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Difference between revisions of "2013 UNCO Math Contest II Problems"

m
m
Line 26: Line 26:
 
(b) Find the smallest positive integer multiple of <math>12</math> that is a perfect cube.
 
(b) Find the smallest positive integer multiple of <math>12</math> that is a perfect cube.
 
(c) Find the smallest positive integer multiple of <math>12</math> that is both a perfect square and a perfect cube.
 
(c) Find the smallest positive integer multiple of <math>12</math> that is both a perfect square and a perfect cube.
−
A B
+
 
−
C
 
  
 
[[2013 UNC Math Contest II Problems/Problem 2|Solution]]
 
[[2013 UNC Math Contest II Problems/Problem 2|Solution]]
Line 41: Line 40:
 
Point <math>C</math> is the center of a large circle that passes through both <math>A</math> and
 
Point <math>C</math> is the center of a large circle that passes through both <math>A</math> and
 
<math>B</math>, and <math>C</math> lies on the small circle whose diameter is <math>AB</math>. The area of the
 
<math>B</math>, and <math>C</math> lies on the small circle whose diameter is <math>AB</math>. The area of the
−
small circle is <math>9\pi</math>. Find the area of the shaded <math>textit{lune}</math>, the region inside
+
small circle is <math>9\pi</math>. Find the area of the shaded <math>\textit{lune}</math>, the region inside
 
the small circle and outside the large circle.
 
the small circle and outside the large circle.
  
Line 103: Line 102:
 
==Problem 9==
 
==Problem 9==
  
−
The standard abbreviation for the non-terminating repeating decimal <math>.34121121121121121\ldots</math> is <math>.34\overline{121}, a string of five digits. How many distinct non-terminating repeating decimals .d1d2d3 . . . have standard abbreviations that have at most six digits? (Consider two nonterminating decimals distinct if they differ in any digit. Nonterminating means that the digits are not eventually all zero.)
+
The standard abbreviation for the non-terminating repeating decimal <math>.34121121121121121\ldots</math> is <math>.34\overline{121}</math>, a string of five digits. How many distinct non-terminating repeating decimals <math>.d_1d_2d_3\ldots</math> have standard abbreviations that have at most six digits? (Consider two nonterminating decimals distinct if they differ in any digit. Nonterminating means that the digits are not eventually all zero.)
−
COMMENTS The standard abbreviation is also the shortest. For example, </math>.34121121121121121\ldots= .34\overline{121}<math> can also be abbreviated as </math>.341\overline{211}<math>, or as </math>.3412\overline{112}<math>, or as </math>.34121\overline{121}<math> by sliding the bar rightward, making longer strings. The nonterminating decimal </math>.34\overline{121}<math> has two parts: a repeating tail </math>T = \overline{121}<math> and a non-repeating head
+
COMMENTS The standard abbreviation is also the shortest. For example, <math>.34121121121121121\ldots= .34\overline{121}</math> can also be abbreviated as <math>.341\overline{211}</math>, or as <math>.3412\overline{112}</math>, or as <math>.34121\overline{121}</math> by sliding the bar rightward, making longer strings. The nonterminating decimal <math>.34\overline{121}</math> has two parts: a repeating tail <math>T = \overline{121}</math> and a non-repeating head
−
</math>H = 34<math>. If the string has no head, the decimal is periodic, which is acceptable. There must be a tail string
+
<math>H = 34</math>. If the string has no head, the decimal is periodic, which is acceptable. There must be a tail string
−
T, which by convention is NOT permitted to be </math>T = \overline{0}<math>, since that corresponds to a terminating decimal. The examples </math>.\overline{345}, .\overline{9}<math>, and </math>.721\overline{9}<math> are all standard abbreviations for nonterminating repeating decimals.
+
<math>T</math>, which by convention is NOT permitted to be <math>T = \overline{0}</math>, since that corresponds to a terminating decimal. The examples <math>.\overline{345}</math>, <math>.\overline{9}</math>, and <math>.721\overline{9}</math> are all standard abbreviations for nonterminating repeating decimals.
  
 
[[2013 UNC Math Contest II Problems/Problem 9|Solution]]
 
[[2013 UNC Math Contest II Problems/Problem 9|Solution]]
Line 114: Line 113:
 
Dav designs a robot, which he calls FrankenCoder, to print nonsense text by scrambling eleven-letter messages.
 
Dav designs a robot, which he calls FrankenCoder, to print nonsense text by scrambling eleven-letter messages.
 
The robot always repeats the same scrambling rule.
 
The robot always repeats the same scrambling rule.
−
</math>\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
+
 
 +
<math>\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
 
\hline
 
\hline
−
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline
+
& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline
 
1st: & E & N & I & G & M & A & C & R & U & S & H \\ \hline
 
1st: & E & N & I & G & M & A & C & R & U & S & H \\ \hline
 
2nd: & G & E & N & I & U & S & C & H & A & R & M \\ \hline
 
2nd: & G & E & N & I & U & S & C & H & A & R & M \\ \hline
 
3rd: & I & G & E & N & A & R & C & M & S & H & U \\
 
3rd: & I & G & E & N & A & R & C & M & S & H & U \\
 
\hline
 
\hline
−
\end{tabular}<math>
+
\end{tabular}</math>
 +
 
 
FrankenCoder scrambles ENIGMACRUSH into
 
FrankenCoder scrambles ENIGMACRUSH into
 
GENIUSCHARM. Using the same rule, Franken-
 
GENIUSCHARM. Using the same rule, Franken-
 
Coder then scrambles GENIUSCHARM into
 
Coder then scrambles GENIUSCHARM into
 
IGENARCMSHU, and so on.
 
IGENARCMSHU, and so on.
 +
 +
<asy>
 +
path Sq=polygon(4);
 +
path He=polygon(6);
 +
draw(rotate(45)*Sq,black);
 +
draw(shift(3.5,0)*rotate(30)*He,black);
 +
draw(circle((1.5,-1.5),.5),black);
 +
MP("1",(0,1),dir(90));
 +
MP("2",(1,0),dir(0));
 +
MP("3",(0,-1),dir(270));
 +
MP("4",(-1,0),dir(180));
 +
 +
MP("5",(3.5,1),dir(90));
 +
MP("11",(3.5+sqrt(3)/2,1/2),dir(30));
 +
MP("8",(3.5+sqrt(3)/2,-1/2),dir(-30));
 +
MP("10",(3.5+0,-1),dir(-90));
 +
MP("6",(3.5-sqrt(3)/2,-1/2),dir(-150));
 +
MP("9",(3.5-sqrt(3)/2,1/2),dir(150));
 +
 +
MP("7",(1.5,-1.5),W);
 +
 +
draw(arc((1.6,-1.5),.2,135,-135),arrow=ArcArrow(TeXHead));
 +
 +
path Ra1=((0,0)--(sqrt(2)/2,sqrt(2)/2));
 +
 +
draw(shift(-1,.4)*Ra1,arrow=Arrow(TeXHead));
 +
draw(shift(0.4,1)*rotate(-90)*Ra1,arrow=Arrow(TeXHead));
 +
draw(shift(1,-.4)*rotate(-180)*Ra1,arrow=Arrow(TeXHead));
 +
draw(shift(-.4,-1)*rotate(90)*Ra1,arrow=Arrow(TeXHead));
 +
 +
path Ra2=((0,0)--(.85*sqrt(3)/2,.45));
 +
 +
draw(shift(3.7,1.2)*rotate(-60)*Ra2,arrow=Arrow(TeXHead));
 +
draw(shift(3.7+1,.4)*rotate(-120)*Ra2,arrow=Arrow(TeXHead));
 +
draw(shift(4.5,-.8)*rotate(-180)*Ra2,arrow=Arrow(TeXHead));
 +
draw(shift(3.3,-1.2)*rotate(-240)*Ra2,arrow=Arrow(TeXHead));
 +
draw(shift(2.3,-.4)*rotate(-300)*Ra2,arrow=Arrow(TeXHead));
 +
draw(shift(2.5,.8)*rotate(0)*Ra2,arrow=Arrow(TeXHead));
 +
 +
</asy>
  
  
−
3
 
−
1
 
−
2
 
−
9
 
−
4
 
−
8
 
−
5
 
−
10
 
−
6
 
−
11
 
−
!
 
−
!
 
−
"
 
−
"
 
−
#
 
−
#
 
−
</math>
 
−
<math>
 
−
%
 
−
&
 
−
! 7
 
 
FrankenCoder’s internal wiring for scrambling letters is diagrammed
 
FrankenCoder’s internal wiring for scrambling letters is diagrammed
 
at left, depicted as a collection of cycles. The arrows
 
at left, depicted as a collection of cycles. The arrows
 
show how each of the eleven letters moves in a single scramble:
 
show how each of the eleven letters moves in a single scramble:
−
letter 7 stays in its place, the first four letters move in a cycle,
+
letter <math>7</math> stays in its place, the first four letters move in a cycle,
 
and the other six letters also trade positions in a cycle.
 
and the other six letters also trade positions in a cycle.
 
Dav sees that the messages printed by FrankenCoder repeat cyclically in paragraphs: eventually, the original
 
Dav sees that the messages printed by FrankenCoder repeat cyclically in paragraphs: eventually, the original
Line 172: Line 192:
 
== Problem 11 ==
 
== Problem 11 ==
  
−
(a) Stages </math>1<math> and </math>2<math> each contain </math>1<math> tile. Stage </math>6<math> contains </math>8<math> tiles. If
+
(a) Stages <math>1</math> and <math>2</math> each contain <math>1</math> tile. Stage <math>6</math> contains <math>8</math> tiles. If
−
the pattern is continued, how many tiles will Stage </math>15<math> contain?
+
the pattern is continued, how many tiles will Stage <math>15</math> contain?
  
 
(b) What is the first Stage in which the number of tiles is a multiple
 
(b) What is the first Stage in which the number of tiles is a multiple
−
of </math>2013$?
+
of <math>2013</math>?
 +
 
 +
<asy>
 +
size(14cm,0);
 +
draw((0,0)--(1,1)--(2,1)--cycle,black);
 +
 
 +
draw((3,0)--(4,2)--(5,0)--cycle,black);
 +
 
 +
draw((6,0)--(7,2)--(8,0)--cycle,black);
 +
draw((6,0)--(5.5,-1)--(8,0)--cycle,black);
 +
 
 +
draw((9,0)--(10,2)--(11,0)--cycle,black);
 +
draw((9,0)--(8.5,-1)--(11,0)--cycle,black);
 +
draw((8.5,-1)--(11,0)--(11.5,-1)--cycle,black);
 +
 
 +
draw((13,0)--(14,2)--(15,0)--cycle,black);
 +
draw((13,0)--(12.5,-1)--(15,0)--cycle,black);
 +
draw((12.5,-1)--(15,0)--(15.5,-1)--cycle,black);
 +
draw((12.5,-1)--(11.5,-3)--(15.5,-1)--cycle,black);
 +
draw((12.5,-1)--(14,-1.75),black);
 +
 
 +
draw((18,0)--(19,2)--(20,0)--cycle,black);
 +
draw((18,0)--(17.5,-1)--(20,0)--cycle,black);
 +
draw((17.5,-1)--(20,0)--(20.5,-1)--cycle,black);
 +
draw((17.5,-1)--(16.5,-3)--(20.5,-1)--cycle,black);
 +
draw((17.5,-1)--(21.5,-3)--(20.5,-1)--cycle,black);
 +
draw((16.5,-3)--(21.5,-3),black);
 +
draw((19,-1.75)--(20,-3),black);
 +
 
 +
MP("Stage 1",(.5,-3));
 +
MP("Stage 2",(3.5,-3));
 +
MP("Stage 3",(6.5,-3));
 +
MP("Stage 4",(10,-3));
 +
MP("Stage 5",(14,-3));
 +
MP("Stage 6",(19,-3));
 +
</asy>
  
 
[[2013 UNC Math Contest II Problems/Problem 11|Solution]]
 
[[2013 UNC Math Contest II Problems/Problem 11|Solution]]

Revision as of 14:03, 16 October 2014

Twenty-first Annual UNC Math Contest Final Round January 19, 2013. Three hours; no electronic devices. Justify your answers. Clear and concise presentations will earn more points. We hope you enjoy thinking about these problems, but you are not expected to solve them all. The positive integers are $1, 2, 3, 4, \ldots$

Problem 1

[asy] filldraw((-4,-3)--(-4,3)--(4,3)--(4,-3)--cycle,grey); filldraw(circle((-4,0),3),white); filldraw(circle((4,0),3),white); draw((-7,3)--(7,3),black); draw((-7,-3)--(7,-3),black); [/asy]

In the diagram, the two circles are tangent to the two parallel lines. The distance between the centers of the circles is 8, and both circles have radius 3. What is the area of the shaded region between the circles?

Solution

Problem 2

EXAMPLE: The number $64$ is equal to $8^2$ and also equal to $4^3$, so $64$ is both a perfect square and a perfect cube. (a) Find the smallest positive integer multiple of $12$ that is a perfect square. (b) Find the smallest positive integer multiple of $12$ that is a perfect cube. (c) Find the smallest positive integer multiple of $12$ that is both a perfect square and a perfect cube.


Solution

Problem 3

[asy] filldraw(circle((0,-sqrt(2)/2),sqrt(2)/2),grey); filldraw(circle((0,0),1),white); draw((-sqrt(2)/2,-sqrt(2)/2)--(sqrt(2)/2,-sqrt(2)/2)--(0,0)--cycle,black); MP("C",(0,0),N);MP("A",(-sqrt(2)/2,-sqrt(2)/2),W);MP("B",(sqrt(2)/2,-sqrt(2)/2),E); [/asy]

Point $C$ is the center of a large circle that passes through both $A$ and $B$, and $C$ lies on the small circle whose diameter is $AB$. The area of the small circle is $9\pi$. Find the area of the shaded $\textit{lune}$, the region inside the small circle and outside the large circle.

Solution

Problem 4

Find all real numbers x that satisfy $(x^2 - \tfrac{7}{2} x + \tfrac{3}{2} )^(x^2+7x+10)= 1.$

Solution

Problem 5

If the sum of distinct positive integers is $17$, find the largest possible value of their product. Give both a set of positive integers and their product. Remember to consider only sums of distinct numbers, and not $3+7+7$ or $2+3+4+4+4$, etc., which have repeated terms. You need not justify your answer on this question.

$\begin{tabular}{|c|c|c|c|} \hline EXAMPLE: & Distinct Integers: {2, 3, 4, 8} & Their Sum: 2+3+4+8=17 & Their Product: 2 \times 3\times 4\times 8=192 \\ \hline \end{tabular}$ (Error compiling LaTeX. Unknown error_msg)


Solution

Problem 6

There is at least one Friday the Thirteenth in every year. (a) What is the latest possible month in which the first Friday the Thirteenth can occur? (b) In a year in which the first Friday the Thirteenth occurs in its latest month, what day of the week is January $1$? The table below shows the number of days in each month. February has $29$ days in leap years and $28$ in others. $\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline JAN & FEB & MAR & APR & MAY & JUN & JUL & AUG & SEP & OCT & NOV & DEC \\ \hline 31 & 28 or 29 & 31 & 30 & 31 & 30 & 31 & 31 & 30 & 31 & 30 & 31 \\ \hline \end{tabular}$

Solution

Problem 7

Suzie and her mom dry half the dishes together; then mom rests, while Suzie and her dad dry the other half. Drying the dishes this way takes twice as long as when all three work together. If Suzie’s mom takes $2$ seconds per dish and her dad takes $5$ seconds per dish, how long does Suzie take per dish?

Solution

Problem 8

EXAMPLE: The non-terminating periodic decimal $0.124124 \ldots = 0.\overline{124}$ has period three and is abbreviated by placing a bar over the shortest repeating block.

(a) If all digits $0$ through $9$ are allowed, how many distinct periodic decimals $0.\overline{d_1d_2 \ldots d_6}$ have period exactly six? Do not include patterns like $0.323$ and $0.17$ that have shorter periods.

(b) If only digits $0$ and $1$ are allowed, how many distinct periodic decimals $0.\overline{d_1d_2\ldots d_{12}}$ have period exactly $12$?

Solution

Problem 9

The standard abbreviation for the non-terminating repeating decimal $.34121121121121121\ldots$ is $.34\overline{121}$, a string of five digits. How many distinct non-terminating repeating decimals $.d_1d_2d_3\ldots$ have standard abbreviations that have at most six digits? (Consider two nonterminating decimals distinct if they differ in any digit. Nonterminating means that the digits are not eventually all zero.) COMMENTS The standard abbreviation is also the shortest. For example, $.34121121121121121\ldots= .34\overline{121}$ can also be abbreviated as $.341\overline{211}$, or as $.3412\overline{112}$, or as $.34121\overline{121}$ by sliding the bar rightward, making longer strings. The nonterminating decimal $.34\overline{121}$ has two parts: a repeating tail $T = \overline{121}$ and a non-repeating head $H = 34$. If the string has no head, the decimal is periodic, which is acceptable. There must be a tail string $T$, which by convention is NOT permitted to be $T = \overline{0}$, since that corresponds to a terminating decimal. The examples $.\overline{345}$, $.\overline{9}$, and $.721\overline{9}$ are all standard abbreviations for nonterminating repeating decimals.

Solution

Problem 10

Dav designs a robot, which he calls FrankenCoder, to print nonsense text by scrambling eleven-letter messages. The robot always repeats the same scrambling rule.

$\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline 1st: & E & N & I & G & M & A & C & R & U & S & H \\ \hline 2nd: & G & E & N & I & U & S & C & H & A & R & M \\ \hline 3rd: & I & G & E & N & A & R & C & M & S & H & U \\ \hline \end{tabular}$

FrankenCoder scrambles ENIGMACRUSH into GENIUSCHARM. Using the same rule, Franken- Coder then scrambles GENIUSCHARM into IGENARCMSHU, and so on.

[asy] path Sq=polygon(4); path He=polygon(6); draw(rotate(45)*Sq,black); draw(shift(3.5,0)*rotate(30)*He,black); draw(circle((1.5,-1.5),.5),black); MP("1",(0,1),dir(90)); MP("2",(1,0),dir(0)); MP("3",(0,-1),dir(270)); MP("4",(-1,0),dir(180)); MP("5",(3.5,1),dir(90)); MP("11",(3.5+sqrt(3)/2,1/2),dir(30)); MP("8",(3.5+sqrt(3)/2,-1/2),dir(-30)); MP("10",(3.5+0,-1),dir(-90)); MP("6",(3.5-sqrt(3)/2,-1/2),dir(-150)); MP("9",(3.5-sqrt(3)/2,1/2),dir(150)); MP("7",(1.5,-1.5),W); draw(arc((1.6,-1.5),.2,135,-135),arrow=ArcArrow(TeXHead)); path Ra1=((0,0)--(sqrt(2)/2,sqrt(2)/2)); draw(shift(-1,.4)*Ra1,arrow=Arrow(TeXHead)); draw(shift(0.4,1)*rotate(-90)*Ra1,arrow=Arrow(TeXHead)); draw(shift(1,-.4)*rotate(-180)*Ra1,arrow=Arrow(TeXHead)); draw(shift(-.4,-1)*rotate(90)*Ra1,arrow=Arrow(TeXHead)); path Ra2=((0,0)--(.85*sqrt(3)/2,.45)); draw(shift(3.7,1.2)*rotate(-60)*Ra2,arrow=Arrow(TeXHead)); draw(shift(3.7+1,.4)*rotate(-120)*Ra2,arrow=Arrow(TeXHead)); draw(shift(4.5,-.8)*rotate(-180)*Ra2,arrow=Arrow(TeXHead)); draw(shift(3.3,-1.2)*rotate(-240)*Ra2,arrow=Arrow(TeXHead)); draw(shift(2.3,-.4)*rotate(-300)*Ra2,arrow=Arrow(TeXHead)); draw(shift(2.5,.8)*rotate(0)*Ra2,arrow=Arrow(TeXHead)); [/asy]


FrankenCoder’s internal wiring for scrambling letters is diagrammed at left, depicted as a collection of cycles. The arrows show how each of the eleven letters moves in a single scramble: letter $7$ stays in its place, the first four letters move in a cycle, and the other six letters also trade positions in a cycle. Dav sees that the messages printed by FrankenCoder repeat cyclically in paragraphs: eventually, the original message ENIGMACRUSH reappears as the start of a new paragraph identical to the first paragraph. (a) How many distinct messages does each paragraph contain? (b) Dav tries to improve the robot, to get an even longer paragraph of distinct messages, by drawing different wiring diagrams for the eleven letter positions. Experimenting with component cycles of various lengths, he perfects his ultimate robot: FrankenCoder-II, a robot that produces the longest possible paragraph of distinct eleven-letter messages. How many distinct messages does FrankenCoder-II produce? (c) Draw a wiring diagram that could describe FrankenCoder-II. There may be ties, since different wiring diagrams can make robots that print paragraphs that have the same length. Draw just one wiring diagram. (d) Dav realizes that because there are ties for the best wiring diagram, he can build an entire army of distinct robots that are as good as FrankenCoder-II at creating long paragraphs. How many distinct robots can he build that are as good as FrankenCoder-II? Include FrankenCoder-II in your count. (Two robots are regarded as distinct if they scramble the starting message ENIGMACRUSH into different messages.)

Solution

Problem 11

(a) Stages $1$ and $2$ each contain $1$ tile. Stage $6$ contains $8$ tiles. If the pattern is continued, how many tiles will Stage $15$ contain?

(b) What is the first Stage in which the number of tiles is a multiple of $2013$?

[asy] size(14cm,0); draw((0,0)--(1,1)--(2,1)--cycle,black); draw((3,0)--(4,2)--(5,0)--cycle,black); draw((6,0)--(7,2)--(8,0)--cycle,black); draw((6,0)--(5.5,-1)--(8,0)--cycle,black); draw((9,0)--(10,2)--(11,0)--cycle,black); draw((9,0)--(8.5,-1)--(11,0)--cycle,black); draw((8.5,-1)--(11,0)--(11.5,-1)--cycle,black); draw((13,0)--(14,2)--(15,0)--cycle,black); draw((13,0)--(12.5,-1)--(15,0)--cycle,black); draw((12.5,-1)--(15,0)--(15.5,-1)--cycle,black); draw((12.5,-1)--(11.5,-3)--(15.5,-1)--cycle,black); draw((12.5,-1)--(14,-1.75),black); draw((18,0)--(19,2)--(20,0)--cycle,black); draw((18,0)--(17.5,-1)--(20,0)--cycle,black); draw((17.5,-1)--(20,0)--(20.5,-1)--cycle,black); draw((17.5,-1)--(16.5,-3)--(20.5,-1)--cycle,black); draw((17.5,-1)--(21.5,-3)--(20.5,-1)--cycle,black); draw((16.5,-3)--(21.5,-3),black); draw((19,-1.75)--(20,-3),black); MP("Stage 1",(.5,-3)); MP("Stage 2",(3.5,-3)); MP("Stage 3",(6.5,-3)); MP("Stage 4",(10,-3)); MP("Stage 5",(14,-3)); MP("Stage 6",(19,-3)); [/asy]

Solution

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