junior mathematics
COMPETITION 2006
TIME ALLOWED: ONE HOUR
Only Year 9 candidates may attempt QUESTION ONE
ALL candidates may attempt QUESTIONS TWO to FIVE
These questions are designed to test ability to analyse a problem and to
express a solution clearly and accurately.
Please read the following instructions carefully before you begin:
(1) Do as much as you can. You are not expected to complete the entire paper. In the past, full
answers to three questions have represented an excellent effort.
(2) You must explain your reasoning as clearly as possible, with a careful statement of the main points
in the argument or the main steps in the calculation. Generally, even a correct answer without any
explanation will not receive more than half credit. Likewise, clear and complete solutions to two
problems will generally gain more credit than sketchy work on four.
(3) Credit will be given for partial solutions and evidence of a serious attempt to tackle a problem.
(4) Textbooks are NOT allowed. Calculators, tables, etc. may be used (but should not be necessary).
Otherwise normal examination conditions apply.
(5) Diagrams are a guide only and are not necessarily drawn to scale.
University of Otago
Department of
Mathematics and Statistics
PLEASE TURN OVER
Question 1 (Year 9, Form 3 only)
The 2006 Kakanui Swimming Carnival was a great success. The only problem came in the Girl’s Diving
Contest. Unfortunately, while the judges were adding up the marks the computer broke down. The five
girls had to wait for 20 minutes before the results were announced.
While they were waiting, each girl made two statements about how they thought the contest had gone:
Aroha said: Beth was first; Dora was last
Beth said: I was second; Aroha was third
Claire said: I was third; Dora was fourth
Dora said: Beth was third; Aroha was fourth
Emma said: I was first; Claire was last
The results were finally announced, and two facts were discovered. First, there were no ties, and second,
to everyone’s surprise it turned out that each girl had made one true statement and one false statement
(not necessarily in that order).
(a) One girl made statements that prove that Beth was not third. Who was she? In a sentence or two
give a brief reason for your answer.
(b) Who was fourth? In a sentence or two give a brief reason for your answer.
(c) Who was last? In a sentence or two give a brief reason for your answer.
(d) Write down, in order from first to last, the placings of the five girls. (In this part of the question
you do not have to give a reason for your answer.)
Question 2
Note: In this question the word “number” means a positive whole number, that is, a member of the set
{1, 2, 3, . . . }.
I am thinking of a number. When I divide the number by 5 the remainder is 4, and when I divide the
number by 7 the remainder is 6.
(a) What is the smallest possible number I could be thinking of?
(b) What is the second smallest possible number I could be thinking of?
(c) Are there an infinite number of such numbers? If the answer is “Yes”, list in order the first five
such numbers. If the answer is “No”, write down the largest possible number.
Now I am thinking of another number. When I divide the number by 2 the remainder is 1. When I divide
the number by 3, the remainder is 2. When I divide the number by 4, the remainder is 3. I continue in
this way until I stop at 9, that is, when I divide the number by 9, the remainder is 8.
(d) What is the smallest possible number I could be thinking of?
(e) Are there an infinite number of such numbers? If the answer is “Yes”, list in order the first three
such numbers. If the answer is “No”, write down the largest possible number.
Question 3
The 2006 Canterbury Duo–Marathon promises to be an exciting race. Each team has two members who
compete together by running and bicycling over the course, which is 42 km long. Here are some of the
rules for the race:
• Each team starts together at the start line.
• Each team of two is allowed only one bicycle. Only one team member may ride the bicycle at any
time. (This usually means that one member bikes at the start, puts the bicycle down at some point
on the course, and then runs to the finish line. The other team member starts by running. When he
or she reaches the bicycle, they pick it up and bike to the finish line.)
• Both team members have to cross the finish line.
• The time recorded for a team is the time for the second member to cross the finish line, or the time
for both members if they cross the line together. (If one team member crosses the line ahead of the
second member, only the second time counts. This means that teams should work out before the
race starts exactly where they should leave the bicycle to give the fastest time for the second
member.)
The table shows the steady running and bicycling
Team Members Running Bicycling
speeds, all in km/h, for three teams. Assume that
A Daniel 12 28
these are the speeds for the team members in the
Susan 12 28
actual 42 km race.
B Ed 16 35
Hazel 10 15
We need to know what distance each member
C Belinda 10 35
should run and bicycle so that each team records
Charles 14 25
their best possible time for the race.
For each team give the distances, the best possible time, and reasoning for your answer.
(a) Team A
(b) Team B
(c) Team C
PLEASE TURN OVER
Question 4
There are 90 four–digit numbers which have the form abba. The largest of them is 9999, where a = 9 and
b = 9, while the smallest is 1001 with a = 1 and b = 0. One interesting property of the number 1001 is
that it is divisible by 11 since 1001 = 7 × 11 × 13.
(a) How many four digit numbers of the form abba are divisible by 11? Explain your answer.
(b) How many four digit numbers of the form abba are divisible by 15? Explain your answer.
(c) How many four digit numbers of the form abba are divisible by 77? Explain your answer.
Question 5
The famous carpet salesman Al Laddin has just purchased a rectangular carpet. Unfortunately he has lost
his measuring tape so he doesn’t know the size of this new carpet. However, when he lays the carpet
down in his rectangular 11 m by 10 m storeroom, the four corners of the carpet just touch the four sides of
the storeroom at points which are a whole number of metres away from the corners of the storeroom.
(Before Al lost his measuring tapes he had placed marks at one metre intervals along the walls.)
One of the corners of the carpet is exactly 4 metres from a corner of the storeroom on the 10 metre side of
the storeroom (see the diagram).
(a) Find the lengths of the base and the height of 11
the carpet.
4
(b) Is your answer to (a) the only possible answer?
Explain your reasoning. 10
The Storeroom (not to scale)
