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Difference between revisions of "2009 AMC 10A Problems"

(→‎Problem 13)
Line 172: Line 172:
 
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?
 
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>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 216</math>
+
<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]]
Line 194: Line 204:
 
label("$B$",B,E);
 
label("$B$",B,E);
 
label("$A$",A,NE);
 
label("$A$",A,NE);
−
</asy></center><math>\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15</math>
+
</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]]
Line 232: Line 254:
  
 
== 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>
 
<math>
−
\mathrm{(A)}\  
+
\mathrm{(A)}\ 401
 
\qquad
 
\qquad
−
\mathrm{(B)}\  
+
\mathrm{(B)}\ 485
 
\qquad
 
\qquad
−
\mathrm{(C)}\  
+
\mathrm{(C)}\ 585
 
\qquad
 
\qquad
−
\mathrm{(D)}\  
+
\mathrm{(D)}\ 626
 
\qquad
 
\qquad
−
\mathrm{(E)}\  
+
\mathrm{(E)}\ 761
 
</math>
 
</math>
  
Line 392: Line 444:
  
 
== 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>
 
<math>
−
\mathrm{(A)}\  
+
\mathrm{(A)}\ 6
 
\qquad
 
\qquad
−
\mathrm{(B)}\  
+
\mathrm{(B)}\ 7
 
\qquad
 
\qquad
−
\mathrm{(C)}\  
+
\mathrm{(C)}\ 8
 
\qquad
 
\qquad
−
\mathrm{(D)}\  
+
\mathrm{(D)}\ 9
 
\qquad
 
\qquad
−
\mathrm{(E)}\  
+
\mathrm{(E)}\ 10
 
</math>
 
</math>
  
 
[[2009 AMC 10A Problems/Problem 25|Solution]]
 
[[2009 AMC 10A Problems/Problem 25|Solution]]

Revision as of 04:16, 13 February 2009

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 triathalon. 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

Solution

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

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?

$\mathrm{(A)}\ \frac{12}{5} \qquad \mathrm{(B)}\ \frac{10}{3} \qquad \mathrm{(C)}\ 9 \qquad \mathrm{(D)}\ 38 \qquad \mathrm{(E)}\ 42$

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 <dollar/>$6.00$, 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

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

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

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

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

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 17

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 18

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 19

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 20

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 21

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 22

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 23

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

Solution

Problem 24

$\mathrm{(A)}\ \qquad \mathrm{(B)}\ \qquad \mathrm{(C)}\ \qquad \mathrm{(D)}\ \qquad \mathrm{(E)}$

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

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