**Individual Contest Problems**

Po Leung Kuk 11th Primary Mathematics World Contest

English Version

- A regular hexagon is given. The vertices of the rectangle lie on the midpoints of the sides of the hexagon. What is the ratio of the area of the rectangle to the area of the hexagon?

- A multiplication of a three-digit number by a two-digit number has the form as shown below. Using only the digits 2, 3, 5 or 7,fill in all boxes to complete the correct multiplication.

3.How many different ways are there to form a three-digit evennumberchoosing the digits from 0, 1, 2, 3, 4 or 5 without repetition?

4.For how many whole numbers between 100 and 999 does the product of the ones digit and tens digit equal the hundreds digit?

5.In a survey of 100 persons, it was found that 28 read magazine A, 30 read magazine B, 42 read magazine C, 8 read magazines A and B, 10 read magazines A and C, 5 read magazines B and C and 3 read all three magazines. How many people do not read any of these magazines?

6.A school has to buy at least 111 pens. The pens are sold in packs of 5 which cost $6 per pack or packs of 7 which cost $7 per pack. What is the lowest cost at which the school can buy the pens?

7.How many digits does the product have?

8.On a wooden rod,there are markings for three different scales. The first set of markings divides the rod into 10 equal parts; the second set of markings divides the rod into 12 equal parts; the third set of markings divides the rod into 15 equal parts. If one cuts the rod at each marking, how many pieces does one get?

9.There are ten identical candies in a jar. Albert can only eat1 or 2of these candies at a time.He does this until there is no more candy left. In how many different ways can he do this?

10.The entrance fee to a museum is $5 per adult and $4 per child. For any group of five people, the entrance fee is $19. Two adults who pay the full entrance fee may take a child for free. Three adults and fourteen children come to visit the museum. What is the least amount they need to spend on the entrance fee?

11.A, B, C, D, A+C, B+C, A+D, B+Drepresentthe eight different natural numbers 1 to 8. If A is the largestnumber amongst A, B, C and D,what is A?

12.A nine-digit number is such that its digits are all distinct and non-zero.The two-digit number is divisible by 2, the three-digit number is divisible by 3, the four-digit number is divisible by 4, and so on so that the nine-digit number is divisible by 9. Find this nine-digit number.

13.In how many ways can seven students A, B, C, D, E, F and Gline up in one row if studentsB and C are always next to each other?

14.A 1001-digit number begins with6. The number formed by any twoadjacent digits is divisible by 17 or 23. Write down the last six digits.

15.The pattern below is formed by drawing semi-circles inside squares. The radii of three types of semi-circles are 4 cm, 2 cm and 1 cm respectively. What is the total area of the shaded regions? (Take = 3.14).

**Team Contest Problems**

Po Leung Kuk 11th Primary Mathematics World Contest

English Version

Question 1:

The diagram below is the street map of a small town. There is a very strange traffic rule. No turns are allowed at any intersection unless it is impossible to drive straight on. Then both left turns and right turns, if possible, are allowed. Entering the town from point E, it is possible to exit from any other point except one. Which exit is impossible?

Question 2:

There are 10 hats. Each hat isa differentcolour.Twohats are cotton ($30 each), five are leather ($50 each) and threeare wool ($10 each). How many ways are there to buy 5 hats such that the total cost is more than $101butless than $149?

Question 3:

On a board are four counters which are white on one side and black on the other side. A counter can only change position by jumping over at least one other counter and landing on the empty space. When a counter has been jumped over, it is flipped over, but the jumping counter itself is not flipped. The configuration in the diagram below on the left must be changed to that on the right in six jumps. Record each jump by indicating the initial position of the jumping counter. Give one possible solution and its corresponding 6-digit number.

Question 4:

At a certain school, four students W, X, Y and Z were predicting their grades before the final examination.

W said:We will all get different grades.

If I get an ‘A’, then Y will get a ‘D’.

**X said:If Y gets a ‘C’, then W** will get a ‘D’.

W will get a better grade than Z.

Y said:If X does not get an ‘A’, then W will get a ‘C’.

If I get a ‘B’, then Z will not get a ‘D’.

Z said:If Y gets an ‘A’, then I will get a ‘B’.

If X does not get a ‘B’, I will not either.

After the final examination was graded, each of the students got his grade as predicted. What grade did each student get?

Answer: /**W : X: Y : Z :**

Question 5:

A circle of radius 1cm rolls along the inside lines of the picture. The side length of each small squarein the picture is 1cm. What is the area in square centimetres that the circle covers when it rolls along the inside lines once?

(Take

Question 6:

In *ABC, E is the midpoint of BC. F is on AE where AE = 3AF. BFmeetsAC at D* as shown in the figure. If the area of ABC = 48, find the area ofAFD.

Question 7:

Consecutive counting numbersare grouped as follows:

(1), (2, 3), (4, 5, 6), (7, 8, 9, 10), . . .

There is one number in the first group, two numbers in the second group, and three numbers in the third group, etc. What is the sum of all numbers in the 2007th group?

Question 8:

A rectangle of area 3456 cm2 lies on the grid lines of a larger grid which is formed by squares of side 1cm as shown below:

We call the points where the grid lines meet “points of intersection” For example, the diagonal of a 2 cm 4 cm rectangle passes through 3 points of intersection.

What is the greatest possible number of points of intersection which a diagonal of the rectangle of area 3456 cm2 can pass?

Question 9:

There are 20 piles of stones.Each has 100 stones. Choose one of the twenty piles, take one stone from each of the remaining 19 piles and put them onto the chosen pile. This is called an operation. In subsequent operations, you may choose any pile amongst the twenty piles, and repeat the above process. After less than 50 operations, there are 66 stones in one of the piles. The number of stones in another pile is between 170 and 200 (inclusive). What is the exact number of stones in this pile?

Question 10:

A palindromic number is a whole number that is the same when written forwards or backwards (for example, 11511, 22222, 10001). Find the ratio, in proper fraction form,of the number of all five-digit palindromic numbers which are multiples of eleven to the number of all five-digit palindromic numbers.

## ANSWERS OF THE CONTEST PROBLEMS

Po Leung Kuk 11th Primary Mathematics World Contest

### Hong Kong, 13 – 18 July 2007

#### Individual TestTeam Contest

1. 1:21. C

2. 775 x 332. 35

3. 52 3. 152415

4. 234. W B, X A. YD, ZC

5. 205. 52.99

6. 1126. 1.6

7. 347. 4042148175

8. 288. 25

9. 899. 186

10. 6410. 41/450

11. 6

12. 381654729

13. 1440

14. 692346

15. 38.88

1