Difference between revisions of "2012 AMC 12A Problems"

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== Problem 15 ==
 
== Problem 15 ==
 +
 +
A <math>3\times3</math> square is partitioned into <math>9</math> unit squares.  Each unit square is painted either white or black with each color being equally likely, chosen independently and at random.  The square is the rotated <math>90^\circ</math> clockwise about its center, and every white square in a position formerly occupied by a black square is painted black.  The colors of all other squares are left unchanged.  What is the probability that the grid is now entirely black?
 +
 +
<math> \textbf{(A)}\ \dfrac{49}{512}
 +
\qquad\textbf{(B)}\ \dfrac{7}{64}
 +
\qquad\textbf{(C)}\ \dfrac{121}{1024}
 +
\qquad\textbf{(D)}\ \dfrac{81}{512}
 +
\qquad\textbf{(E)}\ \dfrac{9}{32}
 +
</math>
  
 
== Problem 16 ==
 
== Problem 16 ==
 +
 +
Circle <math>C_1</math> has its center <math>O</math> lying on circle <math>C_2</math>.  The two circles meet at <math>X</math> and <math>Y</math>.  Point <math>Z</math> in the exterior of <math>C_1</math> lies on circle <math>C_2</math> and <math>XZ=13</math>, <math>OZ=11</math>, and <math>YZ=7</math>.  What is the radius of circle <math>C_1</math>?
 +
 +
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqrt{7}\qquad\textbf{(E)}\ \sqrt{30} </math>
  
 
== Problem 17 ==
 
== Problem 17 ==
 +
 +
Let <math>S</math> be a subset of <math>\{1,2,3,\dots,30\}</math> with the property that no pair of distinct elements in <math>S</math> has a sum divisible by <math>5</math>.  What is the largest possible size of <math>S</math>?
 +
 +
<math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18 </math>
  
 
== Problem 18 ==
 
== Problem 18 ==
 +
 +
Triangle <math>ABC</math> has <math>AB=27</math>, <math>AC=26</math>, and <math>BC=25</math>.  Let <math>I</math> denote the intersection of the internal angle bisectors of <math>\triangle ABC</math>.  What is <math>BI</math>?
 +
 +
<math> \textbf{(A)}\ 15\qquad\textbf{(B)}\ 5+\sqrt{26}+3\sqrt{3}\qquad\textbf{(C)}\ 3\sqrt{26}\qquad\textbf{(D)}\ \frac{2}{3}\sqrt{546}\qquad\textbf{(E)}\ 9\sqrt{3}
  
 
== Problem 19 ==
 
== Problem 19 ==
 +
 +
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts.  Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group.  Each of them has the same number of internet friends.  In how many different ways can this happen?
 +
 +
</math> \textbf{(A)}\ 60
 +
\qquad\textbf{(B)}\ 170
 +
\qquad\textbf{(C)}\ 290
 +
\qquad\textbf{(D)}\ 320
 +
\qquad\textbf{(E)}\ 660
 +
<math>
  
 
== Problem 20 ==
 
== Problem 20 ==
 +
 +
Consider the polynomial
 +
 +
</math>P(x)=\prod_{k=0}^{10}=(x^2^+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)<math>
 +
 +
The coefficient of </math>x^{2012}<math> is equal to </math>2^a<math>.  What is </math>a<math>?
 +
 +
</math> \textbf{(A)}\ 5
 +
\qquad\textbf{(B)}\ 6
 +
\qquad\textbf{(C)}\ 7
 +
\qquad\textbf{(D)}\ 10
 +
\qquad\textbf{(E)}\ 24
 +
<math>
 +
  
 
== Problem 21 ==
 
== Problem 21 ==
  
Let <math>a</math>, <math>b</math>, and <math>c</math> be positive integers with <math>a\ge</math> <math>b\ge</math> <math>c</math> such that
+
Let </math>a<math>, </math>b<math>, and </math>c<math> be positive integers with </math>a\ge<math> </math>b\ge<math> </math>c<math> such that
 
<cmath> a^2-b^2-c^2+ab=2011</cmath> and
 
<cmath> a^2-b^2-c^2+ab=2011</cmath> and
 
<cmath>a^2+3b^2+3c^2-3ab-2ac-2bc=-1997.</cmath>
 
<cmath>a^2+3b^2+3c^2-3ab-2ac-2bc=-1997.</cmath>
  
What is <math>a</math>?
+
What is </math>a<math>?
  
<math> \textbf{(A)}\ 249\qquad\textbf{(B)}\ 250\qquad\textbf{(C)}\ 251\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 253 </math>
+
</math> \textbf{(A)}\ 249\qquad\textbf{(B)}\ 250\qquad\textbf{(C)}\ 251\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 253 <math>
  
 
[[2012 AMC 10A Problems/Problem 24|Solution]]
 
[[2012 AMC 10A Problems/Problem 24|Solution]]
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== Problem 23 ==
 
== Problem 23 ==
  
Let <math>S</math> be the square one of whose diagonals has endpoints <math>(0.1,0.7)</math> and <math>(-0.1,-0.7)</math>.  A point <math>v=(x,y)</math> is chosen uniformly at random over all pairs of real numbers <math>x</math> and <math>y</math> such that <math>0 \le x \le 2012</math> and <math>0\le y\le 2012</math>.  Let <math>T(v)</math> be a translated copy of <math>S</math> centered at <math>v</math>.  What is the probability that the square region determined by <math>T(v)</math> contains exactly two points with integer coefficients in its interior?
+
Let </math>S<math> be the square one of whose diagonals has endpoints </math>(0.1,0.7)<math> and </math>(-0.1,-0.7)<math>.  A point </math>v=(x,y)<math> is chosen uniformly at random over all pairs of real numbers </math>x<math> and </math>y<math> such that </math>0 \le x \le 2012<math> and </math>0\le y\le 2012<math>.  Let </math>T(v)<math> be a translated copy of </math>S<math> centered at </math>v<math>.  What is the probability that the square region determined by </math>T(v)<math> contains exactly two points with integer coefficients in its interior?
  
<math> \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25 \qquad\textbf{(E)}\ 0.32 </math>
+
</math> \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25 \qquad\textbf{(E)}\ 0.32 <math>
  
 
[[2012 AMC 10A Problems/Problem 24|Solution]]
 
[[2012 AMC 10A Problems/Problem 24|Solution]]
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== Problem 24 ==
 
== Problem 24 ==
  
Let <math>\{a_k\}_{k=1}^{2011}</math> be the sequence of real numbers defined by <math>a_1=0.201,</math> <math>a_2=(0.2011)^{a_1},</math> <math>a_3=(0.20101)^{a_2},</math> <math>a_4=(0.201011)^{a_3}</math>, and in general, <cmath>a_k=\left\{\array{c}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if k is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if k is even.}</cmath>
+
Let </math>\{a_k\}_{k=1}^{2011}<math> be the sequence of real numbers defined by </math>a_1=0.201,<math> </math>a_2=(0.2011)^{a_1},<math> </math>a_3=(0.20101)^{a_2},<math> </math>a_4=(0.201011)^{a_3}<math>, and in general, <cmath>a_k=\left\{\array{c}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if k is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if k is even.}</cmath>
  
Rearranging the numbers in the sequence  <math>\{a_k\}_{k=1}^{2011}</math> in decreasing order produces a new sequence  <math>\{b_k\}_{k=1}^{2011}</math>.  What is the sum of all integers <math>k</math>, <math>1\le k \le 2011</math>, such that <math>a_k=b_k?</math>
+
Rearranging the numbers in the sequence  </math>\{a_k\}_{k=1}^{2011}<math> in decreasing order produces a new sequence  </math>\{b_k\}_{k=1}^{2011}<math>.  What is the sum of all integers </math>k<math>, </math>1\le k \le 2011<math>, such that </math>a_k=b_k?<math>
  
<math> \textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\2012 </math>
+
</math> \textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\2012 <math>
  
  
 
== Problem 25 ==
 
== Problem 25 ==
  
Let <math>f(x)=|2\{x\}-1|</math> where <math>\{x\}</math> denotes the fractional part of <math>x</math>.  The number <math>n</math> is the smallest positive integer such that the equation <cmath>nf(xf(x))</cmath> has at least <math>2012</math> real solutions.  What is <math>n</math>?  '''Note:''' the fractional part of <math>x</math> is a real number <math>y=\{x\}</math> such that <math>0\le y<1</math> and <math>x-y</math> is an integer.
+
Let </math>f(x)=|2\{x\}-1|<math> where </math>\{x\}<math> denotes the fractional part of </math>x<math>.  The number </math>n<math> is the smallest positive integer such that the equation <cmath>nf(xf(x))</cmath> has at least </math>2012<math> real solutions.  What is </math>n<math>?  '''Note:''' the fractional part of </math>x<math> is a real number </math>y=\{x\}<math> such that </math>0\le y<1<math> and </math>x-y<math> is an integer.
  
<math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 31\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 62\qquad\textbf{(E)}\64 </math>
+
</math> \textbf{(A)}\ 30\qquad\textbf{(B)}\ 31\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 62\qquad\textbf{(E)}\64 $

Revision as of 15:58, 11 February 2012

Problem 1

A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether?

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15$

Solution

Problem 2

Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?

$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 30$

Solution

Problem 3

A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$?

$\textbf{(A)}\ 120\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 200\qquad\textbf{(D)}\ 240\qquad\textbf{(E)}\ 280$

Problem 4

In a bag of marbles, $\tfrac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?

$\textbf{(A)}\ \dfrac{2}{5} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{4}{7} \qquad\textbf{(D)}\ \dfrac{3}{5} \qquad\textbf{(E)}\ \dfrac{4}{5}$

Problem 5

A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $280$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?

$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 96$

Problem 6

The sums of three whole numbers taken in pairs are $12$, $17$, and $19$. What is the middle number?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8$

Problem 7

Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12$

Problem 8

An iterative average of the numbers $1$, $2$, $3$, $4$, and $5$ is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?

$\textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{8}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16}$

Problem 9

A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born?

$\textbf{(A)}\ \text{Friday} \qquad\textbf{(B)}\ \text{Saturday} \qquad\textbf{(C)}\ \text{Sunday} \qquad\textbf{(D)}\ \text{Monday} \qquad\textbf{(E)}\ \text{Tuesday}$

Problem 10

$\textbf{(A)}\ \frac{3}{10}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{9}{20}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{9}{10}$

Problem 11

Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?

$\textbf{(A)}\ \frac{5}{72}\qquad\textbf{(B)}\ \frac{5}{36}\qquad\textbf{(C)}\ \frac{1}{6}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ 1$[/

Problem 12

A square region $ABCD$ is externally tangent to the circle with equation $x^2+y^2=1$ at the point $(0,1)$ on the side $CD$. Vertices $A$ and $B$ are on the circle with equation $x^2+y^2=4$. What is the side length of this square?

$\textbf{(A)}\ \frac{\sqrt{10}+5}{10}\qquad\textbf{(B)}\ \frac{2\sqrt{5}}{5}\qquad\textbf{(C)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(D)}\ \frac{2\sqrt{19}-4}{5}\qquad\textbf{(E)}\ \frac{9-\sqrt{17}}{5}$

Problem 13

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $\text{8:00 AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at $\text{4:00 PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $\text{2:12 PM}$. On Wednesday Paula worked by herself and finished the house by working until $\text{7:12 PM}$. How long, in minutes, was each day's lunch break?

$\textbf{(A)}\ 30 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 42 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 60$

Problem 14

The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve?

[asy] defaultpen(fontsize(6pt)); dotfactor=4; label("$\circ$",(0,1)); label("$\circ$",(0.865,0.5)); label("$\circ$",(-0.865,0.5)); label("$\circ$",(0.865,-0.5)); label("$\circ$",(-0.865,-0.5)); label("$\circ$",(0,-1)); dot((0,1.5)); dot((-0.4325,0.75)); dot((0.4325,0.75)); dot((-0.4325,-0.75)); dot((0.4325,-0.75)); dot((-0.865,0)); dot((0.865,0)); dot((-1.2975,-0.75)); dot((1.2975,-0.75)); draw(Arc((0,1),0.5,210,-30)); draw(Arc((0.865,0.5),0.5,150,270)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0,-1),0.5,30,150)); draw(Arc((-0.865,-0.5),0.5,330,90)); draw(Arc((-0.865,0.5),0.5,-90,30)); [/asy]

$\textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt3 \qquad\textbf{(C)}\ 3\pi+4 \qquad\textbf{(D)}\ 2\pi+3\sqrt3+2 \qquad\textbf{(E)}\ \pi+6\sqrt3$

Problem 15

A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?

$\textbf{(A)}\ \dfrac{49}{512} \qquad\textbf{(B)}\ \dfrac{7}{64} \qquad\textbf{(C)}\ \dfrac{121}{1024} \qquad\textbf{(D)}\ \dfrac{81}{512} \qquad\textbf{(E)}\ \dfrac{9}{32}$

Problem 16

Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqrt{7}\qquad\textbf{(E)}\ \sqrt{30}$

Problem 17

Let $S$ be a subset of $\{1,2,3,\dots,30\}$ with the property that no pair of distinct elements in $S$ has a sum divisible by $5$. What is the largest possible size of $S$?

$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18$

Problem 18

Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ denote the intersection of the internal angle bisectors of $\triangle ABC$. What is $BI$?

$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 5+\sqrt{26}+3\sqrt{3}\qquad\textbf{(C)}\ 3\sqrt{26}\qquad\textbf{(D)}\ \frac{2}{3}\sqrt{546}\qquad\textbf{(E)}\ 9\sqrt{3}

== Problem 19 ==

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?$ (Error compiling LaTeX. ! Missing $ inserted.) \textbf{(A)}\ 60 \qquad\textbf{(B)}\ 170 \qquad\textbf{(C)}\ 290 \qquad\textbf{(D)}\ 320 \qquad\textbf{(E)}\ 660

$== Problem 20 ==

Consider the polynomial$ (Error compiling LaTeX. ! Missing $ inserted.)P(x)=\prod_{k=0}^{10}=(x^2^+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)$The coefficient of$x^{2012}$is equal to$2^a$.  What is$a$?$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 24

$== Problem 21 ==

Let$ (Error compiling LaTeX. ! Missing $ inserted.)a$,$b$, and$c$be positive integers with$a\ge$$ (Error compiling LaTeX. ! Missing $ inserted.)b\ge$$ (Error compiling LaTeX. ! Missing $ inserted.)c$such that <cmath> a^2-b^2-c^2+ab=2011</cmath> and <cmath>a^2+3b^2+3c^2-3ab-2ac-2bc=-1997.</cmath>

What is$ (Error compiling LaTeX. ! Missing $ inserted.)a$?$ \textbf{(A)}\ 249\qquad\textbf{(B)}\ 250\qquad\textbf{(C)}\ 251\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 253 $[[2012 AMC 10A Problems/Problem 24|Solution]]

== Problem 22 ==

== Problem 23 ==

Let$ (Error compiling LaTeX. ! Missing $ inserted.)S$be the square one of whose diagonals has endpoints$(0.1,0.7)$and$(-0.1,-0.7)$.  A point$v=(x,y)$is chosen uniformly at random over all pairs of real numbers$x$and$y$such that$0 \le x \le 2012$and$0\le y\le 2012$.  Let$T(v)$be a translated copy of$S$centered at$v$.  What is the probability that the square region determined by$T(v)$contains exactly two points with integer coefficients in its interior?$ \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25 \qquad\textbf{(E)}\ 0.32 $[[2012 AMC 10A Problems/Problem 24|Solution]]

== Problem 24 ==

Let$ (Error compiling LaTeX. ! Missing $ inserted.)\{a_k\}_{k=1}^{2011}$be the sequence of real numbers defined by$a_1=0.201,$$ (Error compiling LaTeX. ! Missing $ inserted.)a_2=(0.2011)^{a_1},$$ (Error compiling LaTeX. ! Missing $ inserted.)a_3=(0.20101)^{a_2},$$ (Error compiling LaTeX. ! Missing $ inserted.)a_4=(0.201011)^{a_3}$, and in general, <cmath>a_k=\left\{\array{c}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if k is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if k is even.}</cmath>

Rearranging the numbers in the sequence$ (Error compiling LaTeX. ! Missing } inserted.)\{a_k\}_{k=1}^{2011}$in decreasing order produces a new sequence$\{b_k\}_{k=1}^{2011}$.  What is the sum of all integers$k$,$1\le k \le 2011$, such that$a_k=b_k?$$ (Error compiling LaTeX. ! Missing $ inserted.) \textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\2012 $== Problem 25 ==

Let$ (Error compiling LaTeX. ! Missing $ inserted.)f(x)=|2\{x\}-1|$where$\{x\}$denotes the fractional part of$x$.  The number$n$is the smallest positive integer such that the equation <cmath>nf(xf(x))</cmath> has at least$2012$real solutions.  What is$n$?  '''Note:''' the fractional part of$x$is a real number$y=\{x\}$such that$0\le y<1$and$x-y$is an integer.$ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 31\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 62\qquad\textbf{(E)}\64 $

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