Đề thi Toán quốc tế CALGARY năm 2007

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]

31

JUNIOR HIGH SCHOOL MATHEMATICS CONTEST

May 2, 2007

NAME:

SOLUTIONS

GENDER:

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 pages including this one Each correct answer to PART A will score points You must put the answer in the space provided No part marks are given

Each problem in PART B carries 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 …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 PAGES INCLUDING THIS COVER PAGE

PART A:

SHORT ANSWER QUESTIONS

A1

Exactly two numbers are removed from the setf1;2;3;4;5;6;7;8;9;10g and the sum

8, 10

of the remaining eight numbers is37 Which two numbers were removed?

A2

What is the2007th 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 -

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

A6

Consider the following pattern of …gures

113

, , L , ,

L

, , L , ,

L

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) + = 4; 3+ 18 = 18 36; and

1 4+ 48 = 48 144:

SupposeA andB are positive integers so that

5+ A =

A B:

Find a possible value forA

A8

A; B; C are digits so that the 8-digit (base 10) number

5

37A062BC

is divisible by720:What isA?

A9

A rectangle and a semicircle are drawn as follows:

PART B:

LONG ANSWER QUESTIONS

B1

Richard needs to get from point A to point Dby taking a bus from A to B;a train fromB toC and walking from C toD:He knows the following:

A bus leavesAevery minutes, starting at 8am each day, and takes 10 minutes to reachB:

A train leavesB every minutes, starting at 8am each day, and takes 12 minutes to reachC:

Richard takes minutes to reach Dfrom C:

What is the latest time Richard can catch the bus atA to reachD by 10am?

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

SOLUTION:

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

B2

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 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 bex: 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

10y = 90: Hence y=

100

:

B3

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 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:

A B C D

Begin Nahlah End Train

x 200−x 800−4x

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 times faster, then

CD= 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 =

x+ 200 x+ 800 4x

x =

1000 4x

x =

1000 4x= 4x 8x= 1000

x= 125

Nahlah is

125m

across the bridge

B4

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

F A G

E

H C

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 CG2 = p202 162 = p144 = 12 by Pythagorean

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 12 15 20 = 250cm2:

B5

In Alberta, a 6% tax is added to the cost of all purchases If an item costsx dollars, the tax is computed by calculating0:06x;rounded to the nearest cent (with half cents rounded up) Aprice 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 = $9:54is 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?

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

SOLUTION:

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

B6

The sequence of positive integers1;2;3;4; : : : is written in a spiral in an in…nite grid in the following fashion

17 16 15 14 13 18 12 19 11

20 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?

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

SOLUTION:

The pattern to look for is the diagonal1;9;25; : : :starting at the square containing

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 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 surrounding2007isa:

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 =

1831.

a 1849

(= 432)

2007 2008 2024 2025