MUMS Problem Solving Competition Melbourne University Mathematics and Statistics Society 14 June, 2006
RULES Student teams should have four competitors, while teachers teams should have three. The competition will consist of five rounds, each lasting ten minutes. In each round there are five questions, but only your best three questions will contribute to your score. The questions have 3, 4, 5, 6 and 7 points allocated respectively. Express your answers as exact numbers (surds and fractions, not decimals). You will not lose points for incorrect answers. Prizes will be awarded to the top three school teams and the top teachers team. You must have fun!
ROUND ONE 1. What is the largest number of pieces into which a cylindrical cake can be cut using three straight cuts (in any direction)? 2. Sam pays tax on his income. Had he not paid the tax, he would have 60% more money than he has now. What is the rate of the tax? 3. How many positive factors (including 1 and itself) does the number 360 have? 4. A circle of area 1cm 2 fits just inside a square, which fits just inside another circle. What is the area of the larger circle? 5. Han thrusts a 2m spear into a river, hoping to catch a fish. Instead, he hits the bottom of the river. If 1.6m of the spear is above the surface, with the end 1m above water level, how deep is the river?
ROUND TWO 1. If 3 policemen eat 12 doughnuts in 4 days, how long do 6 policemen take to eat 18 doughnuts? 2. Find a 3-digit perfect square such that the first and last digit add to the middle digit. 3. Norm suspects either Alisa, Michael or Ray ate his chocolate bar. Alisa says: I didn t eat it. Michael says: I didn t eat it. Ray says: Alisa ate it. One of them always tells the truth, and two of them always lie. Who ate the chocolate? 4. If 5 apples and 7 oranges cost twice as much as 7 apples and 2 oranges, and one apple and one orange cost 80 cents, how much does 5 apples and 7 oranges cost? 5. How many rectangles (including squares) are there on a chess board (which is an 8 8 grid of squares)?
ROUND THREE 1. There are positive integers a and b are such that a 4 = b 4 + 65. What is a + b? 2. In a race, Daniel came five places in front of the person who finished last, and two places in front of the person who finished fifth. How many people were in the race? 3. Two identical equilateral triangles of side length 10cm overlap in a regular hexagon. What is the perimeter of this hexagon? 4. Simplify: 32 121 16 151 8 201 4 301 5. Stephen writes down 10 consecutive numbers. He then erases one of them, and finds the remaining numbers add up to 2006. What number did he erase?
ROUND FOUR 1. Sally has a stack of cards. She gives 1 3 of them to Nick, and 3 2 of the remainder to Andrew. If Nick received 30 cards, how many does Sally have left? 2. Adrian owns some cows and chickens, who together have 28 heads and 82 legs. How many chickens does Adrian have? 3. How many numbers between 0 and 300 are not divisible by 3, 5 or 7? 4. In the game Casino War, the dealer deals everyone a card (from a 52-card deck), then takes a card himself. You win if your card is higher than the dealer s, with all suits being equal. What is the probability of winning? 5. James arranges 6 identical coins in a triangle shape so that they all touch each other. If the top of the top coin is 10cm above the bottom of the bottom coin, what is the radius of one coin?
ROUND FIVE 1. How many ways are there of making 50 cents with standard Australian coins? 2. Boris is twice Abdullah s age. In three years time, Boris will be three times Cinderella s age. If the sum of all their ages is 31, how old is Cinderella? 3. How many 3-digit numbers are larger when their digits are reversed? 4. A spherical balloon is being filled with water at a constant rate. At 10:00am, its radius is 1cm, and at 10:05am, its radius is 2cm. When will its radius reach 8cm? 5. Kim is constructing a new building, but he can only afford 7 lifts, each going to 6 floors. If one can go from any floor to any other floor using only one lift, what is the greatest number of floors his building can have?
TIE-BREAKER Question 1 is worth 1 point, while Question 2 is worth 2 points. 1. A 3-digit number leaves a remainder of 15 when divided by 31, and a remainder of 20 when divided by 41. What is the 3-digit number? 2. Maurice has 4 straight pieces of wood, with lengths 1cm, 4cm, 7cm and 8cm. What is the greatest area he can enclose?