Đề thi Toán Olympic AMC Úc cho lớp 4, 5 – Đề 1

Hôm nay Toán Olympic Tiểu Học xin gửi tới các em bộ đề thi Toán Olympic cuộc thi AMC Úc (Australian Mathematics Competition) dành cho khối lớp 4, 5. Đề thi bao gồm 30 câu hỏi, vì là cuộc thi Toán quốc tế nên đề bài sử dụng ngôn nữa tiếng anh. Các em hãy thử làm bài thi này xem nhé.

1) The value of 802 − 208 is

(A) 606

(B) 604

(C) 504

(D) 694

(E) 594

2) Given that 1.08 × 1.8 = 1.944, the value of 108 × 18 is

(A) 194.4

(B) 1944

(C) 19.44

(D) 19 440

(E) 19 400

3) In the diagram, the sides of the triangles are extended and three angles are as shown. The value of x is 

(A) 100

(B) 110

(C) 120

(D) 130

(E) 140

4) The value of frac{mathrm{200 times 8} }{mathrm{200 div 8}} is

(A) 1

(B) 8

(C) 16

(D) 64

(E) 200

5) The digits 5, 6, 7, 8 and 9 can be arranged to form even five-digit numbers. The tens digit in the largest of these numbers is
(A) 5

(B) 6

(C) 7

(D) 8

(E) 9

6) Four consecutive odd numbers add up to 48. What is the largest of these numbers ?

(A) 13

(B) 15

(C) 17

(D) 19

(E) 21

7) A rectangle has an area of 72 square centimetres and the length is twice the width. The perimeter, in centimetres, of the rectangle is

(A) 34

(B) 36

(C) 42

(D) 48

(E) 54

8) What percentage of y is x ?

(A) frac{mathrm{y} }{mathrm{x}}

(B) frac{mathrm{x} }{mathrm{100}}

(C) frac{mathrm{x} }{mathrm{y}}

(D) frac{mathrm{100y} }{mathrm{x}}

(E) frac{mathrm{100x} }{mathrm{y}}

9) In the diagram, triangles PQR and LMN are both equilateral and angleQSM = 20^o . What is the value of x ?

(A) 70

(B) 80

(C) 90

(D) 100

(E) 110

10) When 1000^{2008} is written as a numeral, the number of digits written is

(A) 2009

(B) 6024

(C) 6025

(D) 8032

(E) 2012

11) Anne designs the dart board shown, where she scores P points in the centre circle, Q points in the next ring and R points in the outer ring. She throws three darts in each turn. In her first turn, she gets two darts in ring Q and one in ring R and scores 10 points. In her second turn, she gets two in circle P and one in ring R and scores 22 points. In her next turn, she gets one dart in each of the regions. How many points does she score?

(A) 12

(B) 13

(C) 15

(D) 16

(E) 18

12) How many different positive numbers are equal to the product of two odd one-digit numbers ?

(A) 25

(B) 15

(C) 14

(D) 13

(E) 11

13) Points A, B, C, D and E are nodes of a square grid as shown. Which of these five points forms an isosceles triangle with the other two vertices at X and Y ?

(A) A

(B) B

(C) C

(D) D

(E) E

14) A Fibonacci die has the numbers 1, 1, 2, 3, 5 and 8 on it. Two such dice are thrown. What is the probability that the number on one die is larger than the number on the other?

(A) frac{mathrm{1} }{mathrm{2}}

(B) frac{mathrm{5} }{mathrm{9}}

(C) frac{mathrm{2} }{mathrm{3}}

(D) frac{mathrm{5} }{mathrm{6}}

(E) frac{mathrm{7} }{mathrm{9}}

15) A fishtank with base 100 cm by 200 cm and depth 100 cm contains water to a depth of 50 cm. A solid metal rectangular prism with dimensions 80 cm by 100 cm by 60 cm is then submerged in the tank with an 80 cm by 100 cm face on the bottom.

The depth of water, in centimetres, above the prism is then

(A) 12

(B) 14

(C) 16

(D) 18

(E) 20

16) What is the smallest whole number which gives a square number when multiplied by 2008 ?

(A) 2

(B) 4

(C) 251

(D) 502

(E) 2008

17) The interior of a drinking glass is a cylinder of diameter 8cm and height 12 cm. The glass is held at an angle of 45^o from the vertical and filled until the base is just covered. How much water, in millilitres, is in the glass?

(A) 48pi

(B) 64pi

(C) 96pi

(D) 192pi

(E) 256pi

18) A number is less than 2008. It is odd, it leaves a remainder of 2 when divided by 3 and a remainder of 4 when divided by 5. What is the sum of the digits of the largest such number?

(A) 26

(B) 25

(C) 24

(D) 23

(E) 22

19) PR and QS are perpendicular diameters drawn on a circle centre O. The points T , U, V and W are the midpoints of PO, QO, RO and SO respectively. The fraction of the circle covered by the shaded area is

(A) frac{mathrm{1} }{mathrm{2pi}}

(B) frac{mathrm{1} }{mathrm{pi}}

(C) frac{mathrm{3} }{mathrm{2pi}}

(D) frac{mathrm{2} }{mathrm{pi}}

(E) frac{mathrm{5} }{mathrm{2pi}}

20) Three numbers p, q and r are all prime numbers less than 50 with the property that p + q = r. How many values of r are possible ?

(A) 0

(B) 2

(C) 4

(D) 6

(E) 8

21) Farmer Taylor of Burra has two tanks. Water from the roof of his farmhouse is collected in a 100 kL tank and water from the roof of his barn is collected in a 25 kL tank. The collecting area of his farmhouse roof is 200 square metres while that of his barn is 80 square metres. Currently, there are 35 kL in the farmhouse tank and 13 kL in the barn tank.

Rain is forecast and he wants to collect as much water as possible. He should:

(A) empty the barn tank into the farmhouse tank

(B) fill the barn tank from the farmhouse tank

(C) pump 10 kL from the farmhouse tank into the barn tank

(D) pump 10 kL from the barn tank into the farmhouse tank

(E) do nothing

22) If the tens digit of a perfect square is 7, how many possible values can its units digit have?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

23) Twenty-five different positive integers add to 2008. What is the largest value that the least of them can have?

(A) 65

(B) 66

(C) 67

(D) 68

(E) 69

24) PQR is an equilateral triangle. The point U is the mid-point of PR. Points T and S divide QP and QR in the ratio 1 : 2. The point of intersection of PS, RT and QU is X. If the area of bigtriangleup QSX is 1 square unit, what is the area, in square units, of bigtriangleup PQR ?

(A) 6

(B) 8

(C) 9

(D) 12

(E) 18

25) A two-digit number n has the property that the sum of the digits of n is the same as the sum of the digits of 6n. How many such numbers are there?

(A) 0

(B) 3 (

C) 4

(D) 8

(E) 10

26) In the diagram, angle OPQ = angle OQR = angle ORS = 90^o. OP = 4cm, PQ = 3cm and QR = 12cm. The perimeter of the pentagon OPQRS is 188 cm. What is the area, in square centimetres, of the pentagon OPQRS?

27) A rectangular prism 6 cm by 3cm by 3 cm is made up by stacking 1cm by 1 cm by 1 cm cubes. How many rectangular prisms, including cubes, are there whose vertices are vertices of the cubes, and whose edges are parallel to the edges of the original rectangular prism? (Rectangular prisms with the same dimensions but in different positions are different.)

28) The number 2008! (factorial 2008) means the product of all the integers 1, 2, 3, 4, . . . , 2007, 2008. With how many zeroes does 2008! end?

29) Let us call a sum of integers cool if the first and last terms are 1 and each term differs from its neighbours by at most 1. For example, the sum 1 +2+ 3+4 +3+ 2 + 3+3 + 3+2 + 3+3 + 2+1 is cool. How many terms does it take to write 2008 as a cool sum if we use no more terms than necessary ?

30) All the vertices of a 15-gon, not necessarily regular, lie on the circumference of a circle and the centre of this circle is inside the 15-gon. What is the largest possible number of obtuse-angled triangles where the vertices of each triangle are vertices of the 15-gon ?

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Bên trên là đề thi Toán Olympic quốc tế AMC Úc (Australian Mathematics Competition) dành cho khối học sinh lớp 4 & lớp 5 gồm 30 câu hỏi. Các em hãy cố gắng tự làm bài để thử sức và xem nhiều tài liệu học tập hơn nữa, nếu có gì thắc mắc và cần hỗ trợ có thể liên hệ với Thầy Đào Hưng.