Đề thi Toán quốc tế CALGARY năm 2019

B5 The following hat is made with sides of the same length and with right angles: Starting with a square, we can create a new figure by replacing each side of the square by a hat, so tha[r]

43rd ANNUAL CALGARY JUNIOR HIGH SCHOOL MATHEMATICS CONTEST

MAY 1st, 2019

NAME: GENDER:

PLEASE PRINT (First name Last name) (optional)

SCHOOL: GRADE:

(9,8,7, )

• 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 PART A has a total possible score of 45 points

• 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 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 first 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

PART A: SHORT ANSWER QUESTIONS (Place answers in the boxes provided)

A1

A1 The perimeter of a rectangle with integer edge-lengths is 10cm What is the largest area (in cm2) that the rectangle can have?

A2

A2 A store increases the price of a shirt by 10%, then reduces the cost by $10 The price is then 90% of the original price Find the original price

A3

A3 At 1PM Isaac had done 1/3 of his homework At 2PM he had done 1/2 his homework He works at a constant rate all the time At what time did he finish his homework?

A4

A4 Melissa drives for 11 minutes, the first minute at 10 km/h, the second minute at 20 km/h, and so on until the 11th minute at 110 km/h What is the total distance (in km) she travelled?

A5

A6

A6 Let A(0,0), B(3,5), C(3,0), D(5,0) and E(5,−5) be five points in the Cartesian plane The pentagonABCDE and its reflection in thex-axis are combined to make a seven sided figure What is the area of this figure?

A7

A7 A lawn 10 metres square receives cm of rain over its entire surface Assuming that the volume of each raindrop is cubic millimetre, the number of raindrops that fell on the lawn can be written as a one followed by a number of zeros How many zeros come after the 1?

A8

A8 The number 12 has the strange property that the next number (13) is prime, the number after that (14) is twice a prime (since 14 = 2×7) and the number after that (15) is three times a prime (since 15 = 3×5) Find a number N bigger than 12 so thatN + is prime,N + is twice a prime, andN+ is three times a prime

A9

A9 Three positive integersa, b, c are such that 0< a < b < cand b−a, c−aand c−b

PART B: LONG ANSWER QUESTIONS

B2 An ant is walking along a spiral, as shown in the figure The spiral consists of eight quarter-circles joined together, so that:

• Arc AP1 has its centre in B

• Arc P1P2 has its centre in C

• Arc P2P3 has its centre in D

• Arc P3P4 has its centre in A

• Arc P4P5 has its centre in B

• Arc P5P6 has its centre in C

• Arc P6P7 has its centre in D

• Arc P7P8 has its centre in A

A B C

D P1

P5

P2

P6

P3 P7

P4

P8

B3 A Greek cross is a figure made up of five squares of side 1cm joined along the edges as pictured below:

B5 The following hat is made with sides of the same length and with right angles: Starting with a square, we can create a new figure by replacing each side of the square by a hat, so that each vertical side is replaced by a hat pointing outside the square and each horizontal side is replaced by a hat pointing inside the square, as shown below:

=⇒ =⇒

The following sequence of figures was created applying the same process to each new figure

=⇒ =⇒ =⇒

figure figure figure figure

Suppose the perimeter of the first figure, the square, is cm (a) What is the perimeter (in cm) of figure 4?

B6 Three large spheres sit on the floor of a gymnasium, touching in a row The centres of the spheres are pointsA, B, C in this order, with these points lying in a straight line The radii of spheres with centresAandBare metre and metres respectively

A

B

C

Z Y

X

(a) Find the radius (in metres) of the sphere with centre C

(b) The spheres touch the floor at pointsX, Y, Z respectively Find distance XZ