November 14, 2010

The first one is from Mathematics Olympiad Problems (1996 Vologda). The probability ones are inspired from enrich.maths.org

• There are 3 boxes, each consisting of two balls. One of the boxes contains exactly two white balls, one consists of exactly two red balls, and the third one consists of exactly one white and one red ball. But while labeling the boxes, their labels got mixed up so that none of the boxes has the right label. Is it possible to identify  which  box contains what by just drawing exactly one ball from each box? Can it be done by drawing just two balls or one ball?
• Since one of the kids in our team is a president of his class, he needs to come up with creative ways to do fund raising. What could be a better way then doing some kind of lottery!  Mr. President decided to setup the lottery as follows. In a bag, he places 4 identical hockey pucks in a bag, and numbers them from 1 to 4. Each kid (there are 24 of them who wants to make fortune!) walks up to him, hands him $1, and makes a guess from 1 to 4, and draws the puck. If the guess works, that kid gets $2, otherwise looses his money for the fundraiser. Do you think whether Mr. President will raise any funds? (BTW - kids can't gamble!)
• In the above setup, kids are asked to guess two numbers, instead of one. Then they draw two pucks and if the numbers on the puck matches with their guess, they win. First they draw a puck - record what is the number, places this puck back in the bag and then draw the second one (i.e., with replacement). For example if I guessed 12, then either I draw (1,2) or (2,1), I win. Do you think whether President will raise more money by this scheme. What happens when it is without replacement?
• What is the probability that two kids in the same family have a birthday on the same day?  There are approximately 30,000 families in City of Waterloo with two kids. How many of them will likely have kids having birthdays on the same  day.  There are approximately 14,000 families with three kids. What is the probability that all three of them will have their birthday on the same day.
• You are given a right angled triangle. You need to draw a line parallel to its base, so that the triangle is split in two parts of equal area. Where should one draw the line?
• There is a study done in Ontario schools, in terms of whats the average number of hours/per week the students in Grade 9 spend on their homework. Here is the data: 40 % don't bother, 10% spend 1-5 hours, 10% spend 6-10 hours, 20% spend  11-15 hours,  18% spend 16-20 hours and the rest spend more than 21 hours. Whats the average time a grade 9er spend on their homework?
• Consider the pattern:  The first term when n=0 is 2, when n=1 it is 8, when n=2 it is 14,  when n=3 it is 20, ... Once you figure out this pattern, you will realize that it is a linear pattern. You can graph it and see that it is a straight line.  For a pattern which grows linearly, show that it is sufficient to have two consecutive values to find the pattern rule.  Consider a pattern which follows a quadratic growth, for example the pattern when n=0 it is 1, when n=1 it is 2, when n=2 it is  5, when n=3 it is 10, ...17, .. 26,... Show that it is sufficient have three consecutive values to determine the pattern rule.  How does the graph of these patterns look like?