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The Canadian Mathematical Societyin collaboration withThe CENTRE for EDUCATIONin MATHEMATICS and COMPUTINGTime: 212 hoursCalculators are NOT permitted.Do not open this booklet until instructed to do so.There are two parts to the paper.PART AThis part of the paper consists of 8 questions, each worth 5 marks. You can earn full value for eachquestion by entering the correct answer in the space provided. Any work you do in obtaining an answerwill be considered for part marks if you do not have the correct answer, provided that it is done in thespace allocated to that question in your answer booklet.PART BThis part of the paper consists of 4 questions, each worth 10 marks. Finished solutions must be writtenin the appropriate location in the answer booklet. Rough work should be done separately. If you requireextra pages for your finished solutions, foolscap will be provided by your supervising teacher. Any extrapapers should be placed inside your answer booklet.Marks are awarded for completeness, clarity, and style of presentation. A correct solution poorlypresented will not earn full marks.NOTE: At the completion of the contest, insert the information sheet inside the answer booklet. 2000 Canadian Mathematical SocietyTheCanadian OpenMathematics ChallengeWednesday, November 29, 2000Canadian Open Mathematics ChallengeNOTE: 1. Please read the instructions on the front cover of this booklet.2. Write solutions in the answer booklet provided.3. It is expected that all calculations and answers will be expressed as exact numbers suchas 4 2 7 p, + , etc.4. Calculators are not allowed.PART A1. An operation D is defined by a b abD =1 , b 0.What is the value of 1 2 3 4 D D D ( ) ( ) ?2. The sequence 9, 18, 27, 36, 45, 54, consists of successive multiples of 9. This sequence is thenaltered by multiplying every other term by 1, starting with the first term, to produce the newsequence , , , , , ,... 9 18 27 36 45 54 . If the sum of the first n terms of this new sequence is 180,determine n.3. The symbol n! is used to represent the product n n n 1 2 3 2 1 ( )( ) ( )( )( ) L .For example, 4 4 3 2 1 != ( )( )( ). Determine n such that n!= ( )( )( )( )( )( ) 2 3 5 7 11 1315 6 3 2.4. The symbol x means the greatest integer less than or equal to x. For example,5 7 5 . = , p = 3 and 4 4 = .Calculate the value of the sum 1 2 3 4 48 49 50 + + + + + + + L .5. How many five-digit positive integers have the property that the product of their digits is 2000?6. Solve the equation 4 16 226 sin sin x x = , for 0 2 x p.7. The sequence of numbers , a a a a a a a , , , , , ,3 2 1 0 1 2 3, is defined by a n a nn n+ ( ) = + ( ) 1 322,for all integers n. Calculate a0.8. In the diagram, D ABC is equilateral and the radius of itsinscribed circle is 1. A larger circle is drawn through the verticesof the rectangle ABDE. What is the diameter of the larger circle?E C DA BPART B1. Triangle ABC has vertices A 0 0 , ( ), B 9 0 , ( ) and C 0 6 , ( ). The points P and Q lie on side AB suchthat AP PQ QB = = . Similarly, the points R and S lie on side AC so that AR RS SC = = .The vertex C is joined to each of the points P and Q. In the same way, B is joined to R and S.(a) Determine the equation of the line through the points R and B.(b) Determine the equation of the line through the points P and C.(c) The line segments PC and RB intersect at X, and the line segments QC and SB intersect at Y.Prove that the points A, X and Y lie on the same straight line.2. In D ABC, the points D, E and F are on sides BC, CA and AB,respectively, such that = AFE BFD, = BDF CDE, and = CED AEF.(a) Prove that = BDF BAC.(b) If AB = 5, BC = 8 and CA = 7, determine the length of BD.3. (a) Alphonse and Beryl are playing a game, starting with thegeometric shape shown in Figure 1. Alphonse begins thegame by cutting the original shape into two pieces alongone of the lines. He then passes the piece containing theblack triangle to Beryl, and discards the other piece.Beryl repeats these steps with the piece she receives; that is to say, she cuts along the length ofa line, passes the piece containing the black triangle back to Alphonse, and discards the otherpiece. This process continues, with the winner being the player who, at the beginning of his orher turn, receives only the black triangle. Show, with justification, that there is always awinning strategy for Beryl.(b) Alphonse and Beryl now play a game with the same rulesas in (a), except this time they use the shape in Figure 2and Beryl goes first. As in (a), cuts may only be madealong the whole length of a line in the figure. Is there astrategy that Beryl can use to be guaranteed that she willwin? (Provide justification for your answer.)4. A sequence t t t tn 1 2 3, , , ..., of n terms is defined as follows:t11 = , t24 = , and t t tk k k= + 1 2 for k n = 3 4 , , ..., .Let T be the set of all terms in this sequence; that is, T t t t tn= { }1 2 3, , , ..., .(a) How many positive integers can be expressed as the sum of exactly two distinct elements ofthe set T ?(b) How many positive integers can be expressed as the sum of exactly three distinct elements ofthe set T ?Figure 1Figure 2AEC D BFThe Canadian Mathematical Societyin collaboration withThe CENTRE for EDUCATIONin MATHEMATICS and COMPUTINGTime: 212 hoursCalculators are NOT permitted.Do not open this booklet until instructed to do so.There are two parts to the paper.PART AThis part of the paper consists of 8 questions, each worth 5 marks. You can earn full value for eachquestion by entering the correct answer in the space provided. Any work you do in obtaining an answerwill be considered for part marks if you do not have the correct answer, provided that it is done in thespace allocated to that question in your answer booklet.PART BThis part of the paper consists of 4 questions, each worth 10 marks. Finished solutions must be writtenin the appropriate location in the answer booklet. Rough work should be done separately. If you requireextra pages for your finished solutions, paper will be provided by your supervising teacher. Any extrapapers should be placed inside your answer booklet.Marks are awarded for completeness, clarity, and style of presentation. A correct solution poorlypresented will not earn full marks.NOTE: At the completion of the contest, insert the information sheet inside the answer booklet. 2001 Canadian Mathematical SocietyTheCanadian OpenMathematics ChallengeWednesday, November 28, 2001Canadian Open Mathematics ChallengeNOTE: 1. Please read the instructions on the front cover of this booklet.2. Write solutions in the answer booklet provided.3. It is expected that all calculations and answers will be expressed as exact numbers suchas 4 2 7 p, + , etc.4. Calculators are not allowed.1. An operation is defined by a b a b = +23 .What is the value of 2 0 0 1 ( ) ( )?2. In the given diagram, what is the value of x?3. A regular hexagon is a six-sided figure which has all of its angles equal and all of its side lengths equal.If P and Q are points on a regular hexagon which has a side length of 1, what is the maximum possiblelength of the line segment PQ?4. Solve for x:2 2 4 642x x( )= + .5. Triangle PQR is right-angled at Q and has side lengths PQ = 14and QR = 48. If M is the midpoint of PR, determine the cosine ofMQP.6. The sequence of numbers t t t1 2 3, , , ... is defined by and t ttn nn+ = +111, for every positive integer n.Determine the numerical value of t999.7. If a can be any positive integer and2x a ya y xx y z+ =+ =+ =determine the maximum possible value for x + y + z.DE GFCBA3x4x5x6x2xPQ RMt12 =8. The graph of the function y g x = ( ) is shown.Determine the number of solutions of the equationg x ( ) = 112.PART B1. The triangular region T has its vertices determined by the intersections of the three lines x + 2y = 12,x = 2 and y = 1.(a) Determine the coordinates of the vertices of T, and show this region on the grid provided.(b) The line x + y = 8 divides the triangular region T into a quadrilateral Q and a triangle R.Determine the coordinates of the vertices of the quadrilateral Q.(c) Determine the area of the quadrilateral Q.2. (a) Alphonse and Beryl are playing a game, starting with a pack of 7 cards. Alphonse begins bydiscarding at least one but not more than half of the cards in the pack. He then passes the remainingcards in the pack to Beryl. Beryl continues the game by discarding at least one but not more than halfof the remaining cards in the pack. The game continues in this way with the pack being passed backand forth between the two players. The loser is the player who, at the beginning of his or her turn,receives only one card. Show, with justification, that there is always a winning strategy for Beryl.(b) Alphonse and Beryl now play a game with the same rules as in (a), except this time they start witha pack of 52 cards, and Alphonse goes first again. As in (a), a player on his or her turn must discardat least one and not more than half of the remaining cards from the pack. Is there a strategy thatAlphonse can use to be guaranteed that he will win? (Provide justification for your answer.)3. (a) If f x x x c ( ) = + +26 , where c is an integer, prove that f f 0 1 ( ) + ( ) is odd.(b) Let g x x px qx r ( ) = + + +3 2, where p, q and r are integers. Prove that if g 0 ( ) and g 1 ( ) are both odd,then the equation g x ( ) = 0 cannot have three integer roots.4. Triangle ABC is isosceles with AB = AC = 5 and BC = 6. Point Dlies on AC and P is the point on BD so that = APC 90 . If = ABP BCP, determine the ratio AD:DC. y x024 4 2 2