Grade 7/8 Math Circles November 22 & 23, Math Jeopardy

Transcription

1 Faculty of Mathematics Waterloo, Ontario NL 3G Centre for Education in Mathematics and Computing Grade 7/8 Math Circles November & 3, 06 Math Jeopardy Let s play Math Jeopardy! Today, we will be playing a fun game of jeopardy to review what you have learned so far. Round One Mathematical Puzzles $00 Find three positive whole numbers that have the same answer when added together or when multiplied together.,,3 $00. A number has 3 digits and is odd. Two digits are the same 3. The sum of the digits in the tens and ones places is odd. The sum of the digits is What is the number? $300 Matching Socks Sixteen red socks and sixteen blue socks are mixed up in a dresser drawer. The socks are all identical except for their colour. Suppose Richard wants two matching socks but there is a black out so the room is dark and he can?t see. What is the smallest number of socks that Richard must take out of the drawer to guarantee he has a pair of socks that match?

2 Suppose the first sock Richard takes out is red. Then, the second sock he takes out is red or blue. If it is red, then he is done. If it is blue, Richard now has socks (one red and one blue). The third sock Richard takes out is red or blue. If it is red, then he has a match with the first sock. If it is blue, he has a match with the second sock. So, Richard needed 3 socks to get a matching pair. This is the same outcome if the first sock Richard had chosen was blue. Thus, 3 socks is the smallest number of socks Richard must take out. $00 Cutting the Pie! With one straight cut, you can slice a pie into pieces. With two straight cuts, you can slice a pie into pieces. With three straight cuts, you can slice a pie into as many as 7 pieces, as shown below What is the largest number of pieces that you can make with six straight cuts? The first cut creates pieces. The second cut intersects the first cut to create pieces. The third cut intersects both of the previous cuts to create 7 pieces. Let s organize this into a table. # of cuts # of pieces Difference between # of pieces 0 = = = 3

3 Notice the difference in number of pieces increases by each time. By following this pattern, the fourth straight cut will create 7 + = pieces. The fifth cut will create + 5 = 6 pieces. And finally, the sixth straight cut will create = pieces of pie. $500 Toothpick Geometry Make the fish swim in the opposite direction by moving exactly 3 toothpicks. Solutions may vary. Continued Fractions $00 Express the following as a single improper fraction. + + = + 3 = + 3 = 5 3 = 3 5 $00 Find the gcd(8, 56) and simplify 56 8 to its lowest terms. 8 = gcd(8, 56) = gcd(56, 336) 56 = gcd(56, 336) = gcd(336, 0) 336 = gcd(336, 0) = gcd(0, 6) 0 = gcd(0, 6) = gcd(6, 8) 6 = 8 + gcd(6, 8) = gcd(8, ) 8 = + 0 gcd(8, ) = So, gcd(8, 56) = 56 8 = 3 3 3

4 [ ] $300 Express ;, 5, as a single improper fraction. [ ] ;, 5, = = + + = + + = + 6 = + 6 = 57 6 $00 Express 59 3 as a continued fraction. 59 = = 59 3 = = = = 59 3 = = = 3 + = 59 3 = = 3 + = 59 3 = = $500 Solve for x, y, and z. 6 = x + y + z = = 6 = = = = 6 = Thus, x =, y = 3 and z = 5.

5 Sequences $00 { 8 3, 3, 3, } 3,... is a geometric sequence with a constant ratio of. $00 Which term in the following sequence is equal to 0? {,, 8, 5,...} This is an arithmetic sequence with a common difference of d = 7. The first term of the sequence is t =. Using the formula for the nth term of the sequence, we have... t n = + (n )7 To determine what term number is equal to 0 in the sequence, let t n = 0. Plug this into the formula to get... t n = 0 = + (n )7 98 = (n )7 = n n = 5 So, 0 is the 5 th term of the sequence. (i.e. t 5 = 0) $300 Find the 5 th term of the following recursive sequence. t n = t n + 5t n ; t =, t = t 3 = t + 5t = + 5 = t = t 3 + 5t = + 5 = 58 t 5 = t + 5t 3 = = 36 Thus, t 5 = 36 5

6 $00 What is the 3 rd term in the arithmetic sequence where t 5 = 6 and t 3 = 38? We know that the 5 th term is 6, the 3 nd term is 38, and that there are (3 5) = 7 terms with 38 now being the 8 th term in the sequence. Now, we can solve for the common difference, d. 38 = 6 + (8 ) d 87 = 7 d d = Now, again we know that 6 is the 5 th term and there are 5 3 = terms between the 3 rd and 5 th term. We count 3 terms if we include the 5 th term. Now we can solve for the 3 rd term. 6 = t 3 + (3 ) 6 = t = t t 3 = 9 $500 Without using a calculator, what is the last digit in the 56 th term in the following sequence? { 9, 9, 9 3, 9,... } Notice the pattern of the last digit of every term. t = 9 = 9 t = 9 = 8 t 3 = 9 3 = 79 t = 9 = 656. The last digit of every odd term is 9 and the last digit of every even term is. Following this pattern, the last digit in the 56 th term of this sequence is. 6

7 Visual Group Theory $00 The hexagon below shows a rotation of 60 from the initial starting position, then a rotation of 300 which undoes the first action. Draw the hexagon after the following two rotations: R 60 R 80 $00 Express this rearrangement mathematically. ( )

8 $300 Suppose you have the following five balls. Draw the final configuration of where the five balls are given the following rearrangement rule: ( ) $00 Determine the equivalent action of ( ) ( ) ( ) = $500 Create a rearrangement rule to return all the balls to their initial position given the rearrangement rules below. ( ) ( ) The initial position of the balls is given below. The rearrangement rule below returns the balls to their initial position. ( 3 )

9 Random Questions I $00 There are three people at the dinner table. Two are mothers, and two are daughters. How is this possible? The women at the table are grandmother, mother, and daughter. $00 Express the following as a single improper fraction = = = = = = 0 7 $300 Young Tony Stark asked his grandmother how old she was. Knowing Tony to be quite bright, she replied: Tony, I have four children born three years apart between each one and the next. I was when I had my oldest son. Now my youngest is 3. That s all I m telling you! How old is Tony s grandmother? Tony s grandmother had children in 3 = years since her children were born 3 years apart. She was when she had her first child, so she was + = 36 years old when she had her youngest child. In present day, her youngest is now 3. Therefore, Tony s grandmother is now = 67 years old. $00 There are three gentlemen in a meeting: Mr. Red, Mr. Green, and Mr. Gold. They are wearing red, green and gold ties. Mr. Red: How amazing! Our last names are Red, Green, and Gold, and one of us is wearing a red tie, another is wearing a green tie, and another is wearing a gold tie. Mr. Green: And none of our tie colours match our names! Mr. Gold: You are right! If Mr. Red s tie is not gold, what is the tie colour of each person? We know that no one s last name matches their tie colour. So, Mr. Red s tie is not red and we are told that it is not gold. Thus, Mr. Red must be wearing a green tie. Since Mr. Red is wearing a green tie, Mr. Gold must be wearing a red tie since he cannot be wearing gold. Since gold is the only tie colour left, Mr. Green must be wearing a gold tie. 9

10 $500 Complete the Kenken! Gauss Prep $00 Daniel rode his bicycle at a constant speed. After 0 minutes, he cycled km. How far did he cycle in 30 minutes? km 0 0 min. 0 = 0.6 km min. Therefore, Daniel cycled = 8 km in 30 minutes. $00 Joe is reading a 00 page book. On Monday, he reads 0 pages. On each day after the first, the number of pages that he reads is 0 more than on the previous day. Joe finishes the book on Friday. On Monday, Joe reads 0 pages so there are 00 0 = 360 pages left. On Tuesday, Joe reads = 60 pages so there are = 300 pages left. On Wednesday, he reads = 80 pages so there are = 0 pages left. On Thursday, he reads = 00 pages so there are 0 00 = 0 pages left. And on Friday, he reads = 0 pages so there are 0 0 = 0 pages left. Thus, Joe finishes his book on Friday. 0

11 $300 Which of these values is the largest? (a) (b) (c) (d) (e) Since all options are fractions, we know that the largest value will have the smallest denominator. This eliminates (b) and (d). Since we want the denominator to be as small as possible, we should choose the option with the largest fraction subtracted from. So, < 3 <, therefore (e) is the correct answer. $00 In how many ways can 0 be expressed as the sum of two integers, both greater than zero, with the second integer greater than the first? Beginning with the positive integer as a number in the first pair, we get the sum 0 = +00. From this point we can continue to increase the first number by one while decreasing the second number by one, keeping the sum equal to 0. The list of possible sums is: 0 = = = = After this point, the first number will no longer be smaller than the second if we continue to add to the first number and subtract from the second. There are 50 possible sums in all..

12 $500 In the diagram, ABCD is a square with side length 6, and W XY Z is a rectangle with ZY = 0 and XY = 6. Also, AD and W X are perpendicular. If the shaded area is equal to half of the area of W XY Z, the length of AP is. The area of rectangle W XY Z is 0 6 = 60. Since the shaded area is half of the total area of W XY Z, its area is 60 = 30. Since AD and W X are perpendicular, then the shaded area has four right angles, so is a rectangle. Since square ABCD has a side length of 6, then DC = 6. Since the shaded area is 30, then P D DC = 30 or P D 6 = 30 or P D = 5. Since AD = 6 and P D = 5, then AP =. Round Two The Matrix $00 Add the following matrices: 9 + = $00 Evaluate = 0 $600 Evaluate. [ ] [ ] 9 5 =

13 $800 Scott, Hugh, and Peter are doing extra chores to earn some extra allowance this weekend. On Friday, they all worked for hours. On Saturday, Scott and Hugh worked for hours, and Peter worked for 5 hours. On Sunday, Scott worked for hours, and Hugh worked for 5 hours, and Peter worked for 3 hours. If Scott earns $5 per hour, Hugh earns $6 per hour, and Peter works $7 per hour, on which day did they earn the most money altogether? = Altogether, Scott, Hugh, and Peter earned the most money on Saturday. $000 Hill Cipher Given the key matrix below, encrypt the following message: CIRCLE First, write CIRCLE in matrix form and convert each letter into its corresponding number. (i.e. A = 0, B =, C =,...) C CIRCLE I R 7 C L 8 E 7 Then multiply the key matrix by the message matrix. Subtract 6 from each element of the resulting matrix such that all elements are between and including 0 and Y = P The encrypted message is YPWELL W E L L 3

14 Combinatorial Counting $00 Teodora is getting a new phone! She is deciding between different smartphones and a red, blue, yellow, or green phone case. How many smartphone and case combinations can she choose from? There are = 6 combinations that Teodora can choose from. $00 Calculate the following:!! =! / 3! / = 3 = 8 $600 There s a crisis on Earth! Given a team has members, how many different team arrangements can Batman send from the Justice League if there are 35 available superheroes? ( ) 35 There are = 5360 different teams Batman can send. $800 Calculate the following: 9C 8P 3 = ( ) 9 8! (8 3)! = 9!!(9 )! 5! 8! = 9 /8! 5! /! /5! /8! = 9 = 3 8 $000 Tommy, Chuckie, and three other friends are going to the movies to watch Fantastic Beasts and Where To Find Them. But Tommy and Chuckie had a fight and refuse to sit next to each other. How many different ways can Tommy, Chuckie, and friends sit in the movie theatre? We can solve this by subtracting the number of ways Tommy and Chuckie can sit together from the total number of ways the 5 friends can sit in the theatre. There is a total of 5! = 0 ways they can all sit in the theatre. Next, we will count the number of ways Tommy and Chuckie can sit together. We treat Tommy and Chuckie as one person (or a subgroup). Now we are counting the seating arrangement for friends. So there are! = ways to seats the friends. However, we need to consider that there are! = ways of arranging Tommy and Chuckie s seating arrangement. (i.e. Tommy & Chuckie and Chuckie & Tommy are different seating arrangements.) There are!! = = 8 different seating arrangements such that Tommy and Chuckie are sitting next to each other. Therefore, there are 0 8 = 7 different ways they can sit in the movie theatre.

15 Areas of Triangles $00 Find the area of ABC. A 0 cm 8 cm B cm C ABC is an isosceles triangle so the height of our triangle intersects the base, BC at the midpoint of BC. Half of the base is 6 cm. Now, we can use Pythagorean Theorem to find the height of ABC. 6 + h = 0 h = = 6 = 8cm We now have the base and height of ABC, so using the basic area formula, we get... A ABC = 8 = 8 cm $00 Find the area of DEF. D 7 mm E 6 mm 5 mm F 5

16 Use Heron s formula! s = ( ) = 3 A DEF = 3(3 7)(3 6)(3 5) = 3(7)(8)(9) = 66 A DEF = 0 mm $600 Given the coordinates below, find the area of P QR. P (3, 3), Q(, ), R(8, 7) Use Shoelace Theorem! A P QR = = ( ) ( ) = ( ) ( ) = = 336 = (336) A P QR = 68 units 6

17 $800 The perimeter of XY Z is 7 cm. If XY = XZ and Y Z = 0 cm, what is the area of XY Z? X 6 cm 6 cm Y 0 cm Z XY Z is isosceles since XY = XZ. So, XY = Heron s formula: s = (7) = 36 cm 7 0 = 6 cm. Now we can use A XY Z = 36(36 6)(36 6)(36 0) = 36(0)(0)(6) = A XY Z = 0 cm $000 Given that the perimeter of DEF is, what is the area of hexagon ABCDEF? B E A 5 F D 6 C 7

18 Notice that ABC = BCA, this means BCA is an isosceles triangle and that AB = AC = 5. Use Heron s Formula to find the area of ABC: s ABC = ( ) = 0 A ABC = 0(0 5)(0 5)(0 30) = 0(5)(5)(0) = = 300 Next, since AC = 5, DF = =. We also know that the perimeter of DEF is, so DE = 3 = 5. Use Heron s formula again to find the area of DEF : s DEF = ( ) = A DEF = ( 3)( )( 5) = (8)(7)(6) = 7056 = 8 The area of hexagon ABCDEF is the area of ABC minus the area of DEF so: A ABCDEF = A ABC A DEF = = 6 units Logic $00 James made some cookies. He ate one cookie and gave half of the rest to Guntaas. Then he ate another cookie and gave half of the rest to Daniel. James now has 5 cookies. How many cookies did James start with? Work backwards! Right now, James has 5 cookies. Before this, he gave half his cookies to Daniel so there were 5 = 0 cookies. Then before James ate a cookies, there were 0 + = cookies. Next, James gave half his cookies to Guntaas so then there were = cookies. Then, James ate another cookie. Therefore, James started off with + = 3 cookies. $00 There are three Dalmatian puppies: Spot, Socks, and Patches. Spot has fewer spots than Socks, but more spots than Patches. Which puppy is the spottiest? Spot has fewer spots than Socks and has more spots than Patches, so Spot is in the middle in terms of number of spots. Since Spot has fewer spots than Socks, then Socks must be the spottiest puppy. $600 How can you use only five 5s and only addition to make 565? = 565 8

19 $800 Fill in the blanks! $000 Let s play Clue! Miss Scarlet, Colonel Mustard, Mr. Green, Mrs. Peacock and Mrs. White were involved in a theft. One of the five stole a credit card from one of the other four. The following facts are known:. A man and woman were eating at McDonalds at the time of the theft. The thief and victim were together at the bank at the time of the theft 3. Colonel Mustard was not with a married woman at the time of the theft. Mr. Green was not with Mrs. White at the time of the theft 5. One of the married women was alone at the time of the theft 6. One of the married women was at the bank at the time of the theft 7. The victim is a man Who is the thief and who is the victim? By statements 5 and 6, Miss Scarlet must be the woman eating at McDonalds. By 3, Colonel Mustard was not with a married woman at the time of the theft so he must be the man eating at McDonalds with Miss Scarlet which satisfies. This means Mr. Green is the victim. Then by and 6, Mrs. Peacock must the the married woman at the bank since Mr. Green and Mrs. White cannot be together. This means Mrs. White was alone at the time of the theft. 6 and 7 implies that a married woman is a thief so Mrs. Peacock is the thief. Therefore, the thief is Mrs. Peacock and the victim is Mr. Green. 9

20 Random Questions II $00 What are the missing values for x, y, and z? [ ] [ ] [ ] = $00 Which one is the better deal? 0 chocolates for $5 or 5 chocolates for $8? 0 chocolates for $5 is $.50 a piece and 5 chocolates for $8 is $.60 a piece. Thus, the better deal is 0 chocolates for $5. $600 What is the measure of angle x? B 0 70 C 60 I A D F E 30 G H $800 If the area of a square is 6 m, what is the perimeter? The square has a side length of 6 = 8 m. Thus, the perimeter of the square is 8 = 3 m. $000 There are 5 cards in a standard deck of cards. How many ways can Jiin select cards from the deck if of the cards are spades? (Hint: There are 3 cards per suit. Of the cards, Jiin selects from the deck, of( them ) must be spades. There are 3 3 spades from the deck of 5 cards so there are = 78 ways to choose spades. Now, ( Jiin ) needs to choose cards from the remaining 5 3 = 39 cards. There 39 are = 7 ways to choose the remaining cards. Therefore, in total, there are ( )( ) 3 39 = 78 7 = ways that Jiin can select cards from the deck if of the cards are spades. 0

21 Gauss Prep $00 If 0 x 0 = 9990, then x is equal to Since 0 x 0 = 9990, then 0 x = = If 0 x = 0000, then x = since ends in zeroes. $00 In a class of 0 students, 8 said they liked apple pie, 5 said they liked chocolate cake and said they did not like either. How many students in the class liked both? Of the 0 students, did not like either dessert. Therefore, 0 = 8 students liked at least one of the desserts. But 8 students said they liked apple pie, 5 said they liked chocolate cake, and = 33, so 33 8 = 5 students must have liked both of the desserts. $600 Winifred earns $0/hour and works 8 hours per day for 0 days. She first spends 5% of her pay on food and clothing, and then pays $350 in rent. How much of her pay does she have left? In 0 days, Winifred works 8 0 = 80 hours. So in these 0 days, she earns 80 0 = $800. Since 5% =, she spends $800 = $00 on food and clothing, leaving her with $600. If she spends $350 on rent, she will then have $ = $50 left. $800 The values of r, s, t, and u are, 3,, and 5, but not necessarily in that order. What is the largest possible value of r s + u r + t r? We first recognize that in the products, r s, u r and t r, r is the only variable that occurs in all three. Thus, to make r s + u r + t r as large as possible, we choose r = 5, the largest value possible. Since each of s, u and t is multiplied by r once only, and the three products are then added, it does not matter which of s, u or t we let equal, 3 or, as the result will be the same. Therefore, let s =, u = 3 and t =. Thus, the largest possible value of r s+u r+t r is = = 5.

22 $000 In the diagram, ABCD is a square with area 5 cm. If P QCD is a rhombus with area 0 cm, the area of the shaded region, in cm, is cm Since ABCD is a square and has an area of 5 cm, then the square has a side length of 5 cm. Since P QCD is a rhombus, then it is a parallelogram, so its area is equal to the product of its base and its height. Join point P to X on AD so that P X makes a right angle with AD, and to Y on DC so that P Y makes a right angle with DC. Then the area of the shaded region is the area of rectangle ABZX plus the area of triangle P XD. Since the area of P QCD is 0 cm and its base has length 5 cm, then its height, P Y, must have length cm. Therefore, we can now label DX =, DP = 5 (since P QCD is a rhombus), AX =, and AB = 5. So ABZX is a by 5 rectangle, and so has area 5 cm. Triangle P XD is right-angled at D, and has DP = 5 and DX =, so by Pythagorean Theorem, P X = 3. Therefore, the area of triangle P XD is 3 = 6 cm. So, in total, the area of the shaded region is cm.

23 Final Jeopardy A rectangular piece of paper ABCD is folder so the edge CD lies along edge AD, make a crease DP. It is unfolded, and then folded again so that edge AB lies along the edge AD, making a second crease AQ. The two creases meet at R, forming triangles P QR and ADR, as shown. If AB = 5 cm and AD = 8 cm, what is the area of quadrilateral DRQC, in cm? To find the area of quadrilateral DRQC, we subtract the area of P RQ from the area of P DC. First, we calculate the area of P DC. We know that DC = AB = 5 cm and that DCP = 90. When the paper is first folded, P C is parallel to AB and lies across the entire width of the paper, so P C = AB = 5 cm. Therefore, the area of P DC is 5 5 = 5 =.5 cm. Next, we calculate the area of P RQ. We know that P DC has P C = 5 cm, P CD = 90, and is isosceles with P C = CD. Thus, DP C = 5. Similarly, ABQ has AB = BQ = 5 cm and BQA = 5. Therefore, since BC = 8 cm and P B = BC P C, then P B = 3 cm. Similarly, AC = 3 cm. Since P Q = BC BP QC, then P Q = cm. Also, RP Q = DP C = 5 and RQP = BQA = 5. Using four of these triangles, we can create a square of side length cm (and thus an area of cm ). The area of one of these triangles (for example, P RQ) is of the area of the square, or cm. So the area of quadrilateral DRQC is therefore.5 =.5 cm. 3