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Pre-APMOPS Course
Mock Test
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MOCK TEST
by ChuTieuThichHocToan®
Duration: 100 minutes – 120 points (No calculator used)
P1
(4)
If the product of two natural numbers is 1000 and their difference is
30, find their sum.
P2
In the addition of the three-digit numbers shown, the letters A, B, C,
(4)
D, and E each represent a single digit. Find the value of A + B + C
+D+E
P3
(4)
The area of square ABCD is 64 and AX = BW = CZ = DY = 2.
What is the area of square WXYZ?
P4
(4)
The number 503 is a prime number. How many positive integers are
factors of 2012?
P5
The number N is the product of all positive odd integers from 1 to 99
(4)
that do not end in the digit 5.
That is, N = 1x3x7x9x11x13x17x19x…x91x93x97x99.
Find the unit digit of N?
P6
(4)
In the figure shown, a side of the larger square is a diagonal of the
smaller square. If the area of the smaller square is 1 square unit, what
is the area of the larger square?
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Pre-APMOPS Course
Mock Test
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P7
(4)
As in below figure, The figure are all assembled using sticks of the
same length. The first figure requires 7 sticks, the second figure
require 13 sticks, and so on. How many sticks are required to
assemble the 11th figure?
P8
(4)
The Hope Cup Competition started in 1990. This year is 2011. If we
write
2011
as a continued fraction a 
1990
b
1
1
c
1
d
1
e
a) What is the largest number among the integers a, b, c, d, e ?
b) What is the sum of a, b, c, d, and e ?
P9
(4)
The positive integers are arranged in rows and columns as shown
below.
Row 1
1
Row 2
23
Row 3
456
Row 4
7 8 9 10
Row 5 11 12 13 14 15
Row 6 16 17 18 19 20 21
….
More rows continue to list the positive integers in order, with each
new row containing
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Pre-APMOPS Course
Mock Test
---------------------------------------------------------------------------------------------------------one more integer than the previous row. How many integers less
than 2000 are in the
column that contains the number 2000?
P10
(4)
Chris lies on Fridays, Saturdays and Sundays, but he tells the truth
on all other days.Mark lies on Tuesdays, Wednesdays and
Thursdays, but he tells the truth on all other days. On what day of
the week would they both say: “Tomorrow, I will lie."?
P11
(5)
A square is divided, as shown. What fraction of the area
of the square is shaded?
P12
(5)
There were 9 cards numbered 1 to 9. Four people A, B, C and D
each collected two of them.
A said : “ The sum of my numbers is 6. ”
B said : “ The difference between my numbers is 5. ”
C said : “ The product of my numbers is 18. ”
D said : “ One of my numbers is twice the other. ”
What is the number on the remaining card ?
P13
(5)
Let a, b, c, d be non-zero natural numbers that are not more than 4.
Two of them are identical, and (a+b)(b+c)(c+d)(d+a) = 900.
Find a + b + c + d.
A 1 3
B 1 3

 . Find A
and
B 1 2
2A 5
P14
(5)
Suppose
P15
(5)
Draw two straight lines to divide the figure into four portions whose
areas are in the ratio 1 : 2 : 3 : 4 .
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Pre-APMOPS Course
Mock Test
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P16
(5)
Write natural numbers 1 to 2013 one after another in ascending
order to obtain a long number 1234567891011…20122013. If
divide this number by 9, what is the remainder?
P17
(5)
From the gure shown, three of the nine squares are to be
selected. Each of the three selected squares must share a
side with at least one of the other two selected squares.
In how many ways can this be done?
P18
(5)
How many unit cubes were used to construct this figure?
P19
(5)
What whole number is equal to:
P20
(5)
Doan mixes together 1 litre of 1% butterfat milk, 2 litres of 2%
butterfat milk and 4 litres of 4% butterfat milk. What percentage of
the resulting seven litres of milk is butterfat?
P21
Sixteen coins, numbered 1 to 16, are each red on one side and blue
1 1
1 1
1
1
(1x 2) x(  )  (2 x3) x(  )  ...  (99 x100) x( 
)
1 2
2 3
99 100
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Pre-APMOPS Course
Mock Test
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on the other side. Initially, all coins have their red sides facing up.
The coins that are multiples of 2 are turned over. Then the coins that
are multiples of 4 are turned over. Then the coins that are multiples
of 8 are turned over. Then the coins that are multiples of 16 are
turned over. Afterwards, how many of the coins have the red side
facing up?
P22
(6)
Given that the rectangles has side 6cm and 8cm. The circle has
radius is 7cm. Find the difference between the shaded area of each
shape. Given  
22
7
P23
The mother’s current age is 3 times the sum of ages of her two
(6)
children. However, twenty two years later, her age is the sum of
ages of her two children at that time. If the age difference of her two
children is less than 5 but no less than 2, what are the ages of her
children now?
P24
(6)
A school has less than 300 students but both numbers for boys and
girls are more than 100. If girls are grouped into 3 students a group
or 4 or 5, there is always one student left over. If boys are grouped
into 7 students or 8 students a group, each will be 3 students short.
What is the most number of students this school can have?
P25
Given 4 regions M, N, P, and Q as shown in the figure below. Each
(6)
region can be painted in one of four possible colors. If neighboring
regions (such as M and N) cannot be painted with the same color,
how many different ways these four regions can be painted?
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Pre-APMOPS Course
Mock Test
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P26
(6)
If the sum of all natural numbers between a prime number a and
another prime number b is 280 and there is no prime number in
between a and b, find a+b.
P27
(6)
Steve begins at 7 and counts forward by 3, obtaining the list 7; 10;
13, and so on. Dave begins at 2011 and counts backwards by 5,
obtaining the list 2011; 2006; 2001, and so on. Which of the
following numbers appear in each of their lists?
(A) 1009 (B) 1006 (C) 1003 (D) 1001 (E) 1011
P28
How many different pairs (m; n) can be formed using numbers from
(6)
the list of integers {1; 2; 3; : : : ; 20} such that m < n and m + n is
even?
P29
(6)
How many 3-digit odd numbers abc such that a < b  c and
a +b +c = 21?
P30
(6)
In the following 8-pointed star, what is the sum of the angles
A;B;C;D;E; F; G;H?
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(internal used material)
Pre-APMOPS Course
Mock Test
---------------------------------------------------------------------------------------------------------Result:
No of 4-point question:
No of 5-point question:
No of 6-point question:
Point:
Point:
Point:
Total:
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