Đề thi Toán quốc tế IMSO năm 2013

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Mathematics Short Answer Problems

_________________________________________________________________ 10th International Mathematics and Science Olympiad (IMSO) for Primary School 2013

_________________________________________________________________

Instructions:

* Write down your name and country on the answer sheet.

* Write your answer on the answer sheet.

* You have 60 minutes to work on this test.

* Use pen or pencil to write your answer.

“Smart, Skilled, and Creative In a Joyful Competition for Excellence”

Alfonso, Cavite, Philippines

25 – 29 Nov 2013

Country: Name: No.: Score:

International Mathematics and Science Olympiad 2013

SHORT ANSWER PROBLEMS

(1) Square pieces of sides 0.5 cm are cut from a sheet which is 11 cm long and 2 cm wide. What is the total number of squares that can be cut?

(2) Study the following pattern.

1 1 1 2 2 = × , 1 1 2 1 2 2 3 3 + = × × , 1 1 1 3 1 2 2 3 3 4 4 + + = × × × .

Given that 1 1 1 1 2 1 2 2 3 3 4 2013 2014 3 a a

+ + + + + = × × × × + " , where a is a

positive integer. Find the value of a.

(3) Thirty girls joined a mathematics contest. The first girl scored 70 and the second girl scored 80. The teacher then announced that the score of every girl after the first two was equal to the average of the scores of all the girls before her. What was the score of the last girl?

(4) Five boys, A, B, C, D, and E, attended a meeting. In this meeting : a. A shook hands with one boy. b. B shook hands with two boys. c. C shook hands with three boys. d. D shook hands with four boys. How many boys did E shake hands with?

(5) What is the simplified value of

1 1 1 1 1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 2 3 4 5 6 7 8 9 × + × + × + × + × + × + × + × ?

(6) The sum of the digits of a two-digit number ab is 6. By reversing the digits, one obtained another two-digit number ba . If 18 ab ba − = , find the original two-digit number.

(7) The side length of the biggest square in the given diagram is 10 cm long. As shown in the diagram, the total shaded regions formed by two diagonals inside the circle and two squares is 26 cm2. What is the length side of the smallest square in cm?

(8) The product of 1110, 1111, 1112 and 1113 is the thirteen digit number 152628 755760 x , with one digit replaced by x. What is the value of x?

(9) Each of A, B, C and D either always tells the truth or always tells lies. A says C always tells lies. B says A always tells lies. C says D always tells the truth. D says either A or C always tells lies. Who always tells lies?

(10) In the Figure below each of the interior angles of hexagon PQRSTU is 120° . Given that PQ = 1 cm, QR = RS = 4 cm and ST = 3 cm. Find the perimeter of the hexagon PQRSTU.

(11) PQRSTU is a regular hexagon with side 2 cm. The polygon ABCDEFGHIJKL is obtained by drawing the equilateral triangles of side 4 cm, producing the

sides of the hexagon. Find area of area of

ABCDEFGHIJKL PQRSTU .

(12) Nine lines, parallel to the base of a triangle, divide each of the other sides into 10 equal segments and the area into 10 distinct parts. Find the area of the original triangle, if the area of the largest of these parts is 76 cm2.

(13) The dates of three Sundays in a month are even numbers. What day is the 28th day of the month?

(14) The company Coco has a number of operational cars. The tax for the first car is $2,000, the tax for second car is 5 % more than the tax for the first car, the tax for third car is 10 % …