2003
Australian Intermediate Mathematics Olympiad
1. Whee Bin's Chinese restaurant serves dishes in three sizes: small dishes cost $z, medium
dishes cost Sy and large dishes cost $2, where x < y < 2 and C, y and 2 are positive
integers.
Last Friday, Giovanna, Peter and John ordered a total of 9 small dishes, 6 medium dishes
and 10 large dishes.
John remarked: "This bill is exactly twice as much as when was here 2 nights ago'.
Peter remarked: 'This bill is exactly three times as large as when Iwas here last night'.
Giovanna said, 'It is still a good price; the total is less than $100'.
The prices have not been changed this week. How many dollars was the bill?
2 marks, 597/756]
2. Annie, Bruce and Ken play a game
of cards, each beginning and ending the game with a
whole number of dollars. Annie starts with $3
for every $5 Bruce has, and finishes with
twice as much money as Bruce. Ken starts with the same amount of money as Annie
and ends with
the same amount of money as Bruce. What is the least number of dollars
that Bruce could have lost?
(2 marks, 498/756]
3. For how many integers between n- +4
and 2003 is the improper fraction not in
n+5
simplest form?
[3 marks, 18/756]
4. ABC is a triangle. AB is produced to P so that PB = AB. BC is produced to Q so
that QC = CB. CA is produced to R so that RA = AC. The area of ABC is 51. What
is the area of PQR?
3 marks, 283/756]
5. Rina sets off on her bike to Con's place. At exactly the same moment, Con sets off
to Rina's place along the same straight road in his car. A while later, they pass each
other (neither spotting the other) and shortly after, Con arrives at Rina's place to find
that she is not there. Con waits 22 minutes and then heads back along the same road,
arriving at his place at exactly the same time as Rina. Rina travelled at the same speed
the whole time whereas Con travelled 4 times as fast as Rina on the way to her place
and 5 times as fast on the way back. How many minutes did it take Rina to reach Con's
place?
3 marks, 495/756]
6. Faith designs the following logo for a telephone company. The
logo consists of a T divided into 5 regions. Faith has to colour
each region with one of three colours: red, green or gold. No
two regions with a common edge can have the same colour; and
each colour must be used at least once. How many possible colour
schemes are available to Faith for her logo?
[3 marks, 190/756
11
2003
12
club, Andy, Becky, Chloe
school mathematics
of their is the same.
and
a meeting
number (a positive integer) The
7. During
other
discover that their favourite the number. So, each of the fourfriendsmakes cht
know
members would like to which is true and at least one of three
least one of
statements about
the number,
at
which
false.
The number is less than 12.
Andy: the number.
7 does not divide
is less than 70.
5 times the number
number is greater than 1000.
12 times the
Becky:
10 divides the number.
than 100.
The number is greater
Chloe: 4 divides the number.
11 times the number is less than 1000.
9 divides the number.
Danny: (D1) The number is less than 20.
(D2) The number is a prime number.
(D3) 7 divides the number.
What is their favourite number?
[5 marks, 490/756
8. ABCD is a trapezium with AB parallel to DC. A semicircle is constructed with centre
on AB and with the other three sides of the trapezium are all tangents to it. AB = 1156
and AD = 784. Find BC.
[5 marks, 55/756
9. AB is a diameter of a circle. PQ is a chord of the circle, perpendicular to AB and nearer
toB, cutting AB at V. M is any point on AV. QM produced cuts the circle at R.
[4 marks, mean 0.5
10. The 101 numbers 1, 2, ..., 101 are split into two groups. Group A contains m of the
numbers and group B contains the remaining (101 m). When the number 40
iS
removed from the group it is in and placed in the rises
other mean
group, each group
by 1/2. What is the value of m?
0.3
[5 marks, mean
Investigation
a Find an actual
set of numbers in group to
the other raises each each group for which moving 40 from one
group's mean by 1/2.
Suppose we start with
the numbers 1, 2,... , k.
b
What relationships hold between k m, the number r that is moved, and the amount
by which
y
the average rises?
¢ Is there an
upper limit on the
amount by which the average can rise?
