{4} \qquad \mathrm{(D)}\ \frac{17}{3} \qquad \mathrm{(E)}\ 6$

Solution

Problem 24

Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?

$\mathrm{(A)}\ \frac{1}{4} \qquad \mathrm{(B)}\ \frac{3}{8} \qquad \mathrm{(C)}\ \frac{4}{7} \qquad \mathrm{(D)}\ \frac{5}{7} \qquad \mathrm{(E)}\ \frac{3}{4}$

Solution

Problem 25

For $k > 0$, let $I_k = 10\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?

$\mathrm{(A)}\ 6 \qquad \mathrm{(B)}\ 7 \qquad \mathrm{(C)}\ 8 \qquad \mathrm{(D)}\ 9 \qquad \mathrm{(E)}\ 10$

Solution

See also

2009 AMC 10A (Problems • Answer Key • Resources)
Preceded by
2008 AMC 10A, B 		Followed by
2009 AMC 10B Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10A_Problems&oldid=220472"