Sunday May 30, 2010

1. Ethiopian Tune ran 10K in 32 minutes and 11.5 seconds. What was her speed in Km/Hr? 
2. In comparison, in Vancouver 2010 olympics, 10k speed skating record for men was just under 13 minutes. What the speed in km/Hr.
3. Usain Bolt ran 100m in 9.58 seconds. What is his speed?
4. 100m swimming free style record is 46.91 seconds by Cesar Cielo. Whats the speed?
5. Niemi save percentage in the goal is .916. If Philly wants to score 5 goals in a game, how many shots, on average, needs to be directed at the Chicago's net?
6. In a class, there are 15 young men and 15 young women. When polled, it turns out that each  men dates 4 women, and each women dates 3 men. Is it possible? 
7. You have invited 10 guests for a party.   A custard can serves three guests. How many cans you need?
8. On a windless Sunday morning, you biked on the Colonel By Drive (which is essentially a straight North-South road)  for its entire 10 km length, starting at one end, reaching the other end, and then coming back to the start point. Following Sunday you did exactly the same, but there is a 25 KM/Hr wind blowing from North.   Suppose the amount of effort you put in is identical on both the days -  will the total time of trip on both the Sundays will be same?
9. What music to play when climbing the Eiffel Tower. There are in all 1665 steps to reach the top of the tower (actually we can't since this is an Emergency Exit!).  Suppose you can climb at a constant pace, determine how many steps you can climb in a minute - whats the right beat - which is perfect music - and how long will it take?
10. According to the Canadian Tire Paint Calculator, a room of dimensions 4m times 4m times 3m requires 5.58 l of paint for the walls and 1.86 lit for the ceilings. Given this information how will you determine how much paint you will require for your room?
11. Geometric Construction: Draw a triangle with sides 9cm, 7cm, and 5cm and then determine its area.   

Monday, May 24, 2010 (Victoria Weekend)

The first two problems are motivated from "Our days are numbered" by Brown.

1. Here is a simple game about mind reading:  Bob tells Alice to guess two numbers between 0 and 9.  Bob asks Alice to perform the following four steps (a) multiply the first number by 5 (b)  add 9 to the product (c) double the sum (d) lastly add the second number to the result. Then he asks Alice to tell him the result - and then he subtracts 18 from her answer, and now he knows both the numbers which Alice guessed! Why it works? (For example if Alice guessed 3 and 5, then the steps results in (a) 3x5=15 (b) 15+9=24 (c) 2x24=48 (d) 48+5=53. Bob subtracts 18 from 53 and it equals 53-18= 35 - the two digits which
2. Show that the remainder, when a number is divided by 9, is the same as the sum of its digits (if the sum exceeds 9, then keep taking the sum of the digits, till it is below 9). For example, we know that 345/9 results in a remainder of  3. The sum of the digits of 345= 3+4=5=12, which in turn is 1+2= 3, and it is the same as the remainder.
3. Continuing on the theme of the angle bisector from the last week, how will you draw two lines which are perpendicular to each other (only compass and ruler).
4. Give a line L and point p (p is not on L), how will you find the closest point to p on L geometric construction..
5. Given a triangle ABC, how will you draw an incircle of ABC. An incircle is the largest circle completely contained in the triangle. By the way the center of this circle is the meeting point of the angle bisectors.
6. What about circumcircle of ABC - that is the smallest circle that completely contains ABC. Its center is on the perpendicular bisectors of the  sides of triangle.

May 16, 2010

1. The four circles of equal radius (say 1cm each)  are arranged in such a way that their centers make a perfect square, where each side of the square is 2cm long. What will be the dimension of the little circle that can be placed between all these four circles, such that the little circle just barely touches the four big circles. What is its area compared to the big circles?   (Lean More: Pythagoras Theorem).
2. Actually it may be best that you try to draw the above figure using your geometry set. First draw the square, then the four circles, and if your drawing is perfect then the red circle will touch all the other four circles!
3. How many numbers are there between 1 and 30, which are not prime, not a multiple of 4 or 5, and are  divisble by 2 and 6.
4. Consider two points A and B which are at distance 5cm apart on the plane. Rotate A around B at each possible angle (300,270, 180, 90, 45, 30, ...).  Consider all the images of A. What shape they form?  What happens when A and B are in space - what does the angle of rotation means?
5. In the previous problem - say A is a disc and B is a point. Then what shape the union of the images of B form?
6. How many of the following statements are true? (a) The number of False statements are 1  (b) The number of False statements are 2  (c) The number of False statements are 3 (d) The number of False statements are 4.
7. The sum of the ages of a couple is 77 years. Mr. He is now twice as old as Mrs. She was when he was as old as she is now. What are the current  ages of Mr. He and Mrs. She?
8. How many pennies are there if in your pocket you have 50 coins making a change for a dollar?
9. If 5 is added to 1/3rd of a number, it result in  1/2 of that number. Whats that number?
10. How will you bisect an angle between two lines, without actually measuting the angle (you are allowed to use the compass).

May 9, 2010

1. Five years ago Vic's age was 1/3-rd of the Ash's. Now Vic's age is 17. What is the age of Ash right now?
2. The sum of the ages of the Dad-Son pair is 45 years right now. Five years ago the product of their ages was 4 times the age of the dad at that time. What is the current age of the Dad and the Son?
   
3. In a chess tournament each of the six players will play all other players exactly once. How many games will be played in all?
4. The KidsPlay store bought 6 teddy's for $10, and sold them as 4 for $10. In all they made $60 profit from the teddy's. How many teddy's did they buy?
5. In a quiz, I have 7 times as many correct answers as the wrong ones. In all there were 10 dozen questions in the exam. How many did I get right?
6. There are two tall cedar trees in the Stonecrest park. The trees are 40m apart, and they are 30 m tall. Suppose there is a rope running from the top of each tree to the bottom of the other tree, how high above the ground the two ropes will intersect.
   
7. Suppose in the previous question, the trees were 20 m and 30 m tall, how high above the ground will the ropes meet?
8. Suppose a leaky tap drips two drops of water every second, each drop is about 2ml. How much water is wasted in one year. Suppose you require 2 bucket full of water every day (approx. 40 l), compare that with the amount wasted water.
   

May 2, 2010

1. Four Children A, B, C, and D are natives of four different countries (France, UK, USA, Argentina). Figure out the their native land from the following: (a) B is not from Americas (b) A is not born in Europe (c) D's native language is not English (d) C is not from the Northen Hemisphere.
2. Any natural number can be expressed as sum of at most four squares. For example: 13=1^2 + 2^2 + 2^2 + 2^2; 31=1^2+1^2+2^2+5^2; 39=1^2+2^2+3^2+5^2. Check this for 50; 91. (Learn More: Lagrange's four square theorem.)
   
3. You have only two measuring devices: a 5 liter bucket and a 3 liter bucket. Devise a scheme so that exactly 4 liter of water can be placed in in the 5 liter bucket. You can assume that you have a tap/pipe from which you can draw as much, and as often, water as you want.
4. Find the number if it satisfies the following: (a) If it is not a multiple of 4, then it is between 60 and 69. (b) If it is a multiple of 3, then it is between 50 and 59. (c) If it is not a multiple of 6, it is between 70 and 79.
5. You have in all 72 coins made up of nickels and dimes, totaling to $4.95. How many nickels you have?
6. A4 paper sheet is 8.5 inch by 11 inch. There are two ways to roll it - by long side and by short side - to form a cylinder. Which of these two cylinders will have larger volume?
7. Why are manhole covers made circular and not squares or rectangles or ellipses? (Think of their diameter versus the smallest side.)
8. Spread your fingers (and hand) on a piece of paper, and trace the contour. Between two consecutive fingers, you can measure the angle between two adjacent fingers (say the middle and ring). What is it? Do you think that this angle will be vastly different between a boy and a girl or between an adult and a child?
9. In the Papanack Zoo, you see 30 eyes and 44 legs. How many 2-legged and 4-legged animals are there?