junior mathematics

COMPETITION 2007

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 may be used but students who do not have one
should not be disadvantaged.. 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

DO NOT TURN THIS OVER

UNTIL YOU ARE TOLD.
 8
Question 1 (Year 9 or below only)
1 3
The diagram at the right show a Magic Diamond.
Each circle is supposed to contain one of the numbers 5
from 1 to 9, with no repeats. However, some numbers
are missing from the diagram. 7
In a Magic Diamond the three numbers along each line
always add up to the same total.

(a) Write down in order from smallest to largest the four missing numbers from the Magic
Diamond.
(b) What is the total which the three numbers along each line always add up to?
(c) Carefully copy the Magic Diamond above into your Answer Booklet and write in the
missing numbers so that the three numbers along each line always add up to the total you
wrote down in (b).

Marlon tries to draw a larger Magic Diamond. His Magic Diamond contains all the numbers
from 1 to 13. Marlon’s Magic Diamond is shown below. Once again some of the numbers are
missing. The numbers (sometimes three of them and sometimes four) along each line always
add up to the same total.
10

12 11

13

(d) What is the total which the numbers along each line always add up to?
(e) Carefully copy Marlon’s Magic Diamond into your Answer Booklet and write in the
missing numbers so that the numbers along each line always add up to the total you
wrote down in (d).
Question 2 (All Students)

Sally stores mathematical expressions in her calculator. She uses special function buttons
f1 f2 etc.
to store expressions which she wants to use later.
One day her little brother Robert is looking at her calculator and he asks Sally about the
special function buttons. Sally shows him how they are used (although she doesn’t use any
fractions at all because Robert doesn’t really understand fractions yet).

Sally tells Robert that the expression she stored using f1 is 4x2. This means that if Robert
enters a number into this expression, the number will be squared and multiplied by 4.
(a) Write down the result when Robert enters the number 3 into the expression Sally stored
using f1.

(b) When Robert presses INV f1 the calculator undoes the expression stored using f1. This
means that if Robert uses INV f1 for the result he obtained in part (a) then the answer
would be 3. Write down the result when Robert uses INV f1 for the number 64.

Sally doesn’t tell Robert what expression she stored using f2, but she does tell him that if he
enters 2 and uses f2 the result will be 12, while if he enters 3 and uses f2 the result will be 36.

(c) Find a possible result when the number 4 is entered using the stored result f2. Explain
what expression has been stored using f2 to give your result.
(d) Is the result you found in (c) the only possible result? If it is, explain why. If it isn’t,
write down another possible result when the number 4 is entered using f2 and also give
the expression you used to obtain your new result.

Question 3 (All Students)

For this question, assume that the planet Earth is a perfect sphere. During November 2006,
residents of Dunedin were able to take helicopter rides to look at several icebergs which were
floating off the coast.
(a) One day, when conditions were clear and sharp, viewers were able to see an iceberg on
the horizon from the top of a local hill, Flagstaff, which is 660 m above sea level. If the
distance in a direct line from Flagstaff to the iceberg was estimated to be 90 km, use this
information to show that the radius of the Earth from the centre of the planet to sea level
is between 6000 km and 7000 km.
(b) If the iceberg broke off from Antarctica and then travelled in a straight line (relative to
the surface) covering one twelfth 1 ( )
12
of the circumference of the Earth before it was
seen near Dunedin, find the distance it had travelled before it was seen. (Note that if you
do not have a calculator then you may use the values 6000 km for the radius of the Earth
and π = 3.14.)

PLEASE TURN OVER

Question 4 (All Students)

The 2007 Kakanui Bowls tournament was a knock-out competition. This means that if there
were exactly 32 competitors then the first round of games consisted of each person playing
against someone else, with the loser of each game being eliminated. The 16 competitors left
then played in the next round, with the loser of each of these eight games being eliminated.
This arrangement continued until only two players were left. The winner of this final game
won the tournament.

However if there were extra competitors, for example an extra three (making 35 altogether),
then six competitors first of all had to play a preliminary game (three games would be needed)
to select the three competitors who would go through to be part of the last 32.

(a) How many games were there altogether if there were 35 competitors?

(b) When the entries closed it was discovered that there were 2007 competitors. The knock-
out method described above was employed for this larger number of competitors. Boris,
the eventual winner of the tournament, was one of those who first had to play a
preliminary game. How many games did Boris have to play?

(c) How many games were there altogether for this number (2007) of competitors?

(d) If there are n full rounds of games plus m preliminary games, write down an expression
for the total number of games which must be played in the tournament.

Question 5 (All Students)

(a) Paul has a sheet of graph paper which has 6 squares along each side. He draws squares
of side lengths 1 through to 5 on this paper, along the lines. Show that no matter how he
does this there will be a square on the paper which is part of at least three of the squares
that he has drawn.

(b) Paul now takes another sheet of graph paper which has 40 squares along each side. On
this sheet he draws squares of side lengths 1 through to 39 on this paper, along the lines.
Show that no matter how he does this there will be a square on the paper which is part of
at least 20 of the squares that he has drawn.

(c) For the situation in (b), is it possible for there to be exactly one square of this sort, i.e.
exactly one square which is part of at least 20 of the squares that he has drawn?