Questions for Preliminary (difficulty: easy, problem solving)

1. During Jalan Raya, Najib was given some peanut beans and a pair of bowls by his uncle.                               He was told to divide the beans into the bowls. His uncle will then count the number of                                 beans in both bowls and then multiply the numbers, the result will be the dollar amount                             that his uncle will give to Najib for Hari Raya.

What should Najib do to maximize his $?

2. The number 479 has the interesting property that when it is:a. divided by 6, the remainder is 5b. divided by 5, the remainder is 4c. divided by 4, the remainder is 3d. divided by 3, the remainder is 2e. divided by 2, the remainder is 1

What is the smallest number with the same property?

3. Bill Gates managed to sell a lot of copies (more than a million!) of Microsoft Windows                             such that he gained $1,000,000,000.00 in revenue. The copies were sold at the same                         price each. Interestingly, neither the price nor the number of copies had any zero in the                             numbers.

How many copies were sold at how much price each?

4. Notice that 23 x 96 = 32 x 69

There are several such pairs of two­digit numbers where their product stays the same                         when the order of the digits is reversed. And this does not include the trivial ones like: 22                                 x 55 = 22 x 55, 12 x 21 = 21 x 12, and such.

How many can you find? Investigate!

5. This is a very old puzzle. It tells of a showman travelling the countryside on tour with a                                 wolf, a goat, and a cabbage. He comes to a river bank and the only means of getting                                 across is a small boat which can hold him with only one of the wolf, the goat or the                                   cabbage.

Unfortunately he dare not leave the wolf alone with the goat or the goat alone with the                               cabbage for the wolf would eat the goat and the goat would eat the cabbage. After some                               thought the showman realised that he could use the boat to transport himself and all his                             belongings safely across the river. How did he do it?

6. After a flood three married couples found themselves surrounded by water, and had to                         escape from their holiday hotel in a boat that would only hold three persons at a time.                               Each husband was so jealous that he would not allow his wife to be in the boat or on                                   either bank with any other man (or men) unless he was himself present.

Find a way of getting the couples across the water to safety which requires the smallest                             number of boat crossings. Swimming or helicopters are not allowed!

Questions for Preliminary (difficulty: easy, direct question)

1. A right triangle has its right angle at . is twice as long as . How      ABC ý             ACB /   BA             CA  many degrees is angle  ?ABC /

2. Can a fraction whose numerator is less than its denominator be equal to a fraction whose                             numerator is greater than its denominator? Explain and give example if applicable.

3. There are many ways in which the digits 1, 2, 3, 4, ... 9 can be arranged to form a                                     four­digit number and a five­digit number, for example 5324 and 89716, but only one way                           which maximises their product. Can you find it?

4. This is a riddle attributed to Euclid from AD 300.

A mule and a donkey were walking along laden with sacks of corn. The mule said to the                                 donkey, 'If you gave me one of your sacks, I would be carrying twice as much as you.                                 But if I gave you one, we would both be carrying equal burdens.'

How many sacks of corn were they each carrying?

5. Madam Rose went to supermarket to buy some fruit for her daughter Yasmin and her                           friends who came for their study group. She bought apples at 40 cents each and oranges                             at 70 cents each. She paid for the fruits and got 10 cents change for her $30 cash.                                 Yasmin and her friends each had equal share of the fruits.

How many friends did Jasmine have at the study group?How many apples and oranges did they each have?

6. The sequence below is called Fibonacci sequence. Can you identify the pattern and                       continue the sequence for the next five numbers?

1, 1, 2, 3, 5, 8, 13, ...

7. Four Fours is a famous puzzle playing with four number 4s. The puzzle is actually                           simple, by only using four pieces of number 4s, and any mathematical notation you like,                           make all the numbers from 0 to 100. For example, 0 = 44 ­ 44.

For this competition’s question, make 1 to 10 using these four fours.

Questions for Semifinal (difficulty: medium, problem solving)

1. A hunter was making a camp when he saw a bear. He then chased the bear towards                               South for a kilometer, and then he make a right turn towards West for a kilometer, and                               then make another right turn towards North for a kilometer, suddenly he’s back at his                           camp.

What is the color of the bear?

2. There are five circles with radius of: 50 cm, 40 cm, 20 cm, 20 cm and 10 cm. The four                                     smaller circles are to be put overlapping with the largest circle. Show how to overlap                           them so that the non­overlapping part of the largest circle has the same area as the total                               area of non­overlapping parts of the smaller circles. Explain your answer.

3. Ah Luck loves number 8, it’s his lucky number. Everytime he sees a digit 8 he consider                               he has one luck. So, you can say he has one luck in the sequence 1, 2, 3, ..., 8, 9, 10.

a. How many lucks does he have in the sequence 1, 2, 3, ..., 100?b. How many lucks does he have in the sequence 1, 2, 3, ..., 1,000?c. How many lucks does he have in the sequence 1, 2, 3, ..., 1,000,000?

4. Great Grandmother Bountiful, who only had daughters, realised that each of them had                       produced as many sons as they had sisters, and no daughters. In turn, each of her                             grandsons had produced as many daughters as he had brothers. She was delighted to                         recount this to her friends and, furthermore, that the total number of her daughters,                         grandsons and great­granddaughters was the same as her age!

How old was she?

5. At Zoe's party, Anna, Betty and Candice are standing in line behind each other, with                           Candice at the back able to see Anna and Betty, Betty in the middle able to see Anna,                                 and Anna at the front unable to see the others.

From a collection of two blue and three red paper hats, Zoe places a hat on each of the                                   heads of Anna, Betty and Candice, so they can only see the hats of those in front of                                 them, and do not know which hats are unused.

Zoe then asks each of them, in turn, if they know what colour hat they are wearing.                               Candice, at the back, replies ’No’ followed by a 'No' from Betty. But then Anna, who                             could see none of the hats, was able to give the colour of her hat with confidence. What                                 colour was it?

6. An army on the march through the jungle came to a river which was wide, deep and                               

infested with crocodiles. On the far bank they could see two native boys with a canoe.                             The canoe can hold one man with his rifle and kit or two boys. How does the army cross                                   the river?

8. Ainun stays at Bukit Batok East area which is 3 bus stops away from Bukit Batok                             Interchange. He can take either bus 106 or 77 from the interchange to reach home. As                             Singapore busses are pretty much on time, and both two bus services has 10 minutes                           interval, he never bothered to check their timetables and just wait for whichever service                         starts first whenever he reach Bukit Batok Interchange from the MRT station.

After sometime he realized that he got into bus 106 far more often than 77. He decided to                                 keep a regular check on which service he caught and surprisingly he only caught 77 on                             average once every 10 times. How can this phenomena be explained, considering that                       both bus services has the same 10­minutes interval?

Questions for Semifinal (difficulty: medium, direct question)

1. Is it possible to find a pair of integers a and b such that  ? ba = √2

2. Dzuizz is having a birthday party, eight people were present including his parents and                         himself. How many cuts at minimum do you need to cut his birthday cake into 8 equal                               pieces?

3. Where on earth can you find a triangle each of whose interior angles is 90°?

9. Professor Danzig enjoyed looking for patterns in numbers, and on her birthday in 1992                         she was particularly pleased when she realised that her age in years multiplied by the                           day in the year came to 11,111.

When was she born?

10. The local water company wanted to build a new storage tank to supply the needs of a                               large housing estate. They calculated that they needed the tank to have a capacity of                           

, and they decided to construct it as an open­top rectangular tank. It was to, 00 m4 0 3                            be made from reinforced concrete panels whose cost was proportional to the interior                       surface area of the tank.

What shape should they make it to minimise the cost?

Questions for Final (difficulty: hard, problem solving)

1. Hakim and his friends were playing a game with numbers. They wrote numbers from 1 to                             10 on a white board. They then take turns to choose and erase two numbers from the                               board and replace them with their difference. For example, if Hakim choose 2 and 7, he                             will erase them both and then write 5 on the board.

Prove that when the game ends with only one number remaining on the board, that                           number will always be an odd number.

HINT: try keeping track of sum of the numbers remaining on the board

2. Factorial in mathematics is multiplication of integers from 1 until the number being                       factorial­ed and is represented by ! (exclamation mark). For example,                 

. Note that has one zero at its end. How many zeros! 1 24 4 = × 2 × 3 × 4 =       ! 1205 =                  are there at the end of   ?00!1

3. Triplets of integer numbers that can become lengths of a right triangle’s sides is        a, b, c)(                    called Pythagorean Triple. This means  .a2 + b2 = c2

is the only Pythagorean Triple with three consecutive numbers. However, for3, 4, 5)(                      any odd number , there will always be and such that and are consecutive      a           b     c       b     c    numbers and   is a Pythagorean Triple.a, b, c)(

Investigate how to find the values of such   given an odd number  .a, b, c)( a