- On the boxing day, the sale price of all the items in the Eagle was dropped by 20%. By what percentage they need to be raised, to bring them back to the level they were before the boxing day.
- If a number x is divided by 7, the remainder is 5. When the number y is divided by 7, the remainder is 3. What is the remainder when x+y is divided by 7.
- The average of a set of 5 numbers is 30. The average of three of them is 24. What is the average of the other two numbers?
- Given two numbers x and y, find the average of (x+y)^2 and (x-y)^2. Check your answer by setting x and y as (3,4) , (3-3), ...
- Mr. Generous wants to buy the best possible home theater for his family, so that they can enjoy the vacation by watching best of the best. He goes to future shop, and asks the Salesperson to help him. He needs to buy a TV, DVD Player, Speakers and Amplifier. In all Future Shop carries 20 types of TVs, 10 types of DVD players, 16 types of speakers and 6 types of Amplifiers. Each option takes him approximately a minute to evaluate. How long you think it will take him to make the best possible judgment?
- In Ottawa, all telephone numbers are listed as 613 xxx xxxx. How many different telephone numbers are possible? What about Toronto - do you think that one area code will suffice?
- Ad was assigned the job of lining up 5 kids, ages 2,3,4,5, and 6 in a Q, according to their age, but they are restless - and they keep shuffling their order in the line. What is the chance that Ad will succeed? What is the chance that at least 4 of them are in the right order? What about at least 3?
- If my BBry buzzes every 5 mts and my iphone buzzes every 7 minutes, then when is the first time they will buzz simultaneously, assuming they were turned on at the same time? What if it was 4 and 8 minutes? What this had to do with prime numbers?
- My car typically travels about 500 Kms/week, and out of that 1/5th is on Highway and 4/5th on the city roads. For a liter of a gas, it gives an average of 9Kms in city and 11Kms on Highway. The cost of Gas today is $1.15/liter. How much money should I spend on the gas in a week? The new Honda Hybrid has the rating of 20Kms/l in city and 22.5Kms/l on Hwy. Whats the cost per week for the Hybrid?
- This is based on the TED talk which I heard recently - are more choices good or bad for us? Think about the example of the Home Stero, suppose we had only two choices for each in place of so may possibilities, then what would you have preferred. Think of you set it up - and then you didn't like it - who will you blame?

## Sunday, December 26, 2010

### Sunday, Boxing Day, 2010

Todays problems are based on SAT Exams.

## Sunday, November 21, 2010

### Sunday November 21, 2010

- A thick metal pipe, 2mts long, has inner radius of 12 cms and outer radius of 15cms, is made of steel. How much volume of steel is used to make this pipe? What do you think is the weight of this pipe? Note that weight of 1 cubic meter of steel is 8000 Kg. What if the pipe is made of Bamboo - 1 cubic meter of Bamboo is approximately 350 Kg.
- The new Chapman Mills park is rectangular, where the sides are in the ratio of 4:3. The total area of this park is approximately 3500 sq mts. What is the cost of installing a metal fence around this park, where The Ottawa Fencers charges approximately $45 per linear meter.
- Ms. Quick finishes a job in 8 days, whereas Mr. Slow finishes the same job in 12 days. How long will it take if both of them will work together (assuming that the job can be partitioned nicely (like pulling weeds in the garden)).
- Dr. Light needed to climb his wall to install the newly bought Christmas lights. His ladder is 8mts long, and the wall is approximately 7mts long. How far is the base of the ladder from the wall? Do you think its at a safe distance and will he be stable on the ladder?
- Suppose we toss two unbiased coins 100 times and observe that the number of times we get 2 heads is 25 and number of times we get no heads is 18. How many times we got at least 1 tails? We can try the same problem with 3 coins - lets say #times 3 heads= 22, #times 2 heads=32, #times 1 heads = 24, #times 0 heads = 22. How many times we get at least one tails, at least one tails and one head.
- A disc needs to be cut out from a square sheet. The length of the sheet is 2mts. Whats the area of the largest possible disc? Whats the area of the left out piece of the sheet? Try the same problem for a cube and a sphere. Is the volume of the left out piece larger than that of the sphere?
- The angles of a triangle are in the ratio of 1:2:3. What are the angles? Is it a right angle triangle? What if it is 3:4:5?
- Mr. Too Quick puts all his favorite NHL-team socks in the drawer, and in the morning rush (in the dark in long Northern Winter) - what are chances that he will draw a pair which is of the same team. He has 2 pairs of SENS, 2 pairs of PENS, 2 pairs of DUCKS and 3 pairs of HAWKS. What are the chances that he will draw a pair which does not belong to the same team? First try this for two teams, and lets say one pair from each team, and then start to make it more complex.

## Sunday, November 14, 2010

### 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?

## Sunday, November 7, 2010

### Sunday, November 7, 2010

- Consider three unit disks (i.e., of radius 1) touching each other. I am interested in finding the area of the triangular cavity between the three disks. Draw the triangle joining the center of the disks, and find the area of the part of the triangle, which is outside of the disks.
- Consider a cube, which is painted red, and has side length of 4cms. This cube is divided into unit cubes (i.e., each of side 1cm). How many small cubes will have (a) exactly three sides painted red (b) exactly two sides painted red (c) exactly one side painted red (d) no sides painted red. Repeat the same problem with the starting cube of side length 1cm, 2cm, 3cm, .. Whats the pattern?
- Ad's friend decided to hangout on Friday evening in the Ottawa Expo. They planned carefully and decided that they needed to spend $96 for all the fun activities - and decided to contribute equally. But on the most exciting day, some of his friends (being very unreliable) didn't show up.! Now the remaining ones had to contribute $4 more than what they initially planned! How many of them were eventually at the expo?
- Mr Every T. Wrong is expert in getting everything wrong! He was asked to multiply a number by 4 and then add eight - and he landed up dividing by 4 and then subtracting 8. He got 1 as his answer! What should be the correct answer.
- The overhead water tank on the Moodie drive supplies water to whole of Barrhaven. It requires 4 hours to fill it completely and 6 hours to empty it. Usually the water tank is getting simultaneously filled as well as emptied. How long will it take to fill it? Suppose it takes x hours to fill and y hours to empty, what should be the relationship between x and y. This is a typical producer-consumer problem! Assuming that x is less than y, then how many hours it will take to fill the tank completely in terms of x and y?
- Ad is taller than Rd by 25%. Then, by what percentage Rd is shorter than Ad?
- The height of parliament hill in Ottawa is 92 mts and the height of the National Art Gallery is 60 mts. They are approximately 200 mts apart. Assuming that their base is at the same level, what is the distance between their tops?
- You have a bag consisting of 10 red and 10 blue marbles. What is the smallest number of marbles you need to draw to be sure that you have exactly three of the same color? What happens when you have 10 red and 15 blue? What happens when you have three colors - say consisting of 10 red, 10 blue and 10 green - and you want to draw three of the same color?
- Whats the right length of laces that you need for your ice skates? You need about 20 cms to be left after tightening your skates to tie them up. There are 10 eyelets on each side, 3cms apart, and the distance between two consecutive eyelets on the same side is about 1 cms.

## Sunday, October 17, 2010

### Sunday, October 17, 2010

Some more problems based on the NTSE Exam for Grade 8.

- In a triangle, one of the angle is same as the sum of two others, and those two angles are in the ratio of 4:5. What are three angles?
- In a rectangle, one side is more than the other side by 4 cms. If both the sides are increased by 3 cms each, the total area increases by 81 square-cms. What are the measurement of the sides of the original rectangle.
- A rectangular gallery has sides in the ratio of 5:2, and its total area is 100 square-mts. What are the side lengths?
- You need to find a two digit number, where sum of its digits is 12, and when the digits are reversed, the difference between the two numbers is 18.
- A very keen hockey player signed a contract, where he will get $20/per game which he plays, and -$5/per practice which he lands up missing. There is one practice and one game each day, and at the end of the 30 day season, the player was rewarded $450. How games this player played, and how many practices he missed.
- For her school project, Ms. Chrome, decided to make Kaleidoscope. She decided to make the one where the tube needs to have length of 30 cms, and its diameter to be 7cm. Of course, you roll a sheet of paper, to form this cylinder. What should be the dimensions of the sheet of the paper (assuming no wastage). Understand the relationship between the surface area of a cylinder and area of a rectangle via this problem.
- In our houses, we have a huge hot water tank in the basement. What is the surface area of this tank? How long the sheet of the metal must be rolled to make this tank? How do we find, how much volume it holds?
- Check the reasoning (a) a cow looks, at all the dogs in the barn, and sees that they have a tail. Cow looks at herself, and sees that she has a tail. She concludes that she is dog! (b) All humans are mammals, and all mammals are vertebrates. Are all humans vertebrates? (c) It rains on every Monday. It is raining today. Today is Monday. (d) It hasn't rain today at all, so it can't be Monday.
- Show that the product of any two consecutive even numbers is divisible by 4? Is it divisible by 8?
- What about the product of any three consecutive even numbers? Is it divisible by 16?

## Saturday, September 18, 2010

### Saturday, September 18, 2010

Today's problems are adapted from the NTSE series of books by Tata McGraw Hill for Class 8th (2008).

- How many non-overlapping discs of radius 1cms and fit in a disc of radius 2, 4, and 8cms?
- Given a disc D of radius 2 cm lying on a table, whats the maximum number of discs of the same radius you can place (lying on the table) so that each of them is touching D and none of the discs overlap.
- These are classical packing problems: for example how many balls of radius 1cm can fit in a giant ball of radius 4cms? How can we estimate this number? How can we estimate the volume of the gaps?
- The train coming from Toronto to Ottawa, traveling at a speed of 60Km/Hr made me wait for 10 seconds at the signal. How long (in mts) is the train?
- If Ms. Rich's income is 25% more than Mr. Poor, then by what percentage is Mr. Poor's income less than Ms. Rich?
- A door-to-door salesman sells two cookie jars, each for $10.00 - one to Ms. Tough-Bargain and the other to Mr. What-ever. For Ms. Tough-Bargain he make a loss of 10% and for Mr. What-ever he makes a profit of 10%. Did the salesperson made an overall profit or loss?
- Anant's piggy bank has in all change for $33.00. It consists only of quarters, dimes and nickles, in the ratio of 5:3:2. How many coins of each type Anant has?
- We know that in an equilateral triangle, each interior angle is 60 degrees; in a square it is 90 degrees; what do you think it will be in a regular pentagon; regular hexagon; and how we will go about finding it.
- The Rideau River near my department has pretty fast current. Ottawa police does their rescue drill operations once in a while. They go down-stream, and then up-stream. In 10 minutes, the rescuers travel the downstream in their inflatable boats covering the distance of 1 km. Upstream, for the same distance, it takes them twice as long. What is the speed of the current?
- In the school's Terry Fox Run, I came 19th from the top and 235th from the bottom. How many kids participated in the run?

## Monday, September 6, 2010

### September 6. 2010 (Labour Day)

We met last two weeks, and did problems from the book "Learning Maths 6B by Norrin Hasim (Singapore Asian Publications)". I highly recommend this book for middle school kids for a variety of very nice problems - I haven't seen other levels of this book - but 6B has lots of nice geometric problems. Today's Problems 1-3 are based on this.

- Find the area of the American Football in the square of side 45 cm long, in the above picture.
- 1/5th of Anant's mm's are same as that of 1/3rd of Aditya's. If Aditya gives Anant 24 mm's (unlikely though!), Anant will have thrice as many mm's as Aditya. How many they had to begin with?
- A solid iron ball of volume 576 cubic-centimeter is cast into nine identical solid cubes. What are the dimensions of the cube?
- What about the other way around? Suppose you have 16 identical cubes, each having a side length of 2cm long, is cast as a solid ball. What is its radius?
- A string forms an equilateral triangle, whose sides are 3.14cms long. The same string is now reshaped as a circular ring. What is the radius of the ring?
- A circular race track has a inner radius of 56m and outside radius of 63m. What is the total area of the track itself?
- In the above problem, if the 7mts wide track is divided into 7 equally spaced ones- 1mts each, 7 racers need to run the full length of the track once - how should you place them so that each of them cover the same length?
- The difference between the ages of Mr. Young and Ms. Old is 45 years and the ratio of their ages is 3:8. What are the ages of Mr. Young and Ms. Old.
- Divide 25 into two parts, such that 4 times of one part is the same as 6 times the second.
- In a large movie hall, seats are assigned with respect to row and column numbers. In all there are 1296 seats, and the number of rows are the same as columns - how many rows are there?

## Sunday, June 20, 2010

### Sunday June 20, 2010

This is the last week of the school for the kids, and after this week we will take a break for about 7 weeks - as all of us are travelling to different parts of the world.

- 9 zebras weight same as 4 yaks, 8 yaks weigh same as 15 xantus, 10 xantus weigh same as 27 wolves. How many wolves weigh the same as 4 zebras?
- On a 200m circular race track in your school yard, Bruce and Anant are competing in a race, a long 10KM race. Bruce runs at the speed of 5m/sec and Anant runs at the speed of 4m/sec. How many meters has Anant run when they meet for the first time? Second time? (Both of them are running in the same direction and start at the same time.)
- Aditya and Mark smuggled 72 candies for their trip to the Arrowhead camp. Aditya being scared of getting caught, gives half of his candies to Mark, and since he is still very very scared, he gives in addition 12 more. Mark is thrilled - he has now become the Candy King, he has three times more than what Aditya is left with. How many candies each of Aditya and Mark had to begin with?
- 3/4 of 30 is same as 9/10th of what number?
- Radhika buys her favourite sandal for a discount of 20%. Raju, while strolling in Bayshore, spots the same sandal in two different stores. The one is selling it at an additional discount of 20% on what Radhika paid, the other store is selling it at a discount of 37% at the original price. Which deal is better for Raju?
- Two twins, Dharmesh and Prince, have a birthday on Feb 29th. Geeta, his mentor, decided to give Dharmesh $1 on Feb 1, $2 on Feb 2, $3 on Feb 3, ..., $29 on Feb 29th as the B'day gift. Remo, Prince's mentor, decided to give him $20 everyday from Feb 1 to Feb 29th as his B'day gift. Who had a better present?
- Today's geometric construction - we will draw a parallelogram whose sides are 6cm long, angle is 45-degrees. What is its area?

## Sunday, June 6, 2010

### Sunday, June 6, 2010

- Suppose you have many carpet runners. Their dimension is typically 2ft by 6ft. Your room measure 8ft by 12ft. What kind of interesting patterns you can make to cover your room wall to wall.
- I have tossed a coin 10 times in a row, and so far I haven't got the Heads. (Wow! I am so unlucky!). Does this increases my chance of getting a head in the next toss?
- What's common between golf, notebook and doughnut (or medhu vada).
- Take a loonie (Canadian 1$ coin) and roll it along its edge on a table. Observe the eye of the loon (the bird/ or the Queen), and trace the curve it follows. What shape is this curve? (Learn More: Cycloid)
- Alice and Bob are in two different rooms, and they have no way to communicate with each other as well as they do not trust each other. Think both from Alice and Bobs viewpoint and reason what you will do in the following scenario: Alice gets a $10 bill with the following promise - either she can keep the money or send it to Bob, and if Bob returns the favor then her amount will be doubled (Bob may just walk off). Same is told to Bob, that either he can keep the money which came from Alice or send it back to Alice, and if she returns the favor then the amount which he had will be doubled. They can keep sending back and forth the money and keep doubling! What will you do if you are Alice? or Bob?
- In the news we have heard a lot about the oil spill in the Gulf of Mexico. Here is a simple Math problem related to this. They claim that 26,500 barrels of oil is spilled in a day (1barrel= approx. 160 l). The diameter of the pipe from which the crude oil is oozing out is 20 inches. How many barrels/per day you think they will be spilling out in case they made the diameter of that pipe 24 inches instead of 20 inches.
- An Olympic size swimming pool measures L=50m x W=25m x D=2m. What is the total volume of water it contains? 1 cubic meter requires 1000 liters of water. In terms of the Gulf oil spill, it started 48 days ago - how many pools of oil it is? If the pipe was 24 inches wide then how many swimming pools it would have filled?
- Can you arrange 9 dots on a piece of paper so that there are 10 straight lines, each passing through at least 3 dots.

## Sunday, May 30, 2010

### Sunday May 30, 2010

- Ethiopian Tune ran 10K in 32 minutes and 11.5 seconds. What was her speed in Km/Hr?
- In comparison, in Vancouver 2010 olympics, 10k speed skating record for men was just under 13 minutes. What the speed in km/Hr.
- Usain Bolt ran 100m in 9.58 seconds. What is his speed?
- 100m swimming free style record is 46.91 seconds by Cesar Cielo. Whats the speed?
- 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?
- 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?
- You have invited 10 guests for a party. A custard can serves three guests. How many cans you need?
- 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?
- 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?
- 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?
- Geometric Construction: Draw a triangle with sides 9cm, 7cm, and 5cm and then determine its area.

## Monday, May 24, 2010

### Monday, May 24, 2010 (Victoria Weekend)

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

- 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
- 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.
- 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).
- 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..
- 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.
- 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.

## Sunday, May 16, 2010

### May 16, 2010

- 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).
- 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!
- 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.
- 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?
- 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?
- 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.
- 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?
- How many pennies are there if in your pocket you have 50 coins making a change for a dollar?
- If 5 is added to 1/3rd of a number, it result in 1/2 of that number. Whats that number?
- How will you bisect an angle between two lines, without actually measuting the angle (you are allowed to use the compass).

## Sunday, May 9, 2010

### May 9, 2010

- 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?
- 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?

- In a chess tournament each of the six players will play all other players exactly once. How many games will be played in all?
- 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?
- 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?
- 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.

- Suppose in the previous question, the trees were 20 m and 30 m tall, how high above the ground will the ropes meet?
- 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.

## Sunday, May 2, 2010

### May 2, 2010

- 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.
- 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.)
- 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.
- 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.
- You have in all 72 coins made up of nickels and dimes, totaling to $4.95. How many nickels you have?
- 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?
- Why are manhole covers made circular and not squares or rectangles or ellipses? (Think of their diameter versus the smallest side.)
- 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?
- In the Papanack Zoo, you see 30 eyes and 44 legs. How many 2-legged and 4-legged animals are there?

## Wednesday, April 28, 2010

### April 24, 2010

- You are given an arc of a circle, how will you complete it to a full circle?
- You are given three distinct points on a sheet of paper, label them A , B and C. How will you find a point D that minimizes the maximum distance to the points A, B and C? (Learn More: Fermat-Weber Geometric Medians)
- Given three points on a sheet of paper, how will you draw a circle passing through those three points?
- Let A be a point outside a circle centered at O. Draw a tangent to the circle. There are two tangents - consider just one of them. Let the tangent be incident to the circle at point B. Show (i.e., how to prove) that OB is perpendicular (i.e., makes an angle of 90 degrees) to AB.

Labels:
Geometric Problems - Circles

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