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INTERNATIONL MATHEMATICS AND SCIENCE OLYMPIAD FOR PRIMARY SCHOOLS (IMSO) 2006. Mathematics Contest in Taiwan, Exploration Problems.

Answer the following 5 questions, and show your detailed solution in the answer sheet. Write down the question number in each paper. Each question is worth 8 points. Time limit: 60 minutes.

1. The solution to each clue of this crossnumber is a two-digit number. None of
these numbers begins with zero. Complete the crossnumber, stating the order in which you solved the clues and explaining why there is only one solution.

Clues Across
1. A square number
3. A multiple of 11
Clues Down
1. A multiple of 7
2. A cube number

2. Notice that 2^2 + 2^2 = 2^3 , so two squares can sum to give a cube; however, the two squares here are equal (to 4).
(a) Find two unequal squares whose sum is a cube.
(b) Show that there are infinitely many pairs of unequal squares whose sum is
equal to a cube.

3. Is it possible to find a number 11…11 that is divisible by 19?

4. The menu in the school cafeteria never changes. It consists of 10 different dishes. Peter decides to make his school lunch different everyday (at least 1 dish). For each lunch, he may eat any number of dished, but no two are identical.
(a) What is the maximum numbers of days Peter can do so?
(b) What is the total number of dishes Peter has consumed during this period?

5. A sequence of shapes is made as follows.

(1) Shape S1 is a shaded square of side 1 unit.
(2) Shape S2 is made by dividing S1 into 9 equal squares and removing four
of these, so that only the central and corner squares remain.
(3) Shape S3 is made by applying the process in (2) to each of the squares of
S2.
(4) Shape S4 is made by applying the process in (2) to each of the squares of
S3. And so on.

(a) Find the area and perimeter of shape S3, giving your answers as fractions.
(b) Find the least value of k for which the shape Sk has the area less than 1/30
and also has perimeter greater than 30.

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1. John gives 1/6 of his marbles to Alex and 20 marbles to Simon. John’s remaining marblesvis now 55. How many marbles does John have originally?

2. The total number of ducks and chickens in the garden is 335. If the number of chickensvis 2/3 of the number of ducks, how many chickens are there in the garden?

3. Mr. Sweet has three kinds of books: 120 mathematics, 144 biology, and 100 physics books. He is going to give all the books to the schools in his village such that each school receives the same number of mathematics, biology and physics books. How many schools, at most, receive books from Mr. Sweet?

4. Jerry is riding a motorbike at 60 km/hour while Tom is riding his motorbike at 45 km/hour in the same direction from the same starting point. Jerry left 1 1/3 hours later than Tom did. How long will it take Jerry to catch up Tom?

5. An ant moves on the surface of a cube from vertex A to vertex B. Draw the shortest path from A to B passed by the ant.

6. Jack and Ben are cycling from A to B. Jack travels at a speed of 15 km/hour while Ben travels at a speed of 12 km/hour. It takes Ben 15 minutes more to complete his travel than Jack does. What is the distance between A and B?

7. A student carried out an experiment to compare the lifetime of two types of light bulbs. She obtained the following results:
a. The lifetime of type A bulbs are 140, 125, 135, 141, 135, 142 and 123 days.
b. The lifetime of type B bulbs are 135, 125, 145, 122 and 139 days.
Which type of light bulb has a longer lifetime? Why?

8. Barbara writes numbers consisting of four digits: 3, 5, 7 and 9, according to the following rules:
• Digit 7 does not appear in the first nor the last positions.
• Digit 7 should be to the right of the digit 5. (For example, digit 5 in the number 7395 appears to the right of digits 7, 3 and 9).
Find all such possible numbers.

9. The pages of a book are numbered using 840 digits, starting from page 1. How many pages does the book have? (For example, page 37 uses two digits, namely digits 3 and 7. From page 1 to page 11, thirteen digits are used.)

10. How many positive whole numbers less than 2005 can be found, if the number is equal to the sum of two consecutive whole numbers and also equal to the sum of three consecutive whole numbers? (For example, 21=10+11=6+7+8.)

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1. The product of two three-digit numbers

2. In a right-angled triangle ACD, the area of shaded region is 10 cm^2, as shown in the figure below. AD = 5 cm, AB = BC, DE = EC. Find the length of AB.

3. A wooden rectangular block, 4 cm × 5 cm × 6 cm, is painted red and then cut into several 1 cm × 1 cm × 1 cm cubes. What is the ratio of the number of cubes with two red faces to the number of cubes with three red faces?

4. Eve said to her mother, “If I reverse the two-digits of my age, I will get your age.” Her mother said, “Tomorrow is my birthday, and my age will then be twice your age.” It is known that their birthdays are not on the same day. How old is Eve?

5. Find how many three-digit numbers satisfy all the following conditions:
if it is divided by 2, the remainder is 1,
if it is divided by 3, the remainder is 2,
if it is divided by 4, the remainder is 3,
if it is divided by 5, the remainder is 4,
if it is divided by 8, the remainder is 7.

6. A giraffe lives in an area shaped in the form of a right-angled triangle. The base and the height of the triangle are 12 m and 16 m respectively. The area is surrounded by a fence. The giraffe can eat the grass outside the fence at a maximum distance of 2 m. What is the maximum area outside the fence, in which the grass can be eaten by the giraffe?

7. Mary and Peter are running around a circular track of 400 m. Mary’s speed equals 3/5 of Peter’s. They start running at the same point and the same time, but in opposite directions. 200 seconds later, they have met four times. How many metres per second does Peter run faster than Mary?

8. Evaluate

9. A, B and C are stamp-collectors. A has 18 stamps more than B. The ratio of the number of stamps of B to that of C is 7:5. The ratio of the sum of B’s and C’s stamps to that of A’s is 6:5. How many stamps does C have?

10. What is the smallest amount of numbers in the product 1× 2× 3× 4×…× 26× 27 that should be removed so that the product of the remaining numbers is a perfect square?

11. Train A and Train B travel towards each other from Town A and Town B respectively, at a constant speed. The two towns are 1320 kilometers apart. After the two trains meet, Train A takes 5 hours to reach Town B while Train B takes 7.2 hours to reach Town A. How many kilometers does Train A run per hour?

12. Balls of the same size and weight are placed in a container. There are 8 different colors and 90 balls in each color. What is the minimum number of balls that must be drawn from the container in order to get balls of 4 different colors with at least 9 balls for each color?

13. In a regular hexagon ABCDEF, two diagonals, FC and BD, intersect at G. What is the ratio of the area of ?BCG to that of quadrilateral FEDG?

14. There are three prime numbers. If the sum of their squares is 5070, what is the product of these three numbers?

15. Let ABCDEF be a regular hexagon. O is the centre of the hexagon. M and N are the mid-points of DE and OB respectively. If the sum of areas of ?FNO and ?FME is 3 cm^2, find the area of the hexagon.

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