Difference between revisions of "2005 PMWC Problems"

Latest revision as of 20:49, 2 October 2020

The following is a list of PMWC problems from the year 2005

Problem I1

What is the greatest possible number one can get by discarding $100$ digits, in any order, from the number $123456789101112 \dots 585960$ ?

Solution

Problem I2

Let $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2005}$ , where $a$ and $b$ are different four-digit positive integers (natural numbers) and $c$ is a five-digit positive integer (natural number)ems/Problem I2|Solution]]

Problem I3

Let $x$ be a fraction between $\frac{35}{36}$ and $\frac{91}{183}$ . If the denominator of $x$ is $455$ and the numerator and denominator have no common factor except $1$ , how many possible values are there for $x$ ?

Solution

Problem I4

The larger circle has radius 12 cm. Each of the six identical smaller circles touches its two neighbors and the larger circle. What is the radius of the smaller circles?

$[asy] unitsize(0.5cm); draw((0,3)..(3,0)..(0,-3)..(-3,0)..cycle); for (int i=0;i<6;i=i+1){ draw(dir(60*i)..3*dir(60*i)..cycle); } [/asy]$

Solution

Problem I5

Consider the following conditions on the positive integer (natural number) $a$ :

1. $3a + 5 > 40$

2. $49a \ge 301$

3. $20a \le 999$

4. $101a + 53 \ge 2332$

5. $15a-7 \ge 144$

If only three of these conditions are true, what is the value of $a$ ?

Solution

Problem I6

A group of $100$ people consists of men, women and children (at least one of each). Exactly $200$ apples are distributed in such a way that each man gets $6$ apples, each woman gets $4$ apples and each child gets $1$ apple. In how many possible ways can this be done?

Solution

Problem I7

How many numbers are there in the list $1, 2, 3, 4, 5, \dots, 10000$ which contain exactly two consecutive $9$ 's such as $993, 1992$ and $9929$ , but not $9295$ or $1999$ ?

Solution

Problem I8

Some people in Hong Kong express $2/8$ as 8th Feb and others express $2/8$ as 2nd Aug. This can be confusing as when we see $2/8$ , we don’t know whether it is 8th Feb or 2nd Aug. However, it is easy to understand $9/22$ or $22/9$ as 22nd Sept, because there are only $12$ months in a year. How many dates in a year can cause this confusion?

Solution

Problem I9

There are four consecutive positive integers (natural numbers) less than $2005$ such that the first (smallest) number is a multiple of $5$ , the second number is a multiple of $7$ , the third number is a multiple of $9$ and the last number is a multiple of $11$ . What is the first of these four numbers?

Solution

Problem I10

A long string is folded in half eight times, then cut in the middle. How many pieces are obtained?

Solution

Problem I11

There are 4 men: A, B, C and D. Each has a son. The four sons are asked to enter a dark room. Then A, B, C and D enter the dark room, and each of them walks out with just one child. If none of them comes out with his own son, in how many ways can this happen?

Solution

Problem I12

In the figure below, $BCDE$ is a parallelogram, points $F$ and $G$ are on the segment $ED$ , $BCA$ is a right angled triangle, $AC$ is perpendicular to $BC$ .Suppose that $BC=8cm$ , $AC=7cm$ , and the area of the shaded regions is $12cm^2$ more than that of the triangle $AFG$ . What is the length of $CG$ ?

$[asy] fill((0,0)--(1,1)--(3,1)--(2,0)--cycle,grey); fill((0,0)--(2,1.5)--(2,0)--cycle,white); draw((0,0)--(2,1.5)--(2,0)--cycle,dot); draw((0,0)--(1,1)--(3,1)--(2,0)--cycle,dot); MP("B",(0,0),W);MP("A",(2,1.5),N);MP("C",(2,0),S); MP("E",(1,1),NW);MP("D",(3,1),NE); MP("F",(1.5,1),NW);MP("G",(2,1),NE); draw((1.9,0)--(1.9,.1)--(2,.1),linewidth(.75)); [/asy]$

Solution

Problem I13

Sixty meters of rope is used to make three sides of a rectangular camping area with a long wall used as the other side. The length of each side of the rectangle is a natural number. What is the largest area that can be enclosed by the rope and the wall?

Solution

Problem I14

On a balance scale, three green balls balance six blue balls, two yellow balls balance five blue balls and six blue balls balance four white balls. How many blue balls are needed to balance four green, two yellow and two white balls?

Solution

Problem I15

The sum of the two three-digit integers, $\text{6A2}$ and $\text{B34}$ , is divisible by $18$ . What is the largest possible product of $\text{A}$ and $\text{B}$ ?

Solution

Problem T1

Call an integer "happy", if the sum of its digits is $10$ . How many "happy" integers are there between $100$ and $1000$ ?

Solution

Problem T2

Compute the sum of $a$ , $b$ and $c$ given that $\frac{a}{2}=\frac{b}{3}=\frac{c}{5}$ and the product of $a$ , $b$ and $c$ is $1920$ .

Solution

Problem T3

Replace the letters $a$ , $b$ , $c$ and $d$ in the following expression with the numbers $1$ , $2$ , $3$ and $4$ , without repetition: $a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d}}}$ Find the difference between the maximum value and the minimum value of the expression.

Solution

Problem T4

Buses from town A to town B leave every hour on the hour (for example: 6:00, 7:00, …). Buses from town B to town A leave every hour on the half hour (for example: 6:30, 7:30, …). The trip between town A and town B takes 5 hours. Assume the buses travel on the same road. If you get on a bus from town A, how many buses from town B do you pass on the road (not including those at the stations)?

Solution

Problem T5

Mr. Wong has a $7$ -digit phone number $\text{ABCDEFG}$ . The sum of the number formed by the first $4$ digits $\text{ABCD}$ and the number formed by the last $3$ digits $\text{EFG}$ is $9063$ . The sum of the number formed by the first $3$ digits $\text{ABC}$ and the number formed by the last $4$ digits $\text{DEFG}$ is $2529$ . What is Mr. Wong’s phone number?

Solution

Problem T6

$\begin{eqnarray*} 1+2 &=& 3 \\ 4+5+6 &=& 7+8 \\ 9+10+11+12 &=& 13+14+15\end{eqnarray*}$ $\vdots$ If this pattern is continued, find the last number in the $80$ th row (e.g. the last number of the third row is $15$ ).

Solution

Problem T7

Skipper’s doghouse has a regular hexagonal base that measures one metre on each side. Skipper is tethered to a 2-metre rope which is fixed to a vertex. What is the area of the region outside the doghouse that Skipper can reach? Calculate an approximate answer by using $\pi=3.14$ or $\pi=22/7$ .

Solution

Problem T8

An isosceles right triangle is removed from each corner of a square piece of paper so that a rectangle of unequal sides remains. If the sum of the areas of the cut-off pieces is $200 \text{cm}^2$ and the lengths of the legs of the triangles cut off are integers, find the area of the rectangle.

Solution

Problem T9

Select 8 of the 9 given numbers: 2, 3, 4, 7, 10, 11, 12, 13, 15 and place them in the vacant squares so that the average of the numbers in each row and column is the same. Complete the following table.

$\begin{tabular}{ |c | c | c | c |} \hline 1 & & & \\ \hline & 9 & & 5 \\ \hline & & 14 & \\ \hline \end{tabular}$

Solution

Problem T10

Find the largest 12-digit number for which every two consecutive digits form a distinct 2-digit prime number.

Solution

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Difference between revisions of "2005 PMWC Problems"

Latest revision as of 20:49, 2 October 2020

Contents

Problem I1

Problem I2

Problem I3

Problem I4

Problem I5

Problem I6

Problem I7

Problem I8

Problem I9

Problem I10

Problem I11

Problem I12

Problem I13

Problem I14

Problem I15

Problem T1

Problem T2

Problem T3

Problem T4

Problem T5

Problem T6

Problem T7

Problem T8

Problem T9

Problem T10

See Also

@@ Line 1: / Line 1: @@
-== Problem 1 ==
+The following is a list of [[PMWC]] problems from the year 2005
+== Problem I1 ==
 What is the greatest possible number one can get by discarding <math>100</math> digits, in any order, from the number <math>123456789101112 \dots 585960</math>?
-[[2005 PMWC Individual Test Problems/Problem 1|Solution]]
+[[2005 PMWC Problems/Problem I1|Solution]]
-== Problem 2 ==
+== Problem I2 ==
-Let <math>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2005}</math>, where <math>a</math> and <math>b</math> are different four-digit positive integers (natural numbers) and <math>c</math> is a five-digit positive integer (natural number).  What is the number <math>c</math>?
+Let <math>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2005}</math>, where <math>a</math> and <math>b</math> are different four-digit positive integers (natural numbers) and <math>c</math> is a five-digit positive integer (natural number)ems/Problem I2|Solution]]
-[[2005 PMWC Individual Test Problems/Problem 2|Solution]]
+== Problem I3 ==
+Let <math>x</math> be a fraction between <math>\frac{35}{36}</math> and <math>\frac{91}{183}</math>.  If the denominator of <math>x</math> is <math>455</math> and the numerator and denominator have no common factor except <math>1</math>, how many possible values are there for <math>x</math>?
-== Problem 3 ==
+[[2005 PMWC Problems/Problem I3|Solution]]
-Let <math>x</math> be a fraction between <math>\frac{35}{36}</math> and <math>\frac{91}{183}</math>.  If the denominator of <math>x</math> is <math>455</math> and the numerator and denominator have no common factor except <math>1</math>, how many possible values are there for <math>x</math>?
-[[2005 PMWC Individual Test Problems/Problem 3|Solution]]
+== Problem I4 ==
+The larger circle has radius 12 cm. Each of the six identical smaller circles
+touches its two neighbors and the larger circle. What is the radius of the
+smaller circles?
-== Problem 4 ==
+<asy>
+unitsize(0.5cm);
+draw((0,3)..(3,0)..(0,-3)..(-3,0)..cycle);
+for (int i=0;i<6;i=i+1){
+draw(dir(60*i)..3*dir(60*i)..cycle);
+}
+</asy>
-[[2005 PMWC Individual Test Problems/Problem 4|Solution]]
+[[2005 PMWC Problems/Problem I4|Solution]]
-== Problem 5 ==
+== Problem I5 ==
 Consider the following conditions on the positive integer (natural number) <math>a</math>:
@@ Line 29: / Line 40: @@
 . <math>101a + 53 \ge 2332</math>
-. <math>15a – 7 \ge 144</math>
+. <math>15a-7 \ge 144</math>
 If only three of these conditions are true, what is the value of <math>a</math>?
-[[2005 PMWC Individual Test Problems/Problem 5|Solution]]
+[[2005 PMWC Problems/Problem I5|Solution]]
-== Problem 6 ==
+== Problem I6 ==
 A group of <math>100</math> people consists of men, women and children (at least one of each). Exactly <math>200</math> apples are distributed in such a way that each man gets <math>6</math> apples, each woman gets <math>4</math> apples and each child gets <math>1</math> apple. In how many possible ways can this be done?
-[[2005 PMWC Individual Test Problems/Problem 6|Solution]]
+[[2005 PMWC Problems/Problem I6|Solution]]
-== Problem 7 ==
+== Problem I7 ==
 How many numbers are there in the list <math>1, 2, 3, 4, 5, \dots, 10000</math> which contain exactly two consecutive <math>9</math>'s such as <math>993, 1992</math> and <math>9929</math>, but not <math>9295</math> or <math>1999</math>?
-[[2005 PMWC Individual Test Problems/Problem 7|Solution]]
+[[2005 PMWC Problems/Problem I7|Solution]]
-== Problem 8 ==
+== Problem I8 ==
 Some people in Hong Kong express <math>2/8</math> as 8th Feb and others express <math>2/8</math> as
 nd Aug. This can be confusing as when we see <math>2/8</math>, we don’t know whether it
 is 8th Feb or 2nd Aug. However, it is easy to understand <math>9/22</math> or <math>22/9</math> as 22nd Sept, because there are only <math>12</math> months in a year. How many dates in a year can cause this confusion?
-[[2005 PMWC Individual Test Problems/Problem 8|Solution]]
+[[2005 PMWC Problems/Problem I8|Solution]]
-== Problem 9 ==
+== Problem I9 ==
 There are four consecutive positive integers (natural numbers) less than <math>2005</math> such that the first (smallest) number is a multiple of <math>5</math>, the second number is a multiple of <math>7</math>, the third number is a multiple of <math>9</math> and the last number is a multiple of <math>11</math>. What is the first of these four numbers?
-[[2005 PMWC Individual Test Problems/Problem 9|Solution]]
+[[2005 PMWC Problems/Problem I9|Solution]]
-== Problem 10 ==
+== Problem I10 ==
 A long string is folded in half eight times, then cut in the middle. How many
 pieces are obtained?
-[[2005 PMWC Individual Test Problems/Problem 10|Solution]]
+[[2005 PMWC Problems/Problem I10|Solution]]
-== Problem 11 ==
+== Problem I11 ==
 There are 4 men: A, B, C and D. Each has a son. The four sons are asked to
 enter a dark room. Then A, B, C and D enter the dark room, and each of them
@@ Line 69: / Line 80: @@
 how many ways can this happen?
-[[2005 PMWC Individual Test Problems/Problem 11|Solution]]
+[[2005 PMWC Problems/Problem I11|Solution]]
-== Problem 12 ==
+== Problem I12 ==
+In the figure below, <math>BCDE</math> is a parallelogram, points <math>F</math> and <math>G</math> are on the
+segment <math>ED</math>, <math>BCA</math> is a right angled triangle, <math>AC</math> is perpendicular to <math>BC</math>.Suppose that <math>BC=8cm</math>, <math>AC=7cm</math>, and the area of the shaded regions is <math>12cm^2</math> more than that of the triangle <math>AFG</math>. What is the length of <math>CG</math>?
-[[2005 PMWC Individual Test Problems/Problem 12|Solution]]
+<asy>
+fill((0,0)--(1,1)--(3,1)--(2,0)--cycle,grey);
+fill((0,0)--(2,1.5)--(2,0)--cycle,white);
+draw((0,0)--(2,1.5)--(2,0)--cycle,dot);
+draw((0,0)--(1,1)--(3,1)--(2,0)--cycle,dot);
+MP("B",(0,0),W);MP("A",(2,1.5),N);MP("C",(2,0),S);
+MP("E",(1,1),NW);MP("D",(3,1),NE);
+MP("F",(1.5,1),NW);MP("G",(2,1),NE);
+draw((1.9,0)--(1.9,.1)--(2,.1),linewidth(.75));
+</asy>
-== Problem 13 ==
+[[2005 PMWC Problems/Problem I12|Solution]]
+== Problem I13 ==
 Sixty meters of rope is used to make three sides of a rectangular camping area with a long wall used as the other side. The length of each side of the rectangle is a natural number. What is the largest area that can be enclosed by the rope and the wall?
-[[2005 PMWC Individual Test Problems/Problem 13|Solution]]
+[[2005 PMWC Problems/Problem I13|Solution]]
-== Problem 14 ==
+== Problem I14 ==
 On a balance scale, three green balls balance six blue balls, two yellow balls
 balance five blue balls and six blue balls balance four white balls. How many blue balls are needed to balance four green, two yellow and two white balls?
-[[2005 PMWC Individual Test Problems/Problem 14|Solution]]
+[[2005 PMWC Problems/Problem I14|Solution]]
-== Problem 15 ==
+== Problem I15 ==
 The sum of the two three-digit integers, <math>\text{6A2}</math> and <math>\text{B34}</math>, is divisible by <math>18</math>. What is the largest possible product of <math>\text{A}</math> and <math>\text{B}</math>?
-[[2005 PMWC Individual Test Problems/Problem 15|Solution]]
+[[2005 PMWC Problems/Problem I15|Solution]]
+== Problem T1 ==
+Call an integer "happy", if the sum of its digits is <math>10</math>. How many
+"happy" integers are there between <math>100</math> and <math>1000</math>?
+[[2005 PMWC Problems/Problem T1|Solution]]
+== Problem T2 ==
+Compute the sum of <math>a</math>, <math>b</math> and <math>c</math> given that <math>\frac{a}{2}=\frac{b}{3}=\frac{c}{5}</math> and the product of <math>a</math>, <math>b</math> and <math>c</math> is <math>1920</math>.
+[[2005 PMWC Problems/Problem T2|Solution]]
+== Problem T3 ==
+Replace the letters <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math> in the following expression with
+the numbers <math>1</math>, <math>2</math>, <math>3</math> and <math>4</math>, without repetition:
+<cmath>a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d}}}</cmath>
+Find the difference between the maximum value and the minimum
+value of the expression.
+[[2005 PMWC Problems/Problem T3|Solution]]
+== Problem T4 ==
+Buses from town A to town B leave every hour on the hour
+(for example: 6:00, 7:00, …).
+Buses from town B to town A leave every hour on the half hour
+(for example: 6:30, 7:30, …).
+The trip between town A and town B takes 5 hours. Assume the
+buses travel on the same road.
+If you get on a bus from town A, how many buses from town B
+do you pass on the road (not including those at the stations)?
+[[2005 PMWC Problems/Problem T4|Solution]]
+== Problem T5 ==
+Mr. Wong has a <math>7</math>-digit phone number <math>\text{ABCDEFG}</math>. The sum of
+the number formed by the first <math>4</math> digits <math>\text{ABCD}</math> and the number
+formed by the last <math>3</math> digits <math>\text{EFG}</math> is <math>9063</math>. The sum of the number
+formed by the first <math>3</math> digits <math>\text{ABC}</math> and the number formed by the
+last <math>4</math> digits <math>\text{DEFG}</math> is <math>2529</math>. What is Mr. Wong’s phone number?
+[[2005 PMWC Problems/Problem T5|Solution]]
+== Problem T6 ==
+<cmath>
+\begin{eqnarray*}
++2 &=& 3 \\
++5+6 &=& 7+8 \\
++10+11+12 &=& 13+14+15\end{eqnarray*}</cmath>
+<cmath>\vdots</cmath>
+If this pattern is continued, find the last number in the <math>80</math>th row
+(e.g. the last number of the third row is <math>15</math>).
+[[2005 PMWC Problems/Problem T6|Solution]]
+== Problem T7 ==
+Skipper’s doghouse has a regular hexagonal base that measures
+one metre on each side. Skipper is tethered to a 2-metre rope
+which is fixed to a vertex. What is the area of the region outside
+the doghouse that Skipper can reach? Calculate an approximate
+answer by using <math>\pi=3.14</math> or <math>\pi=22/7</math>.
+[[2005 PMWC Problems/Problem T7|Solution]]
+== Problem T8 ==
+An isosceles right triangle is removed from each corner of a
+square piece of paper so that a rectangle of unequal sides remains.
+If the sum of the areas of the cut-off pieces is <math>200 \text{cm}^2</math> and the
+lengths of the legs of the triangles cut off are integers, find the
+area of the rectangle.
+[[2005 PMWC Problems/Problem T8|Solution]]
+== Problem T9 ==
+Select 8 of the 9 given numbers: 2, 3, 4, 7, 10, 11, 12, 13, 15 and
+place them in  the vacant squares so  that the average of  the
+numbers in  each  row and  column  is the same. Complete the
+following table.
+<math>\begin{tabular}{ |c | c | c | c |}
+     \hline
+&  &  &  \\ \hline
+      & 9 & & 5 \\ \hline
+      & & 14 &  \\ \hline
+   \end{tabular}</math>
+[[2005 PMWC Problems/Problem T9|Solution]]
+== Problem T10 ==
+Find the largest 12-digit number for which every two consecutive
+digits form a distinct 2-digit prime number.
+[[2005 PMWC Problems/Problem T10|Solution]]
 == See Also ==
 * [[Mathematics competition resources]]