MATC Mathematics Club
Problem Set Volume 1 Page

The Puzzle Corner of the MATC Mathematics Club
Madison Area Technical College
Madison, Wisconsin

Volume 1, Spring 2002 Problem Sets
Problem Set for Week #22 ( Click here for the solutions ).
1. A circular table has exactly 60 chairs around it.  There are N people seated at this table in such a way that the next person to be seated must sit next to someone.  Find the smallest possible value for N.
2. In a certain cross-country meet between two teams of 5 runners each, a runner who finishes in the Nth position contributes N to his team‚s score.  The team with the lower score wins.  If there are no ties among the runners, how many different winning scores are possible?
3. The student lockers at MATC are numbered consecutively beginning with locker number 1.  The plastic digits used to number the lockers cost two cents apiece.  If it costs $137.94 to label all the lockers, how many lockers are there at MATC?.
Problem Set for Week #21.
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Problem Set for Week #20 ( Click here for the solutions ).
1. How many whole numbers between 100 and 400 contain the digit 2?
2. In a certain year, January had exactly four Tuesdays and four Saturdays.  On what day did January 1 fall that year?
3. Assume every 7-digit whole number is a possible telephone number except those, which begin with 0 or 1.  What fraction of telephone numbers begin with 9 and end with 0?
Problem Set for Week #19 ( Click here for the solutions ).
1. Imagine a race between Achilles and a tortoise, in which the tortoise is given a head start of 1000 m.  Achilles can run 10 times faster than the tortoise. From this pattern it „appearsš that Achilles will continually gain on the tortoise but never catch it.  Is this reasoning correct?  If Achilles were to pass the tortoise, at what point of the race would it be?
2. Consider 9 small marbles are placed at the nine points of intersections of a 3X3 grid.  You will notice the nine marbles are in 8 rows of 3, viz: the three horizontal lines, three vertical lines and the two diagonals.  Now rearrange the nine marbles into 10 rows of 3.  (Hint: The three marbles need not be placed as before and to make your solution simpler, label the marbles A,B,C,∑H,I)
3. Two poles are 20m and 80m high.  Find the height of the point of intersection of the lines joining the top of each pole to the foot of the other.
Problem Set for Week #18 ( Click here for the solutions ).
1. A bear started from point B and walked one mile due south, then one mile due east, then one mile due north and arrived exactly at the point B where he started.  What color is the bear?
2. OPQ is the first quadrant of a circle with P, Q on the Y, X axes respectively and O at the center.  Semi-circles are drawn in the first quadrant with diameters OP, OQ respectively.  The area common to both these semi-circles is A and the area outside these semi-circles but inside the quadrant of circle OPQ is B.  Find A/B.
3. A cylindrical hole with a radius of six inches is drilled straight through the center of a solid sphere of radius R.  What is the volume remaining in the sphere?
Problem Set for Week #17 ( Click here for the solutions ).
1. There are 1000 lockers numbered 1 to 1000.  Suppose you open all of the lockers, and then close every other locker.  Then, for every third locker, you close each opened locker and open each closed locker.  You follow the same pattern for every fourth locker, every fifth locker, and so on up to every thousandth locker.  Which locker doors will be open when the process is complete?
2. There are eight weights, all the same size, shape, and color.  They all weigh the same except for one, which is heavier than the rest.  Using a balance scale, how can you find the heavier weight in only two weighing?
3. The sums of four numbers, omitting each of the numbers in turn, are 22,24,27 and 20 respectively.  What are the numbers?
Problem Set for Week #16 ( Click here for the solutions ).
1. You have a 24L container of water and three empty containers, a 5L, 11L and 13L containers.  How could you divide the water into three equal portions using just those containers?
2. In how many different ways can 33 people be divided into three teams of 11 each?
3. Derive a formula for the number of regions N formed by connecting n distinct points on a circle.  For instance when n = 1, N = 1 and for n = 2, N = 2 and for n = 3, N = 4 and for n = 4, N = 8 etc.,
Problem Set for Week #15 ( Click here for the solutions ).
1. At the North Pole it is possible to walk due south for 1 mile, due east for 1 mile, and due north for 1 mile, thereby returning to the starting position. Find all other points on the surface of the Earth (assuming it is a perfect sphere) where this happens.
2. Can you divide 40 pounds into 4 (different sized) pieces so that every integer weight from 1 through 40 pounds can be measured using some combination of the pieces and a balance scale? (Hint: any of the pieces can be put on either side of the scale, either with or opposite the item being weighed.)
3. A cube is made from n n n small cubes. Find a formula for the number of cubes you find in the large cube (including the large cube). For example, in a 2 2 2 cube there are 9 "subcubes", namely one that is 2 units on a side and eight that are 1 unit on a side.
Problem Set for Week #14 ( Click here for the solutions ).
1. Start with a square piece of paper.  Draw the largest circle possible inside the square, cut it out and discard the trimmings.  Draw the largest square possible inside this remaining circle and cut the square out and discard the trimmings.  What fraction of the original piece of paper has been cut off and thrown away?
2. What‚s the next number in the series 4,16,37,58,89,145... ?
3. A commuter is in the habit of arriving at her train station each evening exactly at five o‚clock.  Her husband always meets the train and drives her home.  One day she takes an earlier train, arriving at the station at four.  The weather is pleasant, so instead of telephoning home she starts walking along the route always taken by her husband.  They meet somewhere on the way.  She gets into the car and they drive home, arriving at their house ten minutes earlier than usual.  Assuming that her husband always drives at a constant speed, and that on this occasion he left just in time to meet the five o‚clock train, can you determine how long the wife walked before her husband meets her?
Problem Set for Week #13 ( Click here for the solutions ).
1. Derive a formula for the relationship between the number of chords C drawn through a circle and the maximum number of parts P into which chords can divide a circle.
2. A rope of one unit long is randomly cut in two places.  What is the probability that the three resulting pieces can be arranged to form a triangle?
3. A certain substance doubles in volume in every minute.  At 9:00 a.m. a small amount is placed in a container and at 10:00 a.m. the container is just full. At what time was the container one quarter full?
Problem Set for Week #12 ( Click here for the solutions ).
1. Bush and Gore bought three horses of equal value for $840.  Bush contributed $450 and Gore contributed $390.  By the end of the day, they had sold one horse to a passerby for $840, and decided to end their partnership.  How should they divide the cash fairly if Gore keeps the two horses?
2. In an irregular five-pointed star the sum of the measures of the angles at the points (vertices) of the star is constant.  Find the sum and verify your answer.
3. A friend takes two cards from a standard deck and lays them face down in front of you.  She announces, "One of these cards is red."  What is the probability that the other card is red?
Problem Set for Week #11 ( Click here for the solutions ).
1. When between 12:00 noon and 1:00 P.M. are the minute and hour hands of a clock 180° apart?
2. A lottery sells 45 tickets for a total of $279.  If some tickets are sold at full price and the rest are sold for half price, how much is brought in by the full price ticket?
3. Each correct answer on Student Math League counts 2 points and each incorrect answer counts Ų0.5 points.  What possible scores can one get if he/she answers 14 questions?
Problem Set for Week #10 ( Click here for the solutions ).
1. Find:
2. What is the value of the continued fraction ?
3. Determine
Problem Set for Week #9 ( Click here for the solutions ).
1. A hundred bushels of grain are to be distributed among 100 persons in such a way that each man receives 3 bushels, each woman 2 bushels and each child half a bushel.  What are the possible combinations?
2. Find all right triangles with sides of integral length whose areas are equal to their perimeters.
3. Find the minimum value of the expression   for positive real a,b.
Problem Set for Week #8 ( Click here for the solutions ).
1. How many numbers in the sequence end in a 6?
2. Two locomotives 300 miles apart are heading toward each other on the SAME track.  The first locomotive travels at 100 mph and the second travels at 50 mph.  As the locomotives depart a bird flying 200 mph leaves the first locomotive and heads for the second.  Upon reaching it, the bird instantaneously turns around and heads back toward the first.  The bird continues flying back and forth in this manner.  How far will the bird fly before being crushed between the two locomotives?
3. If x, y, z are integers such that 2x3y5z = 1350, then evaluate 2x + 3y+ 5z.
Problem Set for Week #7 ( Click here for the solutions ).
1. In how many zeroes does the number 17! (factorial 17) end?
2. What is the smallest composite number generated by , where P is a prime?
3. Express as a rational number.
Problem Set for Week #6.  ( Click here for the solutions ).
1. How many odd numbers with four distinct digits can be formed using the digits 2, 3, 4, 5, 6 and 7?
2. What are the positive integer divisors of 1998?
3. Tom goes into a store and says "Lend me as much money as I already have, and I will spend $20 in your store."  The owner agrees, and Tom spends $20.  Tom does the same thing at a second, third and fourth store with the owner agreeing each time and Tom spending $20 each time.  After this Tom has no money.  What is the total of his debt to the four storeowners?
Problem Set for Week #5 ( Click here for the solutions ).
1. At the North Pole it is possible to walk due south for 1 mile, due east for 1 mile, and due north for 1 mile, thereby returning to the starting position. Find all other points on the surface of the Earth (assuming it is a perfect sphere) where this happens.
2. Can you divide 40 pounds into 4 (different sized) pieces so that every integer weight from 1 through 40 pounds can be measured using some combination of the pieces and a balance scale? (Hint: any of the pieces can be put on either side of the scale, either with or opposite the item being weighed.)
3. A cube is made from n n n small cubes. Find a formula for the number of cubes you find in the large cube (including the large cube). For example, in a 2 2 2 cube there are 9 "subcubes", namely one that is 2 units on a side and eight that are 1 unit on a side.
Problem Set for Week #4 ( Click here for the solutions ).
1. You are aware that the number of diagonals of a quadrilateral is 2. Similarly the number of diagonals of a  pentagon, hexagon, heptagon,∑are respectively 5,9,14∑Using pattern recognition or otherwise, derive the number of diagonals of a  n-gon (polygon with n sides).
2. How many odd numbers with four distinct digits can be formed using the digits 2, 3, 4, 5, 6 and 7? Justify your answer without brute force method.
3. In a list of nine scores, the mean of the first five scores is 3.  The mean of the last five scores is 4.  The mean of all nine scores is 32/9.  What is the fifth score?
Problem Set for Week #3.
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If you are interested, email Jeganathan "Sri" Sriskandarajah ( jsriskandara@madisoncollege.edu ) or contact him in room 211G.    Watch this page and the student bulletin for further announcements.

 


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