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 p[r]
Trang 131 JUNIOR HIGH SCHOOL MATHEMATICS CONTEST
May 2, 2007
(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
drawn to scale They are intended as visual hints only
When the teacher tells you to start work you should
read all the problems and select those you have the
best chance to do …rst You should answer as many
problems as possible, but you may not have time to
answer all the problems
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
Please return the entire exam to your supervising teacher
at the end of 90 minutes
Trang 2PART A: SHORT ANSWER QUESTIONS
A1 Exactly two numbers are removed from the set f1; 2; 3; 4; 5; 6; 7; 8; 9; 10g and the sum
8, 10
of the remaining eight numbers is 37 Which two numbers were removed?
A2 What is the 2007th letter of the sequence
H
ILOVEMATHILOVEMATHILOVEMATH : : :?
A3 The date August 28, 2006 has the property that when this date is written in the Feb 2,
2008
format MMDDYYYY, all eight digits are even, i.e 08282006 What is the next date
after this one with this same property?
A4 Aesha and Aris play a game where they take turns choosing positive integers Aesha
1003
starts by choosing a positive integer smaller than 2007 After that the players take
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
winning?
A5 In the following diagram, ABCD is a square and the curved lines are quarter circles 16 - 4
3:43 centred at B and D Find the area in square cm of the shaded region
C D
4 cm
C D
4 cm
Trang 3A6 Consider the following pattern of …gures.
113
How many little shaded squares are there in the 8th …gure in this pattern?
A7 Notice that
100
(600 or 3100 are also cor-rect)
1
2 +
2
4 =
4
4;
1
3+
3
18 =
18
36; and
1
4+
4
48 =
48
144: Suppose A and B are positive integers so that 1
5+
5
A =
A
B: Find a possible value for A
A8 A; B; C are digits so that the 8-digit (base 10) number
5
37A062BC
is divisible by 720:What is A?
A9 A rectangle and a semicircle are drawn as follows:
17
A
B
C
D
A
B
C
D
so that AB = 4 and AC = 1: Find BD
Trang 4PART B: LONG ANSWER QUESTIONS
B1 Richard needs to get from point A to point D by taking a bus from A to B; a train from B to C and walking from C to D: He knows the following:
A bus leaves A every 8 minutes, starting at 8am each day, and takes 10 minutes
to reach B:
A train leaves B every 6 minutes, starting at 8am each day, and takes 12 minutes
to reach C:
Richard takes 5 minutes to reach D from C:
What is the latest time Richard can catch the bus at A to reach D by 10am?
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SOLUTION:
To reach D by 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 reach C; the 9:42 train reaches C at 9:54 and is the latest train Richard can catch
To reach B by 9:42, since the bus from A to B takes 10 minutes, Richard must make the latest bus leaving A before 9:32 am Buses leave A at 8:00, 8:08, 8:16 This continues to 9:12, 9:20, 9:28, which is the latest bus
Therefore, the latest time Richard can catch the bus is 9:28 am.
Trang 5B2 There is a jar of candies Diyao takes 10% of the candies plus 20 more candies Then Dori takes 30% of the remaining candies plus 40 more candies There are then just
9 candies left How many candies were in the jar at the beginning?
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SOLUTION:
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 be x: Then 107x = 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 =100:
This problem can also be done by guess and check
Trang 6B3 Nahlah is walking on the footpath of a train track crossing a 200m bridge She sees a train coming towards her from ahead of her and immediately deduces the following: 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?
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SOLUTION:
Let x be the distance that Nahlah is across the bridge
Therefore AB = x and BC = 200 x; since the bridge is 200m long Since Nahlah can reach C 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 A the same time the train does, then
(distance from D to A) (distance from B to A) =
AB + BC + CD
x + 200 x + 800 4x
1000 4x
1000 4x = 4x
8x = 1000
x = 125 Nahlah is 125m across the bridge
This problem can also be done by guess and check
Trang 7B4 A water tank with dimensions 20cm 20cm 40cm, as shown, has an open top It
is tilted so that AB is touching the ground and C is 16cm above the ground What
is the greatest volume of water (in cubic cm) that can be placed in the tank in this state, without spilling?
C
B A
20 cm
20 cm
40 cm
C
B A
20 cm
20 cm
40 cm
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SOLUTION:
ground
E
D
16 20
Label the points A; C; D; E; F; G; H as shown The goal is to …nd the area of the shaded region CHEA and multiply the answer by the length of AB; which is 40cm,
to get the desired volume Note that CH is parallel to the ground
First note that AG = p
AC2 CG2 = p
202 162 = p
144 = 12 by Pythagorean Theorem
Note that \DHC = \CAG by parallel lines, and \HDC = \AGC = 90 : Hence HDC and AGC are similar
DH =
GC
GA; so
20
DH =
16
12; so 16 DH = 240: Therefore DH = 15: The area of CHEA is [area of ACDE] [area of CDH] = 20 20 12 15 20 = 250cm2: Therefore the greatest volume of water that can be placed in the tank
is 250 40 =10,000cm3:
Trang 8B5 In Alberta, a 6% tax is added to the cost of all purchases If an item costs x dollars, the tax is computed by calculating 0:06x; rounded to the nearest cent (with half cents rounded up) A price is called impossible if it cannot be the price of an item after tax
is added
(a) Show that $9.98 is an impossible price
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SOLUTION:
We start by listing possible prices For example $9:00+0:54 = $9:54 is 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% = $0:5652 which 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?
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SOLUTION:
Note that $10:00 = $9:43+ tax, since $9:43 6% = $0:5658 which is rounded up
to 57 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 are 943 such prices, all between 1/c and $10:00 = 1000/c
Therefore, the number of impossible prices in this range is 1000 943 = 57.
Trang 9B6 The sequence of positive integers 1; 2; 3; 4; : : : is written in a spiral in an in…nite grid
in the following fashion
17 16 15 14 13 .
21 22 23 24 25 26 Somewhere in this grid, the number 2007 is surrounded by eight numbers as shown
d 2007 e
What is the smallest of these eight numbers?
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SOLUTION:
The pattern to look for is the diagonal 1; 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 entry 452 = 2025:
The numbers 1; 2; 3; : : : ; 2025 form a 45 45 square in the spiral and 2007 is 18 away 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 above, each of the three rows is increasing from left to right and each of the three columns is increasing from top to bottom Therefore, the smallest of the eight numbers surrounding 2007 is a:
Now look at the diagonal containing the odd perfect squares The entry on this diagonal in the same row as a is 432= 1849: Furthermore, the square containing a is also 18 squares to the left of the square containing 1849:
Therefore a = 1849 18 = 1831.
(= 432)
(= 452)