November 20, 2011

Today's problems are inspired from the set of Problems from the Waterloo's Centre for Education "Problem of the Week" collections.

1. Consider the above figure. Assuming that the angle BAC is x-degrees, and |AB|=|BC|=|CD|=|DE|=....   How many triangles like ABC,  BCD,  CDE, DEF, .. one can form?  Of course this will depend upon the angle x. Try it for x=15, 30, 45, .. (Smaller angles are more interesting than the large ones!) 
2. How many different numbers are there between 10 and 99, so that the sum of their digits equals 10? Try for numbers between 100 and 999?
3. Take a right angled triangle, say with vertices ABC, where |AB|=3, |AC|=4, |BC|=5. Imagine that this triangle is standing on the edge AB, and you rotate it full circle (360) by keeping A fixed, but moving B the full circle. What is the volume of the swept figure? What if the triangle is standing on just the vertex A, and you rotate it full circle. What will be the volume of the swept figure?
4. There are six houses on a street, where the average income of 1st and 2nd is 70k, 2nd and  3rd is 80k, 3rd and 4th is 90k, 4th and 5th is 100k, 5th and  6th is 80k. What is the average income of this street? Whats the average income of 1st and 6th? If I tell you that the first household made 80k, then can you determine the income of each of the house? [In many survey statistics, they  collect several types of information - family income, age, ethinic group, education qualification, etc., and then one can query the database in terms of finding out average statistics about a neighborhood.  For example you can look at study of immigrants in Ottawa Area.  There is a possibility that by looking at different statistical tables, you may be able to gather some extra information. For example, from the study of immigrants, a grocery store in a particular locality may carry more Caribbean delicacies, or Chinese, or South Asian. Similarly knowing the average age and income, the marketing and advertisement may be targeted. In our problem, if I didn't tell you that the first house made 80k, what extra piece of information can help me to deduce the income of each of the families?] 
5. Consider a five digit number abcde. The digits 1, 2, 3, 4 and 5 are used to form this number (all digits are being used). You need to find what is the number, and the extra information  provided to you is that abc is divisible by 4, bcd is divisible by 5, and cde is divisible by 3. [Recall that a number is divisble by 5, if the last digit is either a 0 or 5. A number is divisible by 4 if the last two digits are divisible by 4, and a number is divisible by 3 if the sum of its digits are divisble by 3.]
6. A number is split into several equal parts in the following way. 1st part is 10 + 10% of the remaining. 2nd part is 20 + 10% of the remaining now. 3rd part is 30 + 10% of the remaining now. 4th part is 40 + 10% of the remaining now. and so on.  If each  part has the same value, then how many parts are there, and what is the value of each part? You can try this with 20% in place of 10% and see what happens?
7. Mr A., being an A student, received 94% marks average among the six subjects in his mid-term exam. But just before he was supposed to receive the great honor from the School's principal, his French teacher realized that his marks in place of being 96/100 should have been 69/100. (May be the teacher read them upside down!). Whats Mr A's right average then?  
8.  Two airplanes are trying to land at Ottawa Airport on the same runway! The control tower needs to determine how to ensure that they can safely land. Control tower knows which aircrafts are coming, and it knows the location of the two aircrafts in the sky (say their distance and what angle they are with respect to some reference). The tower needs to tell the aircrafts what path to follow,  ensuring that they are separated by 5 Nautical Miles (approximately 10Kms) from each other at any point of time till they land. Can you think of some strategy for figuring out these paths. Now think about worlds busiest airports like the one in Atlanta or in London or in Chicago.  Approximately 200 flights/hour arrive in Atlanta (thats 3 per minue!).