JUNIOR HIGH SCHOOL MATHEMATICS CONTEST May 2, 2007
ST THE CALGARY MATHEMATICAL ASSOCIATION
31 JUNIOR HIGH SCHOOL MATHEMATICS CONTEST May 2, 2007
NAME: GENDER:
SOLUTIONS PLEASE PRINT (First name Last name) M F
SCHOOL: GRADE:
(7,8,9)
You have 90 minutes for the examination. The test has MARKERS’ USE ONLY two parts: PART A — short answer; and PART B — long answer. The exam has 9 pages including this one.
PART A Each correct answer to PART A will score 5 points. You must put the answer in the space provided. No
5 part marks are given. Each problem in PART B carries 9 points. You should
B1 show all your work. Some credit for each problem is based on the clarity and completeness of your answer.
B2 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.
B3 PART 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 of the remaining eight numbers is 37. Which two numbers were removed?
8, 10 th
What is the 2007 letter of the sequence
A2
ILOVEMATHILOVEMATHILOVEMATH : : :?
H
The date August 28, 2006 has the property that when this date is written in the
A3 Feb 2,
format MMDDYYYY, all eight digits are even, i.e. 08282006. What is the next date
2008
after this one with this same property? Aesha and Aris play a game where they take turns choosing positive integers. Aesha
A4
Consider the following pattern of …gures.
A6 113
, , , , , , , , L L th L L
How many little shaded squares are there in the 8 …gure in this pattern? Notice that
A7
1
2
4
1
3
18
1
4
48 = + + + ; = ; and = :
2
4
4
3
18
36
4 48 144
100
A
1
5 :
Suppose A and B are positive integers so that + = A B 5 (600 or 3100
Find a possible value for A. are also cor- rect)
Richard needs to get from point A to point D by taking a bus from A to B; a train
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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
There is a jar of candies. Diyao takes 10% of the candies plus 20 more candies. Then
B2
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 7 x this number be x: Then = 49: Hence x = 70: 10 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:
9 y : Then = 90: Hence y =
100
10 This problem can also be done by guess and check. Nahlah is walking on the footpath of a train track crossing a 200m bridge. She sees a
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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: x 200 x 800 4 x − −
A B C D Begin Nahlah End Train Let x be the distance that Nahlah is across the bridge.
x; Therefore AB = x and BC = 200 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 x
= 4 BC = 4(200 ) = 800 4x: Finally, since Nahlah can reach A the same time the train does, then
(distance from D to A) AB + BC + CD A water tank with dimensions 20cm 20cm 40cm, as shown, has an open top. It
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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 C
20 cm 20 cm 20 cm 20 cm
B B
40 cm 40 cm
A A
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SOLUTION: D H
20 C E In Alberta, a 6% tax is added to the cost of all purchases. If an item costs x dollars,
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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, The sequence of positive integers 1; 2; 3; 4; : : : is written in a spiral in an in…nite grid
B6 in the following fashion.
..
17
16
15
14 13 .
..
18
5
4
3 12 .
..
19
6
1
2 11 .
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 e
2007 f g h 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 2 squares. Particularly, this diagonal contains the entry 45 = 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