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INDIVIDUAL CONTEST PROBLEMS
Po Leung Kuk 11th Primary Mathematics World Contest
English Version
1. 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?
2. 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 even number
choosing 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 2516  238 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
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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 eat 1 or 2 of 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+D represent the eight different natural
numbers 1 to 8. If A is the largest number amongst A, B, C and D, what is
A?
12. A nine-digit number abcdefghi is such that its digits are all distinct and
non-zero. The two-digit number ab is divisible by 2, the three-digit
number abc is divisible by 3, the four-digit number abcd is divisible by 4,
and so on so that the nine-digit number abcdefghi is divisible by 9. Find
this nine-digit number.
13. In how many ways can seven students A, B, C, D, E, F and G line up in
one row if students B and C are always next to each other?
14. A 1001-digit number begins with 6. The number formed by any two
adjacent 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).
8 cm
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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 is a different colour. Two hats are cotton ($30
each), five are leather ($50 each) and three are wool ($10 each). How many
ways are there to buy 5 hats such that the total cost is more than $101 but
less than $149?
Question 3:
On a 1 5 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.
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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 1 cm rolls along the inside lines of the picture. The side
length of each small square in the picture is 1 cm. What is the area in square
centimetres that the circle covers when it rolls along the inside lines once?
(Take   3.14)
Question 6:
In  ABC, E is the midpoint of BC. F is on AE where AE = 3AF. BF meets AC
at D as shown in the figure. If the area of  ABC = 48, find the area of
 AFD.
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Question 7:
Consecutive counting numbers are 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 1 cm 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.
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ANSWERS OF THE CONTEST PROBLEMS
Po Leung Kuk 11th Primary Mathematics World Contest
Hong Kong, 13 – 18 July 2007
Individual Test
1. 1:2
2. 775 x 33
3. 52
4. 23
5. 20
6. 112
7. 34
8. 28
9. 89
10. 64
11. 6
12. 381654729
13. 1440
14. 692346
15. 38.88
Team Contest
1. C
2. 35
3. 152415
4. W B, X A. YD, ZC
5. 52.99
6. 1.6
7. 4042148175
8. 25
9. 186
10. 41/450
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