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NAME:
PLEASE PRINT (First name Last name) M F
SCHOOL: GRADE:
(7,8,9)
You have 90 minutes for the examination. The test has
two parts: PART A — short answer; and PART B —
long answer. The exam has 9 pages including this one.
Each correct answer to PART A will score 5 points.
You must put the answer in the space provided. No
part marks are given.
Each problem in PART B carries 9 points. You should
show all your work. Some credit for each problem is
based on the clarity and completeness of your answer.
You should make it clear why the answer is correct.
PART A has a total possible score of 45 points. PART
B has a total possible score of 54 points.
You are permitted the use of rough paper.
Geome-try instruments are not necessary. References
includ-ing mathematical tables and formula sheets are not
permitted. Simple calculators without programming
or graphic capabilities are allowed. Diagrams are not
MARKERS’USE ONLY
PART A
5
B1
B2
B3
B4
B5
B6
TOTAL
(max: 99)
BE SURE TO MARK YOUR NAME AND SCHOOL AT THE TOP OF
THIS PAGE.
THE EXAM HAS 9 PAGES INCLUDING THIS COVER PAGE.
after this one with this same property?
turns (Aris going next), but each player’s choice of positive integer must be greater
than the previous player’s choice and no greater than twice the previous player’s
choice. The …rst player who can choose the number 2007 is the winner. What is
the largest number Aesha could choose at the beginning so that she would be sure of
3:43
centred atB and D. Find the area in square cm of the shaded region.
A B
C
D
4 cm
A B
C
D
, , L , ,
, , L , ,
How many little shaded squares are there in the 8th …gure in this pattern?
1
4+
4
48 =
48
144:
SupposeA andB are positive integers so that 1
5+
5
A
B:
Find a possible value forA.
37A062BC
is divisible by720:What isA?
A bus leavesAevery 8 minutes, starting at 8am each day, and takes 10 minutes
to reachB:
A train leavesB every 6 minutes, starting at 8am each day, and takes 12 minutes
to reachC:
Richard takes 5 minutes to reach Dfrom C:
What is the latest time Richard can catch the bus atA to reachD by 10am?
To reachDby 10 am, Richard must reach C by 9:55 am.
Trains leave B at 8:00, 8:06, 8:12, etc. This pattern continues to 9:36, 9:42, 9:48.
Since it takes 12 minutes to reachC;the 9:42 train reaches C at 9:54 and is the latest
train Richard can catch.
To reachB by 9:42, since the bus from AtoB takes 10 minutes, Richard must make
the latest bus leavingA before 9:32 am. Buses leaveA at 8:00, 8:08, 8:16 . . . . This
continues to 9:12, 9:20, 9:28, which is the latest bus.
We work backwards. Before Dori took his 40 candies, there are 49 candies. Since
Dori took 30% of what was remaining, 49 candies is 70% of what was remaining. Let
this number bex: Then <sub>10</sub>7x= 49: Hence x= 70:
Hence, before Diyao took 20 candies, there were 90 candies in the jar. Since Diyao
took 10% of the original number of candies, 90 candies is 90% of what was in the jar
originally. Let this number be y:
Then 9
10y = 90: Hence y=
The train is moving 4 times faster than Nahlah can run.
If she runs towards the train, they both get to the end of the bridge at the same
time.
If she runs away from the train, they both get to the beginning of the bridge at
the same time.
How far across the bridge is Nahlah?
A B C D
Begin Nahlah End Train
<i>x</i> 200−<i>x</i> 800−4<i>x</i>
Letx be the distance that Nahlah is across the bridge.
ThereforeAB =x and BC = 200 x; since the bridge is 200m long. Since Nahlah
can reachC the same time the train does, and the train is moving 4 times faster, then
CD= 4 BC = 4(200 x) = 800 4x:
Finally, since Nahlah can reach Athe same time the train does, then
(distance from DtoA)
(distance from B toA) =
AB+BC+CD
AB = 4
x+ 200 x+ 800 4x
x = 4
1000 4x
x = 4
1000 4x= 4x
8x= 1000
x= 125
Nahlah is
C
B
A
20 cm
20 cm
40 cm
C
B
A
20 cm
20 cm
40 cm
ground
F <b>A</b> G
E
H <sub>C</sub>
D
20 20
16
20
Label the points A; C; D; E; F; G; H as shown. The goal is to …nd the area of the
shaded regionCHEA and multiply the answer by the length of AB; which is 40cm,
to get the desired volume. Note thatCH is parallel to the ground.
First note that AG = pAC2 <sub>CG</sub>2 <sub>=</sub> p<sub>20</sub>2 <sub>16</sub>2 <sub>=</sub> p<sub>144 = 12</sub> <sub>by Pythagorean</sub>
Theorem.
Note that \DHC = \CAG by parallel lines, and \HDC = \AGC = 90 : Hence
HDC and AGC are similar.
Hence DC
DH =
GC
GA; so
20
DH =
16
12; so 16 DH= 240: Therefore DH= 15:
The area ofCHEAis [area ofACDE] [area of CDH]= 20 20 1<sub>2</sub> 15 20 = 250cm2:
(a) Show that $9.98 is an impossible price.
We start by listing possible prices. For example$9:00+0:54 = $9:54is a possible
price.
Let us try$9:40 as an original price. The tax is$9:40 6% = $0:564;which is
rounded down to 56 cents. The total price is then$9:40 +:56 = $9:96;which is
getting closer to$9:98:
Computing $9:41+ tax yields a tax of $9:41 6% = $0:5646 which is again
rounded down to 56 cents. The total price is$9:41 +:56 = $9:97.
Computing $9:42+ tax yields a tax of $9:42 6% = $0:5652which this time is
rounded up to 57 cents. The total price is$9:42 +:57 = $9:99:
The value $9.98 is skipped and thus is an impossible price.
(b) How many impossible prices are there less than or equal to $10.00? That is,
how many of the prices
1/c; 2/c; : : : ; 99/c; $1:00; $1:01; : : : ; $9:99; $10:00
are impossible?
Note that$10:00 = $9:43+tax, since $9:43 6% = $0:5658which is rounded up
to57 cents, so$9:43+tax is$9:43 +:57 = $10:00.
Thus the possible prices are 1/c+ tax, 2/c+ tax, 3/c+ tax, . . . up to $9:43+ tax.
There are943such prices, all between 1/c and $10:00 = 1000/c.
17 16 15 14 13 ...
20 7 8 9 10 27
21 22 23 24 25 26
Somewhere in this grid, the number 2007 is surrounded by eight numbers as shown.
a b c
d 2007 e
f g h
What is thesmallest of these eight numbers?
The pattern to look for is the diagonal1;9;25; : : :starting at the square containing 1
and going in the down-right direction. This pattern consists of all of the odd perfect
squares. Particularly, this diagonal contains the entry452 = 2025:
The numbers1;2;3; : : : ;2025form a45 45square in the spiral and 2007is18away
from 2025: Since the spiral approaches 2025 from the left, the square containing
2007 is 18 to the left of the square containing 2025: In the 3 3 subgrid shown
Now look at the diagonal containing the odd perfect squares. The entry on this
diagonal in the same row asais432= 1849: Furthermore, the square containingais
also 18squares to the left of the square containing 1849:
Thereforea= 1849 18 =
a 1849
(= 432)
2007 2008 2024 2025