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Difference between revisions of "2005 PMWC Problems"

m (2005 PMWC Individual Test moved to 2005 PMWC: To put the team contests and individual contests on the ame page)
(individual probs I1,I2,I3...)
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== Problem 1 ==
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== 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>?
 
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>?
  
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[[2005 PMWC Individual Test Problems/Problem 1|Solution]]
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[[2005 PMWC Problems/Problem I1|Solution]]
  
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== Problem 2 ==
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== 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).  What is the number <math>c</math>?
  
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[[2005 PMWC Individual Test Problems/Problem 2|Solution]]
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[[2005 PMWC Problems/Problem I2|Solution]]
  
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== Problem 3 ==
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== 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>?
 
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>?
  
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[[2005 PMWC Individual Test Problems/Problem 3|Solution]]
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[[2005 PMWC Problems/Problem I3|Solution]]
  
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== Problem 4 ==
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== Problem I4 ==
  
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[[2005 PMWC Individual Test Problems/Problem 4|Solution]]
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[[2005 PMWC Problems/Problem I4|Solution]]
  
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== Problem 5 ==
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== Problem I5 ==
 
Consider the following conditions on the positive integer (natural number) <math>a</math>:
 
Consider the following conditions on the positive integer (natural number) <math>a</math>:
  
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If only three of these conditions are true, what is the value of <math>a</math>?
 
If only three of these conditions are true, what is the value of <math>a</math>?
  
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[[2005 PMWC Individual Test Problems/Problem 5|Solution]]
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[[2005 PMWC Problems/Problem I5|Solution]]
  
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== Problem 6 ==
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== 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?
 
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?
  
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[[2005 PMWC Individual Test Problems/Problem 6|Solution]]
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[[2005 PMWC Problems/Problem I6|Solution]]
  
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== Problem 7 ==
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== 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>?
 
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>?
  
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[[2005 PMWC Individual Test Problems/Problem 7|Solution]]
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[[2005 PMWC Problems/Problem I7|Solution]]
  
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== Problem 8 ==
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== Problem I8 ==
 
Some people in Hong Kong express <math>2/8</math> as 8th Feb and others express <math>2/8</math> as
 
Some people in Hong Kong express <math>2/8</math> as 8th Feb and others express <math>2/8</math> as
 
2nd Aug. This can be confusing as when we see <math>2/8</math>, we don’t know whether it
 
2nd 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?
 
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?
  
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[[2005 PMWC Individual Test Problems/Problem 8|Solution]]
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[[2005 PMWC Problems/Problem I8|Solution]]
  
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== Problem 9 ==
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== 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?
 
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?
  
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[[2005 PMWC Individual Test Problems/Problem 9|Solution]]
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[[2005 PMWC Problems/Problem I9|Solution]]
  
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== Problem 10 ==
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== Problem I10 ==
 
A long string is folded in half eight times, then cut in the middle. How many
 
A long string is folded in half eight times, then cut in the middle. How many
 
pieces are obtained?
 
pieces are obtained?
  
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[[2005 PMWC Individual Test Problems/Problem 10|Solution]]
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[[2005 PMWC Problems/Problem I10|Solution]]
  
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== Problem 11 ==
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== Problem I11 ==
 
There are 4 men: A, B, C and D. Each has a son. The four sons are asked to
 
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
 
enter a dark room. Then A, B, C and D enter the dark room, and each of them
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how many ways can this happen?
 
how many ways can this happen?
  
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[[2005 PMWC Individual Test Problems/Problem 11|Solution]]
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[[2005 PMWC Problems/Problem I11|Solution]]
  
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== Problem 12 ==
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== Problem I12 ==
  
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[[2005 PMWC Individual Test Problems/Problem 12|Solution]]
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[[2005 PMWC Problems/Problem I12|Solution]]
  
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== Problem 13 ==
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== 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?
 
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?
  
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[[2005 PMWC Individual Test Problems/Problem 13|Solution]]
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[[2005 PMWC Problems/Problem I13|Solution]]
  
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== Problem 14 ==
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== Problem I14 ==
 
On a balance scale, three green balls balance six blue balls, two yellow balls
 
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?
 
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?
  
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[[2005 PMWC Individual Test Problems/Problem 14|Solution]]
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[[2005 PMWC Problems/Problem I14|Solution]]
  
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== Problem 15 ==
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== 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>?
 
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>?
  
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[[2005 PMWC Individual Test Problems/Problem 15|Solution]]
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[[2005 PMWC Problems/Problem I15|Solution]]
  
 
== See Also ==
 
== See Also ==
  
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]

Revision as of 11:54, 30 September 2007

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). What is the number $c$?

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

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

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

See Also

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