CompetitionMathsQuestions

8/3/2019 CompetitionMathsQuestions

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MATHEMATICS COMPETITIONS

Many mathematicians tell us that research results in mathematics are achieved after longexperience and a deep understanding of mathematical concepts, that progress is madeslowly and collectively and that flashes of inspiration come only occasionally during longperiods of sustained effort.

Mathematics competitions, in contrast, require a relatively brief period of intenseconcentration, ask for quick insights on specific questions and require a concentrated butisolated effort. Yet there is a significant amount of evidence that participants in thesecompetitions have gone on to become outstanding mathematicians, scientists, engineers,and have attached great significance to their early competition experiences.

For many of these students, the competition problem is an introduction, a glimpse into theworld of mathematics not generally available in the usual classroom situation. A goodproblem tries to capture in miniature the process of creating mathematics. It is all there: aperiod of reading and understanding the problem, the exploration of possible approachesand the pursuit of various paths to a solution. Many times there is a dead end. Some pathsoffer new perspectives which (hopefully) lead to a better solution. A good problem introducesmany students to mathematical creativity which is the essence of real experiences andleaves the student wanting still more. Students need the opportunity to experience good non-standard problems and to be made aware of the similarities among problems.

Mathematics and Mathematics competitions should be an important part of the Mathematicscurriculum at all secondary school levels. The School Mathematics Competition is but thestart of a chain of competitions that eventually can lead to selection in the Australian

Mathematics Olympic team to compete on the world scene at the IMO. There are studentsout there with this potential and we should work to identify and encourage them.

While selection in the AMO team is the pinnacle for a few it is not the only reward. Thiscountry needs more mathematicians, scientists, engineers, We need to know that ourstudents can compete on a level playing ground with students throughout Australia andindeed the world, especially in this fast changing technological world where mathematicsplays a key role. We must continue to lift our game in order that our students can be playersin this game and not merely spectators. And it is only on the basis of raising the standard ofthe majority that the emergence of exceptionally bright students will occur.

THE VISION

That all students have access to mathematics competitions that provide a climate thatgenerates interest and enthusiasm for learning mathematics, encourages creativity andrewards excellence.


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THE COMPETITION

Should be an educational experience that complements classroom activities. Be of sufficient variety and type to provide of different approaches. Serve a diverse population of students.

THE FORMAT OF THE COMPETITION

Junior (year 8 and below)Intermediate (year 9 10)Senior (year 11-12)

Each paper to contain

i) 9 short answer questions (Normally 1-step problems)ii) 3 essay style questions (Which require students to follow an argument )

We recognize that the Competition have critics that argue that we should recognize andreward excellence in its many forms. But our resources (both financial and support) arelimited. We welcome comments on what we do and how we do it and will always welcometeachers who wish to become involved and offer other extension and enrichment activities.

The questions that follow are meant to provide opportunities for students to discuss a numberof different non-standard problems. They range between short answer style and essay styleand are somewhat more difficult than problems that will be in the SMC where we aim to give

all students an opportunity to achieve some success while providing some sting in somequestions which will allow us to discriminate our final winners.

Some students might prefer to work alone on these problems while others may find morecomfort with a group approach. The set of answers given are not the only way in which theproblem can be done and we encourage you to seek better approaches but hopefully youcan learn by considering the answers and the methods used.


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INTRODUCTORY PROBLEMS

1. The distance between A-town and B-town is 999 km. Kilometre posts along the roadconnecting A and B show the distances from the posts to A and B.They read

A 0 A1 A 2 A 3 A 998 A 999B999 B998 B997 B 996 B 1 B 0

i) Complete the following kilometer postsA 118 A .. A 636 A .B .. B 336 B B 544

ii) How many posts have only two digits on the posts?

2. The longest side of a triangle is 10 cm. in length. If the shortest side is 3 cm. inlength, what can you say about the length of the third side?

3. Albert, Brian and Charles have money in the ratio 3 : 2 : 1 at the start of a cardgame. After the game the money they had was in the ratio 1 : 1: 2. If Albert lost $9,how much did Charles start with?

4. Given 3 numbers a, b and c, such thata + b = 15 a + c = 20 b + c = 21

What is the value of a, b and c?

5. One pipe can fill a tank in 3 hours, another pipe can fill the same tank in 4 hours,while a third pipe would take 6 hours to fill the same tank. How long would it take tofill the tank if all 3 pipes were open?

6. With the digits 3, 5 and 7 you can form 27 three digit numbers. (In some of these thedigits may be repeated.) What is the sum of these 27 numbers?

7. Jill has a number of 2 x 1 tiles.In how many ways can she tile an area which is 2 x 10 ?

8, The sum of three positive integers is 387. If a number is subtracted from each ofthem we will obtain 107, 109 and 111. What are the three numbers?

9. Find the largest number which is the product of positive integers whose sum is 18.What about 19? 99? 100?

10. It is now 2.00 am. In exactly k minutes, the hour hand and the minute hand of anaccurate clock will make an angle of 38 degrees with each other. What time is it?


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11. Delete 50 digits from the number12345678910111213141516..383940

so that the resultant number is as large as possible.

Note: you cannot change the order of the numbers.

12. For every 2-digit number, Tyrone took the product of its digits. Then he added all ofthese products together. What number did Tyrone get?Lisa then did the same with all the 3-digit numbers. What result did Lisa get?

13. You are given a set of 27 different odd numbers all less than 100. Can you find a pairof these numbers whose sum is 102? Can you always do this?

14. Consider the two lists (sequences) of numbers1, 4, 7, 10, 13, 16, 19,

3, 14, 25, 36, 47, 58,

a. What is the smallest number which appears on both lists?b. What are the first 5 numbers that appear on both lists?c. What is the 100th

d. What is the smallest number greater than 1000 which appears on both lists?. number on each list?

15. 8 points are equally spaced around a circle.

a. If each point is connected to every other point by a straight line, how many lineswill be drawn?

b. By choosing 3 points a triangle can be formed. How many?c. By choosing 4 points a rectangle can be formed. How many?

15. All the integer multiples of 9 are written down in a sequence9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117,

and for each of these numbers the sum of the digits is found

9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 18, 9, 9,

a. In what place in the second sequence does the number 81 first appear?b. In what place in the second sequence does 4 consecutive 27s occur for the

first time?

16. 100 people have $1000 between them but no 10 of them have more than $190.What is the maximum possible amount of money any one of them can have ?

17. N children want to divide 6 identical pieces of chocolate into equal amounts, eachpiece being broken not more than once.For what values of N is this possible?


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SAMPLE JUNIOR PROBLEMS

1. Use the prime digits 2, 3, 5, 7, to solve the following addition problems in which eachletter stands for a different prime digit ;i) BA + AC = CD ii) BA + AA + AB = CC iii) CA + CB + CB = DDA

2. Two numbers are called mirror numbers if one is obtained from the other byreversing the order of the digits. Thus , for example, 123 and 321 are mirrornumbers.Find two mirror numbers whose product is 92 565.

3. Hops, skips and jumps are units of length in Kanga-land. If b hops equals c skips , dskips equals c jumps and f jumps equals g metres , how many hops are there in 1metre?

4. More than 93% of the students in a class are girls, and there is a least one boy in theclass. What is the smallest possible size of the class?

5. The number 2222 is a palindromic number. The number 22 is also palindromic. Howmany palindromic numbers are there between 22 and 2222?

6. The two digit numbers from 10 to 99 are written consecutively and the number

N = 1011121314151617979899

is formed.

i) What is the sum of the digits in this number?ii) There are 180 digits in this number. Without changing the order of these

digits remove 120 of them so as to make the largest 30 digit number youcan.

7. How many numbers in the set {1,2,3,4,5,,999} are divisible by neither 8or 12?

8. A dog and a cat club found that it could achieve a membership ratio of 2 dogs to 1 catby enrolling 24 more dogs or by expelling x cats. What is the value of x?

9. How many 3-letter sequences can be made using the letters of the wordsi) MATHS ii) TENNIS iii) BOOK-KEEPER ?


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10. 7 black unit tiles and 2 red unit tiles on a table are to be assembled into a

3x3 square. How many different designs are possible? (Two designs are different ifthey look different no matter how you rotate them on the table. Flipping isforbidden.)

11. A,B and C paint a long line of fence posts. A paints the first post then every athpost. B paints the second post then every bth post while C paints the third post thenevery cth post. Every post gets painted just once.Find all the possible ways this can be done? (There is more than one answer)

12. A set of distinct positive numbers includes the number 68. The average ofthese numbers is 56. When the number 68 is removed the average of the remainingnumbers is 55.

i) How many numbers in the set?

ii) What is the largest number that can occur in this set?

13. The numbers 1,2,3,, 100 are partitioned into two sets A and B. What is themaximum number of different sums (a + b) you can have , where a is in A and b isin B?

14. A student entered a competition in which 20 problems were given. For eachproblem answered correctly the student received 8 points but for every incorrectanswer 5 points were deducted. For a problem not answered zero marks were

given. Given that the student scored 13 points, how many problems did the studentanswer and how many were correct?

15. 3 pipes lead into a dam. The pipes are called Upper, Middle and Lower.The dam can be filled in 3 ways :

a. the lower and upper pipes flow for 3 daysb. the upper and middle pipes flow for 4 daysc. the lower and the middle pipes flow for 6 days

Assume that each pipe has water flowing at a constant rate which is notaffected by the flow of the other pipes.How long will it take to fill the dam if all 3 pipes are flowing?

16. Two identical jars are filled with a mixtures of water and vinegar in the ratios of 2:1and 3:1 respectively. Both jars are now emptied into another larger jar.What is the ratio of water to vinegar in this new mixture?


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17. i) What is the next palindromic number after 1331 , 2002, 3993, ii) How many palindromic numbers are less than 10,000 ?iii) What is the 2009th

palindromic number?

18. Find the value of the digit n for which the number 55555n66666is divisible by i) 9, ii) 7 ?

19. Exactly 3 of the interior angles of a convex polygon are obtuse. What isthe maximum number of sides such a polygon can have?

20. In the grid below, each row is assigned the values A, B, C, D, E and each columnis assigned the values a, b, c, d, e respectively. The number in each cell is the sumof its row and column values.Thus, for example, for the cell with 8 in it we have C + d = 8.

Find the values of x and y in the grid.a b c d e

A 3 0 5 6 -2B -2 -5 0 1 yC 5 2 x 8 0D 0 -3 2 3 -5E -4 -7 -2 -1 -9

21. The letters in the word ZEBRA are arranged in dictionary order, as if each were anordinary 5 letter word. (They do not all make sense.)

a. How many words in the list?b. How many of these words start with the letter E?c. What is the 86th

word in the list?


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SAMPLE INTERMEDIATE PROBLEMS

1. From a group of boys and girls, 20 girls leave. There are now 3 boys for each girl.

After this, 40 boys leave and there are now 2 girls for each boy.How many students in the original group?

2. P is a convex polyhedron with 26 vertices, 60 edges and 36 faces. 24 of the faces aretriangular and 12 are quadrilaterals. A space diagonal is a line segment connectingtwo vertices which do not belong to the same face.How many space diagonals does P have?

3. A licence plate for motor cars has 3 letters followed by 3 digits. If all possible licenceplates are allowed, how many have either a letter palindrome or a digit palindrome (orboth)?

4. Consider an analog clock with an hour hand and a minute hand but no numerals(face numbers). At time T am, a mirror image of the clock shows a time of X am thatis 5 hours 28 minutes later than T.Find the time T in terms of hours and minutes.

5. Concentric circles with radii 1,2,3,4,,100 are drawn. The interior of the smallestcircle is coloured red and the annular rings are coloured alternately green and red, sothat no two adjacent regions are the same colour. Find the total area of the green

regions.

6. ABCD is a parallelogram. P is a point on DA extended. PC meets AB in Q and DB inR such that PQ = 735, QR = 112. Find the length of RC.

7. A school has 2009 students. Between 80% and 85% study Spanish, between 30%and 40% study French, and all students take at least one of these subjects. Find thesmallest and largest number of students that could study both languages.

8. How many right triangles exist whose sides have integer length and whose perimeter

is 2009?

9. How many pairs of consecutive integers in the sequence1000,1001,1002,,2000

can be added without a carry?For example 1004 and 1005 can but 1005 and 1006 cannot.


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10. Each angle of a regular m-gon is 7/5 times larger than each angle of an n-gon.What values of m and n are possible?

11. How many even numbers between 4000 and 7000 have all digits different?

12. Find the sum of all positive rational numbers of the form k/30 (in lowestterms) which are all less than 10.

13. Water taxis which take 6, 10 or 15 passengers are used to transportpassengers from an airport to a hotel. Six water taxis of each size are

available when a party of 120 arrived. In how many different ways is itpossible to use the taxis so that no taxi being used has an empty seat?

14. A, B, C and D all weigh different amounts. A is 8 kg heavier than C and D

is 4 kg heavier than B. The sum of the weights of the heaviest and thelightest is 2 kg less than the sum of the weights of the other two. The totalweight of all four is 402 kg.What is As weight?

15. Let A1, A

2, A

3, A

4, ,A

15be a regular pentadecagon (a 15 sided polygon). Line L

contains the interval A1A

2and each of the sides of the pentadecagon and not adjacent

to A1A

2is extended to intersect L.

The acute angle of each of the 12 intersections is calculated.Find the sum (in degrees) of the 12 acute angles so formed?

16. The points A, B and C lie on a line (in that order) with AB = 9, BC = 21.Let D be a point not on AC such that AD = CD and the distances AD and

BD are integers.

Find all the possible perimeters of the triangle ACD.

17. a and b are integers with 2 a 11 and 3 < b < 9. What is the largestand smallest value of

i)aba + ii)

abba + iii)

ba +1

18. The cyclic quadrilateral ABCD has sides AB = 25, BC = 39, CD = 52, andDA = 60. Find the radius of its circumcircle.


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19. i) Find the first positive integer whose square ends in three 4s.

ii) Find all the positive square integers that end in three 4s.

iii) Explain why no perfect square can end in four 4s.

20. After finishing a very boring book , David noticed that the page numberscontained exactly 29 zeroes and exactly 137 ones. The book started with

page 1.

How many pages were in the book?

21. Each stick in a pile has an integer length. If any 3 sticks are selected from the pilethey will NOT form a triangle. The longest stick in the bag has a length of 100 units.What is the maximum number of sticks in the bag?

22. Consider the number (11111111) 2 where there are 20 ones in the bracket.i) How many digits in this number?ii) What is the 15th

iii) What is the middle digit in this number?digit from left in this number?

23. You have an unlimited supply of cubes of volume 1, 8 and 27 and a box thatmeasures 3 x 3 x 223. What is the smallest N 100 for which it is possible to fillthe box with exactly N of these cubes?

24. An elongated Pentagonal Orthocupolarotundais a polyhedron with exactly 37faces., 15 of which are squares, 7 of them are regular pentagons and 15 of whichare triangles.How many vertices does it have?Hint Remember Eulers rule F + V = E + 2

25. A fractionb

ais in lowest terms if a and b have no common factor larger than 1.

How many of the 71 fractions

72

1,72

2,72

3, ,

72

70,72

71

are in lowest terms?


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SENIOR PROBLEMS

1. What is the greatest positive integer n which makes n 3+ 90 divisible by n + 9

2. The set A contains x consecutive integers with sum 2x. The set B consistsof 2x consecutive integers with sum x. The difference between the largest

element of A and B is 99. Find x.

3. Find the coefficient of x 2 in the expansion of (1-x)(1-2x)(1-3x),,, (1-7x)

4. How many two digit positive integers N have the property that the sum of N andthe number obtained by reversing the digits of N is a perfect square.

5. n is an integer such that f(n) = (n-3) for n > 999and f(n) = f(f(n+5)) for n< 1000Find the value of f(237)

6. Two squares of equal sides are placed so that their centres coincide and thefigure inside both squares is a regular octagon of side 2 units.What is the area of this octagon?

7. ABCD is a quadrilateral with AB = 8, BC = 6, BD = 10, angle BAD = angle CDA,angle ABD = angle BCD.Find the length of CD.

8. k is a positive integer such that 36+k, 300+k, 596+k are the squares of threeconsecutive terms of an arithmetic sequence.Find the value of k?

9. For what values of n (n an integer) is (n-5) a factor of (n 2 - 7n)?

10. Find all the integer solutions of the equation (x-5) 2x

= 1

11. Let a, b and c be three consecutive positive integers. Prove thata. ab cannot be the square of an integer.b. ac cannot be the square of an integerc. abc cannot be the square of an integer


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12. ABCD is a rectangle and E is the midpoint of BC. DB and DE meet AC inP and Q respectively. If the area of the rectangle ABCD is 1 sq unit what is thearea of the triangle PQD?

13. There are 720 different six digit numbers that can be formed by using all

the digits 1,2,3,4,5 and 7, for example 431751 and 731452.How many of these numbersi) are divisible by 11?ii) lie between 431751 and 731452?

14. Two perpendicular chords intersect in a circle. The lengths of the segments ofone chord are 3 and 4. The lengths of the segments of the other chord are6 and 2. Find the diameter of the circle?

15. Find the greatest integer that will divide 13,511, 13,903 and 14,589 and leave

the same remainder.

16. a, b and c are prime numbers such thata(b+c) = 234 and b(a+c) = 220

What are the values of a, b and c?

17. In the rectangle ABCD, AD = 10 and CD = 15. P is a point inside therectangle such that PB = 9 and PA = 12.Find the length of PD.

18. Find all the positive integers n for which n 2 - 19n + 99 is a perfect square.

19. Jill has a rectangular block of wood 8 x a x b (a,b integers). She paintsthe entire surface of the block, then cuts the block into unit cubes and

notices that exactly one-half of the unit cubes are completely unpainted.Find all the possible values of a and b.

20. BC is an isosceles triangle with AB = BC. Points K and L are chosen on thesides AB and BC respectively so that AK + LC = KL. A line parallel to BC is

drawn through the midpoint of the segment KL intersecting AC at the point N.

Find the size of the angle KNL.


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21. Let f(x) be a function defined on the closed interval [0,1] having thefollowing properties :

a) f is an increasing functionb) f(3x) = 2 f(x)c) f(x) + f(1-x) = 1

Find the value of (a) f(13

1) (b) f(

27

19)

22. Find all the positive integers a, b, m and n (with m and n relatively prime) suchthat

(a 2 + b 2 ) m = (ab) n

23. P is a point in the interior of an equilateral triangle ABC with PA = 5,

PB = 7 and PC = 8. Find all the possible values of AB.

24. i) In how many ways can three As, three Bs and three Cs bearranged in a line?

ii) In how many ways can they be arranged if no three adjacent lettersare the same in any arrangement?


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SOME OF MY FAVOURITE PROBLEMS

Some easy, some difficult. These are problems that I have enjoyed. They offer a rangeof challenges and results that are useful in developing an interest and motivation to domore mathematics.

1. Susie has a large number of 2 x 1 tiles and wishes to tile a floor.In how many distinct ways can she tile a

i) 10 x 2 area,ii) 10 x 3 area?

Hint : start by considering a smaller area . say 2 x 3 area and then

move onto a 4 x 3, a 6 x 3 and an 8 x 3 area and establishing a pattern.

2. You should all be familiar with Eulers Rule ( V + F = V + 2 ) for polyhedratopologically equivalent to a sphere and having V vertices, F faces and E edges. The

proof is quite simple and it is left for you to do.

The extension problem for you is to prove that the sum S of all the face angles of the

polyhedron is given by the formula

S = 2 (V 2)

where V equals the number of vertices in the polyhedron.

And now a problem : the sum of all the angles on the faces of a polyhedron, exceptone, is equal to 3150 degrees. How large is the missing angle?

3. Prove that a quadratic equation with integer coefficients cannot have a discriminantof 31.

4. An ARBELOS consists of three points A, B, C (in order) whichare collinear, with AB < BC, together with the semicircles with diameters AB, BC andAC. It was so named because of its shape like that of a bootmakers knife orarbelos (Greek).BT is a tangent to the two small circles with T on the large semicircle. The lines AXT

and CYT meet the semicircle on AB and BC at X and Y respectively. .Prove that

i) XY is a tangent to the small semicirclesii) XY and BT bisect each otheriii) The area of the arbelos is equal to the area of the circle on BT as

diameter


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5. Jack and Jill went to a dinner attended by 4 other couples, making a total of 10

people. As people arrived, a certain number of handshakes took place. While thehandshaking was unpredictable two conditions were met:

i) no one shook their own hand, and

ii) no partners shook hands.After the dinner, Jack became curious and asked each person, including his wife,

how many hands they shook. He was surprised that he got 9 different answers.

How many hands did Jill shake ?

6. We have 3 identical buckets : the first one contains 3 litres of syrup, the secondcontains n litres of water and the third bucket is empty. You can perform anycombination of the following operations :

i. you can pour away the entire contents of any bucketii. you can pour the entire contents of one bucket into another oneiii. you may choose 2 buckets and then pour from the remaining bucket into

one of the chosen buckets until the chosen bucket contains the sameamount

i) How can we obtain 10 litres of 30% syrup if n = 20?ii) Find all the values of n for which it is possible to obtain 10 litres of 30% syrup.

7. A1

A2

An

is a regular polygon with n sides. B is a point outside the polygon such

that the triangle A1A

2B is equilateral.

i) If n = 24 prove that An, A

1and B are consecutive vertices of a regular polygon.

ii) What is the largest n for which A n , A1 and B are consecutive vertices of a regular

polygon?

8. A right circular cone has a base radius of 1 unit and a vertical height of 3 units.A cube is inscribed in the cone so that one face of the cube is contained in the baseof the cone.Find the dimensions of the cube.

9. Rachael has an interesting quadratic equation that she shows to Stacey. The rootsof the equation are two positive integers, one of which is my age and the other theage of my younger cousin Emilie. All the coefficients are integers and their sum is aprime numberHelp Stacey determine the age of Rachael and Emilie.

10. ABC is an equilateral triangle with P, Q, R being the midpoints of sidesBC, AC, AB respectively and X, Y, Z being the midpoints of CP, AQ,BR respectively. By joining XR, YP and ZQ a small triangle is enclosedFind the ratio of the area of this enclosed triangle to the area of thetriangle ABC.


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11. In the gambling game craps, two dice are tossed (rolled) and the resulting sum

of the spots is noted. The player wins if, on the first roll, the sum equals 7 or 11 andhe loses if, on the first roll, the sum of the spots is either 2, 3 or 12.If another sum results, the player continues to roll the dice until that sum results and

he wins, or, a sum of 7 results and he loses.

Show that the probability of winning is 244/495 = .493

Note that this probability is slightly less than .5 which means that in thelong run the player stands to lose. So the advice is say NO to craps.

12. When Xerxes marched in Greece his army dragged out for 80 km. A dispatch rider(on horseback) had to ride from its rear to its head, deliver a message and returnwithout delay. By the time the rider returned to the rear, the army had moved on

80km.How far did the dispatch rider travel (in km)?

13. n points are placed on a circle. Each point is joined to every otherpoint by a straight line so that no three such lines are concurrent.

a. How many intersection points are there inside the circle?b. How many regions are there inside the circle when n = 4, 5 ?c. Find a rule for the number of regions inside the circle in terms of n

14. An A x B x C rectangular box has half the volume of an (A+2) x (B+2) x (C+2)Rectangular box, where A, B and C are integers and A < B < C.What is the largest possible value of C?

15. S is a set of positive integers containing 1 and 666. No element is larger than 666.For every n in S, the arithmetic mean of the other elements of S is an integer.What is the largest possible number of elements in S?Give an example of such a set.

16. The surface of a right circular cone is painted black. The cone hasheight 4 and its base radius is 3. It is cut into 2 parts by a planeparallel to the base, so that the volume of the top part (the small cone)divided by the volume of the bottom part (the frustrum) equals k andthe painted area of the top part divided by the painted area of thebottom part also is equal to k.Find the value of k. (Note : do not forget the bottom of the cone).


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17. Given the sequence an

=

2009

2n

for n = 1, 2, 3, , 2009

i) How many zeros in the sequence?ii) How many different integers in the sequence?

Note: the square brackets denote the greatest integer function

18. Find the number of numbers between 1 and 250 that are not divisibleby 2 or 3 or 5 or 7?

19. In how many ways can the integers 1, 2, 3, 4, 5, 6, 7, 8, 9, be arrangedsuch that each odd integer will be in its correct place?

NOTE Question 18 and 19 will require a little researchQ 19 -- The principle of inclusion/exclusionQ 20 --- Derangements an arrangement in which objects are not

in original positions.

THERE IS NO LACK OF PROBLEMS. THIS IS JUST A SET THATHOPEFULLY WILL SPARK YOUR INTEREST AND MOTIVATE YOU TOSOLVE MORE PROBLEMS AND TO LEARN MORE MATHEMATICS.

A SEPARATE SET OF SOLUTIONS IS BEING PREPARED AND WILL BEAVAILABLE SHORTLY

Questions, comments, answers, to

Keith Hamann MASA

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