Đề thi thử toán AMO lớp 7 năm 2019

Segment of drive Distance Average driving speed with (miles) no traffic delay (mph) From home to freeway entrance 0.. 6 25?[r]

AMO MOCK TEST #1

Grade 7

INSTRUCTIONS

1 Please DO NOT OPEN the contest booklet until the Proctor has given

per-mission to start

2 TIME: 1 hour 30 minutes

3 Attempt all 25 questions Each question scores 1 point No points are

de-ducted for incorrect answers

4 Write your answers neatly on the answer sheet

5 PROCTORING : No one may help any student in any way during the contest

6 No calculators are allowed

7 All students must fill in your Name and School

8 MINIMUM TIME: Students must stay in the exam hall at least 1h 15 min

9 Students must show detailed working and put answers on the answer sheet

10 No spare papers can be used in writing this contest Enough space is provided

for your working of each question

Remark: Counting numbers are whole numbers except 0, i.e 1, 2, 3, 4, 5,

Question 1 Find the sum of the counting numbers from 1 to 35 inclu-sive In other words, if

S = 1 + 2 + 3 + · · · + 34 + 35,

find the value of S

Question 2 In three bowling games, Alice scores 139, 143,and 144

What score will Alice need in the fourth game in order to have an av-erage score of 145for all four games?

2

Question 3 For all integers P and Q, P ⊗ Qis defined as P ⊗ Q =

P × Q

2 What is the value of 8 ⊗ (1 ⊗ 6)?

Question 4 In a group of 30 secondary school students, 8 students

take French, 12 students take Spanish and3 students take both

lan-guages How many students of the group take neither French nor

Span-ish?

Question 5 Each of the smallest boxes in the below figure is a square.

How many different squares can be formed by using the lines in the below figure?

Question 6 The product of some whole numbers is 120 Find their least possible sum

4

Question 7 Soshana looks in a mirror and sees the reflection of a

clock behind her as shown in the figure on the left How many minutes

later will the reflection in the mirror of the same clock next show the

image shown in the figure on the right?

Question 8 Let A and B be the following sets:

A = {21, 22, 23, 24, 25, 26, 27, 28, 29}

B = {9, 10, 11, 12, 13, 14, 15, 16}

Let C be the set of all possible positive differences when an number

from set A is subtracted by an number from set B How many elements

are contained in set C?

Question 9 A number has a remainder of 1 when divided by 4, a re-mainder of 2 when divided by 5, and a remainder of 3 when divided by

6 What is the smallest number that has the above properties?

Question 10 The perimeter of a rectangle is 20cm and the number of centimetres in each of its length and width is a whole number How many rectangles with different shapes satisfy these conditions?

6

Question 11 In a maths contest of 20 problems, 5 points were given

for each problem with correct answer and 2 points were deducted for

each problem with incorrect answer or no answer If Nancy answered

all 20 problems and scored 72 points, how many correct answers did

she have?

Question 12 In the "magic square" at the right, the four numbers in

each column, in each row, and in each of the two diagonals, have the

same sum What is the value of N?

10 13

N 12 1

2 15 8

14 9

Question 13 Thirteen dots are arranged on a square grid in the

pat-tern below How many different squares can be formed by connecting

at least four of these dots?

Question 14 A landscaper is designing a rectangular garden The

length of the garden is to be 5 yards longer than the width If the area

of the garden will be 104 square yards, what will be the length, in yards,

of the garden?

8

Question 15 The complete outside including the bottom of a wooden 4

-cm cube is painted red The painted cube is then cut into 1cm cubes as

in the below figure How many of the 1-cm cubes do not have red paint

on any face?

Question 16 In a school library, 10% of the fiction books and 15% of

the non-fiction books are new books Given that the ratio of fiction books

to non-fiction books is 9 : 5 and that there are 66 new books, what is the

total numbers of books in the library?

Question 17 A train can hold 78 passengers The train starts out empty and picks up 1 passenger at the first stop, 2 passengers at the second stop, 3passengers at the third stop, and so forth Given that no one

leaves the train during the trip, after how many stops will the train be full?

Question 18 Find the least value of whole number N, with N > 10, so that the expression 2N − 7 is both a perfect square and a perfect cube

10

Question 19 In a deck of 54 playing cards, there are four different

suits of 13 cards each and two Joker cards What is the minimum

num-ber of cards that must be drawn from the deck to ensure that there are

at least 5 cards of the same suit?

Question 20 If a kindergarten teacher places her children 4 on each

bench, there will be 3 children who will not have a place However, if

5 children are placed on each bench, there will be 2empty places on

the last bench What is the smallest number of children the class could

have?

Question 21 How many numbers in the set {1, 2, 3, , 28, 29, 30} can-not be represented by adding two or more different numbers in the set

{0, 2, 3, 5, 8, 13}? (Note that 4cannot be represented as the sum of num-bers from the above list.)

Question 22 The ten-digit number 3872649A0B is divisible by 72 The letters A and B each represent single digit even numbers Find the prod-uct (A × B)

12

Question 23 Twelve identical machines, running continuously at the

same constant rate, take 8 days to complete a shipment How many

additional machines, each running at the same constant rate, would be

needed to reduce the time required to complete a shipment by 2 days?

Question 24 In the addition problem below, each letter represents

a digit and different letters represent different digits What three-digit

number does RID represent?

I N

+ R I D

D E E R

Question 25 Ms Simon drives her car from her home to her workplace

every workday morning The table below shows the distance, in miles,

and her average driving speed, in miles per hour (mph), when there is

no traffic delay, for each segment of her drive

Ms Simon’s Workday Morning Drive

Segment of drive Distance Average driving speed with

(miles) no traffic delay (mph)

From freeway entrance to freeway exit 15.4 50

From freeway exit to workplace 1.4 35

If Ms Simon starts her drive at 6 : 30a.m., she can drive at her average

driving speed with no traffic delay for each segment of the drive If she

starts her drive at 7 : 00 a.m., the travel time from the freeway entrance

to the freeway exit increases by 33% due to slower traffic, but the travel

time for each of the other two segments of her drive does not change

Based on the table, how many more minutes does Ms Simon take to

arrive at her workplace if she starts her drive at 7 : 00 a.m than if

she starts her drive at 6 : 30 a.m.? (Round your answer to the nearest

minute.)

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