TOÁN QUỐC tế violympic

TOÁN QUỐC TẾ
VÒNG 1
EX1. Fill in the blank with the correct number
Question 1:There are two parallel lines in the figure. What is the value of x?

Answer: x =
Question 2:

ABC and RTS are similar triangles.The value of x is
Question 3:If x<0 and 5x+8+|3x|=0 then x=
Question 4:I am a multiple of 70 between 400 and 600. My tens digit is odd. Who am I?

Answer:
Question 5:Mary is 24 years old. She is twice as old as Bella was when Mary was as old as Bella is now.
How old is Bella now?Answer:
years.
Question 6:The sum of 2 positive integer numbers is greater than their product if the smaller number
is
Question 7:How many different real numbers satisfy the following equation?
Answer:
Question 8:How many triples of integers (a,b,c) are there such that
Answer:
triples.

?


Question 9:There are 115 cars in a village of 26 families. Each family has either 4 or 5 cars.
How many families have 4 cars? Answer:
families.
Question 10:The area of trapezoid ABCD is 180

cm.What is AB,in centimeters?
Answer:

.The altitude is 8 cm,AD is 10 cm,and BC is 17

cm.

Question 11:Jane draws a circle and a triangle on each page of her booklet, and near each drawing she
writes the number of common points of the circle and the triangle.How many different numbers can she
write?Answer:
numbers
EX2. ARRANGE: Use the mouse cursor. Choose consecutively the cells with increasing value to
remove them from the table. If you choose wrong more than 3 times the test will end.

EX3. TALENTED GOLDEN TOAD: Choose the most suitable answer in 4 available answer
1. The product of 2015 and any number is always…
2015
Odd number
2. In this figure, AB is perpendicule to BC.
Find the measure of the angle ABD?

3. Express the given phrase into algebraic expression
“6 less than the sum of 3 the product of x and 5”

Even number

Prime number


3 + 5x - 6

6 – 3 + 5x
6 + 3 – 5x
6 – 3 – 5x
4. Which to following points on the number line represents a number that has a square root greater than
X
itself?
X

2.25
1.21
1.69
0.64
5. Three girls put $7; $8; and $9 in a piggy bank. How can they share the money in ratio 1:2:3
$2:$4:$12

6. Find the value of n if

$3.5:$7:$13.5
$7:$8:$9
$4:$8:$12
1
1
1
1
+
+
+ ... +
= 11
1+ 2
2+ 3

3+ 4
n −1 + n

Answer: n = ……..
7. Start with a unit square. Connect midpoints and vertices as shown in the figure below. Find the area of
the shared square.
1
5

1
2

1
8

1
3

8. Which of the following gives the greatest value
3200
9.

2100

2300

975

?
A

B

C

D

VÒNG 2
EX1. Fill in the blank with the correct number
Question 5:A book uses 2568 digits altogether for its page numbers.How many pages does the book
have?Answer: The book has
pages.
Question 6:A bus was travelling from Town A to Town B. If it increased its speed by 18 km/h then it
would take 3h, instead of 4h, to reach Town B.Find the distance between the two towns.
Answer: The distance between the two towns was
km.


Question 1:Fill the suitable number in the following blank:
1, 3, 7, 13, 21,
.
Question 2:The ratio of x and y is 3:4 and the sum of them is 28.
Find the product of x and y.Answer: Product of x and y is

.

Question 3:Given the isosceles triangle ABC, BC is base segment. If
measure of angle at A?Answer: The measure of angle at A is
Question 4:The ratio of x to y is 3:7. If y=112 then x=

then find the
degrees.

.

Question 5:Given the numbers sequence: 2, 8, 14, 20, 26, …Which term is number 668?
Answer: Number 668 is the
term.
Question 6:The sum of twelve consecutive whole numbers is 330. Find the greatest number.
Answer: The greatest number is
.
Question 7:The ratio of the age of Mimi to the age of her mother is 1:6 and the ratio of the age of her
grandmother to the age of her mother is 8:3. If her grandmother is 80 years old then how old is Mimi?
Answer: Mimi is
years old.
Question 8:A number is added to 7. The sum is then multiplied by 5. The product is divided by 3. 9 is
then subtracted from the quotient. The answer is 11.Find the number.
Answer: The number is
.
Question 9:The sum of 21 consecutive numbers is 567. Find the sum of the next 12 consecutive numbers.
Answer: The sum of the next 12 consecutive numbers is
.
Question 10:What is the sum of all digits of the page numbers of a 76-page book?
Answer: The sum is
.
EX2. FIND EQUAL PAIRS:


EX3. OBSTACLE
1. Suppose 3a

×

b = 217. Find the value of a

×

(-3b) = ……….

2. In the summer, Mark get a part – time job. He work 8 hours per day, 6 days a week and his hourly
salary is $7.5. How much is his weekly salary rate?
$45
$105
$60
$360
3. Given that 3x < 28. Find the greatest possible value of x if x is a prime number
…………………….
4. Convert the following speed 216km/h = …………..m/s
5. Suppose -2 < a < 5 and 3 < b < 9, a and b are interger numbers. The smallest value of a + b
is……………….


6. Given a%b = (a – 3)

×

b + b – a. Find the value of b if 7%b = 13

b=6
b=4

b=5
b=3
7. There are 60 trees along a stretch of road and the trees are planted at opposite ends of the road. If each
tree is 3 metres away from another how long is road? ………………
8.

9. 150 postcard are given away to a group of children in this manner
Anna
1
5
…
Who will get the 102th postcard?

Cindy
2
6
…

Tom
3
7
…

Andy
4
8
…

10.6 bakers take 15 minutes to base 30 pieces of cake. How many pieces of cake can 11 baker in 1 hour?
220 pieces

200 pieces
55 pieces
110 pieces
11. 3 kg of apple and 5 kg of guava cost $84. 6kg of guava and 2kg of apple cost $80. How much is one
kg of apple, one kg of guava?
12. The ratio of the distance from Anna’s house to Marry’s house and from Mary’s house to Tom’s house
is 3: 5. Anna drives at a contant speed of 60km/h from her house and she takes 12 minutes to reach
Mary’s house. Find the greatest possible distance from Anna’s house to Tom’s house?
VÒNG 3
EX1. FIND EQUAL PAIRS:


EX2. TALENTED GOLDEN TOAD:

1. How many whole numbers lie in the interval between

5
3

and

2π

?....................

2. Aunt Anna is 42 years old. Caitlin is 5 years younger than Brianna and Brianna is half as old as aunt
Anna. How old is Caitlin?
17
15
21

16
3. Each principal of Lincoln High School server exactly one 3-year term. What is the maximum number
of principals this school could have during an 8-year period?
4
2
3
5
4. What is the minimum possible product of three different numbers of the set
{-8;-6;-4;0;3;5;7}
-336
0
-280
-210
5. At Victoria Secondary School the mathematics teachers are Miss Anna, Mr. Jonh, and Mrs. Jeny. There
are 11 students is Miss Anna’s class, 8 students in Mr. Jonh’s class, and 9 students in Mr. Jeny’s class
taking the Gobal test this year. How many mathematics students ats Victoria Secondary School are taking
the contest?
6. The lengths of the sides of a triangle in inches are three consecutive intergers. The length of the shoterst
side is 30% of the perimeter. What is the length of the longest side?
9
11
12
10
7. A collector to buy state quarters for 2000% of their face value. At that rate how much will Bryden get
for his four state quarters?
500 dollars

20 dollars

200 dollars

50 dollars


8. Using the function below, evaluate g(16)
G(x) = 4x3 – 3x(x2 + 4) + 4(3x + 8). Answer:………………………..
9. Three circular arcs of radius 5 units
bound the region shown. Arcs AB and AD
are quarter-circles and arc BCD is a semicircle.
What is the area, in square units, of the region?
50 + 5
10.

π

25

A
B and D
EX3. OBSTACLE

10 + 5

B
A and C

C

π

D
A and D

50

B and C

1. In the diagram below AB = AC = AD

·ABC = 40o and ·ACD = 80o
. Find

·
BAD

≤ x ≤ 7 and − 1 ≤ y ≤ 2

2. Given -6
Find the smallest value of x

.
×

y ?................................

3. What line is paraller to 2x + 3y = 6 at (3;2)?Answer: ………………………………….
4. The sum of 3 exterior angles of a triangle is 360o
The ratio of these angles is 2:3:4.
Find the ratio of the 3 interior angles respectively

1:5:3
5:1:3
3:1:5
5:3:1
5. There are five red balls, six yellow balls and seven bue balls hidden in a box. You are allowed to take
out some balls without looking at their colors. At least how many balls must you take to guarantee that
you have two bue balls among them?
6. If the line through the point (5;-3) and (-2; p) is parallel to the line y = -2x – 3, what is the value of p?
Answer:……………………………
7. ABCD is a square and BCE is equilateral


Find

·AED

?

Answer:…………………..
8. Find the missing number x in the following number sequence
2, 9, -18, -11, x, 29, -58, -51, …
9. As shown in the diagram, the points L and M
Lie on QP and QR respectively. O is the point
Os intersection of the lines LR and PM. Given
That MP = MQ, LQ = LR; PL = PO and

·
POR
= xo

.

Find the value of x?..........................
VÒNG 4
EX1. TALENTED GOLDEN TOAD:
1. A shopper buys a 100 dollar coat on sale 20% off. An addition 5 dollars are taken off the sale price by
using a discount coupon. A sale tax of 8% is paid on the final selling price. The total amount the shopper
pays for the coats
%82.00

$81.00

$82.08

2. For how many positive interger values of N make the expression

$81.40
36
N +2

an interger?

7
10
3. Which of the following is the largest?

8

9

5
12

1
4

3
8

4. Calculate:

1
3

1 2 3 4 5 6 7 8 9 55
+ + + + + + + + +
= ..................
10 10 10 10 10 10 10 10 10 10

5. The number halfway between
5
24

1
5

1
1
and
6

4

is
7
24

1
10

6. Calculate: 33 + 33 + 33 = ?
273
34
39
93
7. The perimeter of a square is 3 times larger than the perimeter of the other square. The area of the larger
square is how many times larger than the area of the smaller one?


6
9
3
4
8.The arithmetic mean (average) of four number is 85. If the largest of these numbers is 97, then the mean
of the remaining three numbers is
84

85
1
1+

9.

81

= ..............

1
2+

83

1
3

10.

EX2. INTELLECTUAL PEAKS
1. The average of n whole numbers is 80. One of the number is 100. After removing the number 100 the
average of the remaining numbers is 78. Find the value of n?...............
2. Find the 10th term of this sequence: 1; 4; 9; 16; 25; ….
3. The side of the square ABCD are 15cm long.
E and F are respectively the midpoints of AB and AD
Find the area of the shaded region?
75cm2
35 cm2
60 cm2
50 cm2
4. The list price of an article is $6000. If is sold at half price, the profit is 25%. At what price must it be
sold to have the profit of 50%
$4500

$3600
5. What is the next number of the sequence?

$2400

$3000

1, 2, 4, 7, 11, …..
6. In 2014, both John and Mary have the same amount of pocket money per month. In 2015, John had an
increase of 10% and Mary a decrease of 10% in their pocket money. In 2016, John had a decrease of 10%
and Mary an increase of 10% in their pocket money. Compare the amount of pocket money between John
and Mary, in 2016?
Mary’s money is 2 times John’s money
John’s money is 2 times Mary’s money
John’s money = Mary’s money
John’s money is 3 times Mary’s money
7. If the first three Fibonacci numbers are given as 1, 1, and 2, then what is the eleventh number?


8. Find the next number of this sequence: 2, 6, 18, 54, ….
9. Jessica has taken 4 exams and her average score so far is 79. If she gets 100, a perect score, on the next
2 exams, what will her new average be?
10. Sally receiver the following scores on her Math tests: 78, 92, 83, 99. What score does she need on the
next in order to have an average of 90 on her 5 math tests?
11. A language school has 100 pupils in which 69% of the pupils study French, 79% study German, 89%
study Japanese and 99% study English. Given that at least x% of the sudent study all four languages. Find
the value of x.
EX3. ARRANGE

VÒNG 5

EX1. Fill in the blank with the correct number
Question 1:In a group of 100 persons, 72 people can speak English and 43 can speak French. How many
can speak English only? Answer: There are
persons.
Question 2:Mary has six cubes. When she arranges them from smallest to largest, the difference between
the heights of any two neighbouring cubes is 3 cm. The largest cube is as high as tower built from the two
smallest cubes. How high is a tower built from all cubes?Answer:
cm.
Question 3:Find the value of x if: (x+ 70)3 = 357911.Answer: x =

.

Question 4:Tom can build a wall in 6 hours, Peter can build it in 10 hours and Mark can build it in 15 hours,
separately. In how many hours can they do it together?Answer: They can do it together in
hours.


·
RPQ
= 60o
Question 5:In triangle PQR, PQ=PR,

. PQS is a straight line.

·
RQS
Find .

Answer:

=

.

·
WFG
Question 6:The figure shown below is not drawn to scale. EFGH and WXYZ are parallelograms. Find

·
WFG
Answer:
Question 7:How many prime numbers are there from 20 to 69?
Answer: There are
prime numbers.

Question 8:In the figure below, ABCD is a parallelogram and CDE is an isosceles triangle.Find

Answer:

.

Question 9:How many composite numbers are there from 20 to 29?Answer:
There are
composite numbers.
Question 10:The height of two building is 34 m and 29 m respectively. If the distance between the two
building is 12 m, find the distance between their tops.
Answer: The distance between their tops is
m.

EX2. OBSTACLE

.


1. When three diffierent numbers from the set {-3; -2; -1; 4; 5} are multiplied, the largest possible product
is……….
10
40
30
2. Which digit of 12345, when changed into 9, gives the largest number?

20

1

2

3

4

1
1+

3.

1
3+

1

4

=………………..

4. Compute the following equation 2016 – 1 – 2 – 3 – 4 - …- 50 = …………………
5. A ladder 13m long is placed on the ground in the such a way that it touches the top of a vertical wall
12m high. Find the distance of the foot of the ladder from the bottom of the wall
4m
5m
6m
7m
6. If the square of the hypotenuse of the an isosceles right triangle is 128cm 2 find the area of the triangle.
32cm2
64cm2
28cm2
10cm2
7. The sum of three numbers is 98. The ratio of the first to the second is 2:3; and the ratio of the second to
the third is 5:8. The second number is:
30
60
EX3. TALENTED GOLDEN TOAD:

50

40

VÒNG 6
EX1. Fill in the blank with the correct number
Question 1:
What is the difference between the smallest 5-digit odd number and the largest 4-digit even number?

Answer:
.
Question 2:
Given the following addition


What is the sum of the missing digits?
Answer:
.
Question 3:
Find two consecutive prime numbers with a sum of 144.
Answer:
.
(write your answer from least to greatest and use ";" between two numbers)
Question 4:
A right triangle
has
. The circumradius of

is

cm.

Question 5:
How many two-digit numbers have the property of being equal to 7 times the sum of their digits?
Answer:
.
Question 6:
ABCD is a rectangle with a length of 22cm and an area of
and AB .

. E and F are respectively midpoints of AC

Find the area of the shaded triangle?
Answer:
.
Question 7:
If an arc of

of circle A has the same length as an arc of

the area of circle B is

then the smallest value of a+b is

of circle B. The ratio of the area of circle A to
.

Question 8:
A Ferris wheel has 32 passenger cabins. If you are sitting in cabin 14 which is currently at the top of the
wheel, what is the number of the cabin at the bottom of the wheel?
Answer:
.
Question 9:
A box of erasers cost 4 dollars. A box of pencils cost 6 dollars more. Darwin spent 240 dollars on some boxes
of erasers and pencils. The ratio of the number of pencil boxes to the number of eraser boxes was 4:5. How
many boxes of pencils did he buy?
Answer:
.
Question 10:

Ann has walked 8km at a speed of 4 km/h. She then run at a speed of 8 km/h. How long does she have to run
in order to have an overall average speed of 5 km/h?
Answer:
minutes.


Fill in the blank with the correct number
Question 1:
Find two consecutive prime numbers with a sum of 144.
Answer:
.
(write your answer from least to greatest and use ";" between two numbers)
Question 2:
A right triangle
has
. The circumradius of

is

cm.

Question 3:
What is the difference between the smallest 5-digit odd number and the largest 4-digit even number?
Answer:
.
Question 4:
Given the following addition

What is the sum of the missing digits?
Answer:

.
Question 5:
ABCD is a rectangle with a length of 22cm and an area of
and AB .

. E and F are respectively midpoints of AC

Find the area of the shaded triangle?
Answer:
.
Question 6:
How many two-digit numbers have the property of being equal to 7 times the sum of their digits?
Answer:
.


Question 7:
If an arc of

of circle A has the same length as an arc of

the area of circle B is

then the smallest value of a+b is

of circle B. The ratio of the area of circle A to
.

Question 8:
A Ferris wheel has 32 passenger cabins. If you are sitting in cabin 14 which is currently at the top of the

wheel, what is the number of the cabin at the bottom of the wheel?
Answer:
.
Question 9:
Ann has walked 8km at a speed of 4 km/h. She then run at a speed of 8 km/h. How long does she have to run
in order to have an overall average speed of 5 km/h?
Answer:
minutes.
Question 10:
A box of erasers cost 4 dollars. A box of pencils cost 6 dollars more. Darwin spent 240 dollars on some boxes
of erasers and pencils. The ratio of the number of pencil boxes to the number of eraser boxes was 4:5. How
many boxes of pencils did he buy?
Answer:
.