Difference between revisions of "2005 PMWC Problems"
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Revision as of 02:44, 6 October 2014
The following is a list of PMWC problems from the year 2005
Contents
- 1 Problem I1
- 2 Problem I2
- 3 Problem I3
- 4 Problem I4
- 5 Problem I5
- 6 Problem I6
- 7 Problem I7
- 8 Problem I8
- 9 Problem I9
- 10 Problem I10
- 11 Problem I11
- 12 Problem I12
- 13 Problem I13
- 14 Problem I14
- 15 Problem I15
- 16 Problem T1
- 17 Problem T2
- 18 Problem T3
- 19 Problem T4
- 20 Problem T5
- 21 Problem T6
- 22 Problem T7
- 23 Problem T8
- 24 Problem T9
- 25 Problem T10
- 26 See Also
Problem I1
What is the greatest possible number one can get by discarding digits, in any order, from the number
?
Problem I2
Let , where
and
are different four-digit positive integers (natural numbers) and
is a five-digit positive integer (natural number). What is the number
?
Problem I3
Let be a fraction between
and
. If the denominator of
is
and the numerator and denominator have no common factor except
, how many possible values are there for
?
Problem I4
The larger circle has radius 12 cm. Each of the six identical smaller circles touches its two neighbours and the larger circle. What is the radius of the smaller circle?
Problem I5
Consider the following conditions on the positive integer (natural number) :
1.
2.
3.
4.
5.
If only three of these conditions are true, what is the value of ?
Problem I6
A group of people consists of men, women and children (at least one of each). Exactly
apples are distributed in such a way that each man gets
apples, each woman gets
apples and each child gets
apple. In how many possible ways can this be done?
Problem I7
How many numbers are there in the list which contain exactly two consecutive
's such as
and
, but not
or
?
Problem I8
Some people in Hong Kong express as 8th Feb and others express
as
2nd Aug. This can be confusing as when we see
, we don’t know whether it
is 8th Feb or 2nd Aug. However, it is easy to understand
or
as 22nd Sept, because there are only
months in a year. How many dates in a year can cause this confusion?
Problem I9
There are four consecutive positive integers (natural numbers) less than such that the first (smallest) number is a multiple of
, the second number is a multiple of
, the third number is a multiple of
and the last number is a multiple of
. What is the first of these four numbers?
Problem I10
A long string is folded in half eight times, then cut in the middle. How many pieces are obtained?
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?
Problem I12
In the figure below, is a parallelogram, points
and
are on the
segment
,
is a right angled triangle,
is perpendicular to
.Suppose that
,
, and the area of the shaded regions is
more than that of the triangle
. What is the length of
?
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?
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?
Problem I15
The sum of the two three-digit integers, and
, is divisible by
. What is the largest possible product of
and
?
Problem T1
Call an integer "happy", if the sum of its digits is . How many
"happy" integers are there between
and
?
Problem T2
Compute the sum of ,
and
given that
and the product of
,
and
is
.
Problem T3
Replace the letters ,
,
and
in the following expression with
the numbers
,
,
and
, without repetition:
Find the difference between the maximum value and the minimum
value of the expression.
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)?
Problem T5
Mr. Wong has a -digit phone number
. The sum of
the number formed by the first
digits
and the number
formed by the last
digits
is
. The sum of the number
formed by the first
digits
and the number formed by the
last
digits
is
. What is Mr. Wong’s phone number?
Problem T6
\begin{eqnarray*} 1+2 &=& 3 \\ 4+5+6 &=& 7+8 \\ 9+10+11+12 &=& 13+14+15 (Error compiling LaTeX. Unknown error_msg)
If this pattern is continued, find the last number in the
th row
(e.g. the last number of the third row is
).
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 or
.
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 and the
lengths of the legs of the triangles cut off are integers, find the
area of the rectangle.
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.
Problem T10
Find the largest 12-digit number for which every two consecutive digits form a distinct 2-digit prime number.