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

Revision as of 13:32, 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. A B C

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 $.34\overline{121}, a string of five digits. How many distinct non-terminating repeating decimals .d1d2d3 . . . have standard abbreviations that have at most six digits? (Consider two nonterminating decimals distinct if they differ in any digit. Nonterminating means that the digits are not eventually all zero.) COMMENTS The standard abbreviation is also the shortest. For example,$ .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.

[[2013 UNC Math Contest II Problems/Problem 9|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.$ (Error compiling LaTeX. Unknown error_msg)\begin{tabular}{|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.

3 1 2 9 4 8 5 10 6 11 ! ! " "

$ (Error compiling LaTeX. Unknown error_msg)

$% & ! 7 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.)

[[2013 UNC Math Contest II Problems/Problem 10|Solution]]

== Problem 11 ==

(a) Stages$ (Error compiling LaTeX. Unknown error_msg)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$ (Error compiling LaTeX. Unknown error_msg)2013$?

Solution

@@ Line 48: / Line 48: @@
 ==Problem 4==
-Find all real numbers x that satisfy <math>(x^2 - \frac{7}{2}x + \frac(3}{2})^(x^2+7x+10)= 1.</math>
+Find all real numbers x that satisfy <math>(x^2 - \tfrac{7}{2} x + \tfrac{3}{2} )^(x^2+7x+10)= 1.</math>
 [[2013 UNC Math Contest II Problems/Problem 4|Solution]]
@@ Line 60: / Line 60: @@
 <math>\begin{tabular}{|c|c|c|c|}
 \hline
-EXAMPLE: & Distinct Integers: {2, 3, 4, 8} & Their Sum: 2 + 3 + 4 + 8 = 17 & Their Product: 2 × 3 × 4 × 8 = 192 \\
+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}</math>
@@ Line 72: / Line 72: @@
 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?
+(b) In a year in which the first Friday the Thirteenth occurs in its latest month, what day of the week is January <math>1</math>?
-The table below shows the number of days in each month. February has 29 days in leap years and 28 in others.
+The table below shows the number of days in each month. February has <math>29</math> days in leap years and <math>28</math> in others.
-JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
+<math>\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
-28 or 29 31 30 31 30 31 31 30 31 30 31
+\hline
+JAN & FEB & MAR & APR & MAY & JUN & JUL & AUG & SEP & OCT & NOV & DEC \\ \hline
+& 28 or 29 & 31 & 30 & 31 & 30 & 31 & 31 & 30 & 31 & 30 & 31 \\
+\hline
+\end{tabular}</math>
 [[2013 UNC Math Contest II Problems/Problem 6|Solution]]
@@ Line 82: / Line 86: @@
 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
+Drying the dishes this way takes twice as long as when all three work together. If Suzie’s mom takes <math>2</math> seconds
-per dish and her dad takes 5 seconds per dish, how long does Suzie take per dish?
+per dish and her dad takes <math>5</math> seconds per dish, how long does Suzie take per dish?
 [[2013 UNC Math Contest II Problems/Problem 7|Solution]]
@@ Line 89: / Line 93: @@
 ==Problem 8==
-EXAMPLE: The non-terminating periodic decimal 0.124124 . . . = 0.124 has period three and is abbreviated by
+EXAMPLE: The non-terminating periodic decimal <math>0.124124 \ldots = 0.\overline{124}</math> has period three and is abbreviated by placing a bar over the shortest repeating block.
-placing a bar over the shortest repeating block.
-(a) If all digits 0 through 9 are allowed, how many distinct periodic decimals 0.d1d2 . . . d6 have period exactly
+(a) If all digits <math>0</math> through <math>9</math> are allowed, how many distinct periodic decimals <math>0.\overline{d_1d_2 \ldots d_6}</math> have period exactly six? Do not include patterns like <math>0.323</math> and <math>0.17</math> that have shorter periods.
-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.d1d2 . . . d12 have period exactly 12?
+(b) If only digits <math>0</math> and <math>1</math> are allowed, how many distinct periodic decimals <math>0.\overline{d_1d_2\ldots d_{12}}</math> have period exactly <math>12</math>?
 [[2013 UNC Math Contest II Problems/Problem 8|Solution]]
@@ Line 99: / Line 103: @@
 ==Problem 9==
-The standard abbreviation for the non-terminating repeating decimal .34121121121121121 . . . is .34121, a string
+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.)
-of five digits. How many distinct non-terminating repeating decimals .d1d2d3 . . . have standard abbreviations
+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
-that have at most six digits? (Consider two nonterminating decimals distinct if they differ in any digit. Nonterminating
+</math>H = 34<math>. If the string has no head, the decimal is periodic, which is acceptable. There must be a tail string
-means that the digits are not eventually all zero.)
+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.
-COMMENTS The standard abbreviation is also the shortest. For example, .34121121121121121 . . . = .34121 can
-also be abbreviated as .341211, or as .3412112, or as .34121121 by sliding the bar rightward, making longer
-strings. The nonterminating decimal .34121 has two parts: a repeating tail T = 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 = 0, since that corresponds to a terminating decimal. The
-examples .345, .9, and .7219 are all standard abbreviations for nonterminating repeating decimals.
 [[2013 UNC Math Contest II Problems/Problem 9|Solution]]
@@ Line 116: / Line 114: @@
 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.
+</math>\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
+\hline
+& 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline
+st: & E & N & I & G & M & A & C & R & U & S & H \\ \hline
+nd: & G & E & N & I & U & S & C & H & A & R & M \\ \hline
+rd: & I & G & E & N & A & R & C & M & S & H & U \\
+\hline
+\end{tabular}<math>
 FrankenCoder scrambles ENIGMACRUSH into
 GENIUSCHARM. Using the same rule, Franken-
 Coder then scrambles GENIUSCHARM into
 IGENARCMSHU, and so on.
-2 3 4 5 6 7 8 9 10 11
-st: E N I G M A C R U S H
-nd: G E N I U S C H A R M
-rd: I G E N A R C M S H U
@@ Line 140: / Line 144: @@
 #
 #
+</math>
 <math>
-</math>
 %
 &
@@ Line 168: / Line 172: @@
 == Problem 11 ==
- (a) Stages 1 and 2 each contain 1 tile. Stage 6 contains 8 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 15 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
-of 2013?
+of </math>2013$?
 [[2013 UNC Math Contest II Problems/Problem 11|Solution]]

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

Revision as of 13:32, 16 October 2014

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9