Problem Solving Problems for Group (Due by EOC Sep. 3) Caution, This Induction May Induce Vomiting! 3 35. a) Observe that 3, 3 3, and 3 3 56 3 3 5. 3 Use inductive reasoning to make a conjecture about the value of 3 3 nn. Use your conjecture to determine the value of 33 00,000 00,00. 3 b) Observe that 3, 5 3 5, and 7 3 5 7. Use inductive reasoning to make a conjecture about the value of 3 5 n. Use your conjecture to determine the value of 3 5,999,999. c) Observe that, 3, and 3 3 3 3. Use inductive reasoning to make a conjecture about the value of 3 3 nn. Use your conjecture to determine the value of. 3 3 999,000 Interesting Is In The Eye Of The Beholder. There is an interesting five-digit number. With a after it, it is three times as large as with a before it. What is the number? {Hint: If x is the five-digit number, then x abcde, abcde 0x,abcde 00,000 x.} Who Needs Logarithms? 3. If x 5 and 5 y 3, then find the value of xy. {Hint: Substitute the first equation into the second equation, and use an exponent property.}
. What are the final two digits of The Last Two Standing 08 7? {Hint: Look for a pattern: Power of 7 Final two digits 7 07 07 7 9 9 3 7 33 3 7 0 0 5 7 6807 07 6 7 769 9 } A Lot Of Weeks, But How Many Days Left Over? 5,0 5. What is the remainder when is divided by 7? {Hint: Look for a pattern in the remainders: Power of Remainder when divided by 7 3 8 6 5 3 } Seven Heaven or Seven 6. Find the largest power of 7 that divides 33!. 33! 3 3 33 {Hint: The multiples of 7 occurring in the expansion of 33! are 7,,,8,,7 9. The multiples of 7 9 occurring in the expansion of 33! are 9,98,,9 7 3 The multiple of 7 33 occurring in the expansion of 33! is just 33. There are no multiples of higher powers of 7 occurring in the expansion of 33!}
Just Your Average Joe. 7. If Joe gets 97 on his next math quiz, his average will be 90. If he gets 73, his average will be 87. How many quizzes has Joe already taken? {Hint: Let n be the number of quizzes he has already taken, and T the total number of points T 97 T 73 he has already earned on the quizzes. Then 90, 87.} n n Happy 08! 8. Find the 08 th digit in the decimal representation of 7. {Hint: 7 0.857, so use a pattern.} A European Sampler. 9. A box contains 8 French books, Spanish books, 9 German books, 5 Russian books, 8 Italian books, and 0 Chinese books. What is the fewest number of books you can select from the box without looking to be guaranteed of selecting at least 0 books of the same language? {Hint: What is the largest number of books you can select and still not have 0 books of the same language? The answer to the problem is more than the answer to the previous question.} The Beast With Many Fingers And Toes. 6,666 0,000 0. How many digits does the number 8 5 have? {Hint: Zeroes come from factors of 0. Factors of 0 come from 5 s and s.} Odds, Evens, What s The Difference?. a) What do you get if the sum of the first 8,000,000,000 positive odd integers is subtracted from the sum of the first 8,000,000,000 positive even integers? {Hint: 6 8 6,000,000,000 3 5 7 5,999,999,999 } b) What do you get if you subtract 8,000,000,000 from the sum of the first 8,000,000,000 even numbers?
Destination Cancellation.. Express as a fraction, in lowest terms, the value of the following product of,999,999 factors 3,000,000. {Hint: Look for a pattern: 3 3 3 3 3 3 } Just Gimme An A While I m Hanging Out In The Library. 3. a) A class of fewer than 5 students took a test. The results were mixed. One-third of the class received a B, one-fourth received a C, one-sixth received a D, one-eight of the class received an F, and the rest of the class received an A. How many students in the class got an A? {Hint: The number of students in the class must be a multiple of 3,, 6, and 8, and must be smaller than 5.} b) The library in Johnson City has between 000 and 000 books. Of these, 5% are fiction, are biographies, and are atlases. How many books are either biographies 3 7 or atlases? Cogswell Cogs Or Spacely Sprockets?. In a machine, a small gear with 5 teeth is engaged with a large gear with 96 teeth. How many more revolutions will the smaller gear have made than the larger gear the first time the two gears are in their starting position? {Hint: A revolution of the smaller gear is a multiple of 5 teeth, and a revolution of the larger gear is a multiple of 96 teeth. So the gears are again in the starting positions at common multiples of 5 and 96.}
Life Is Like A Box Of Chocolate Covered Cherries. 5. a) Assume that chocolate covered cherries come in boxes of 5, 7, and 0. What is the largest number of chocolate covered cherries that cannot be ordered exactly? {Hint: If you can get five consecutive amounts of cherries, then you can get all amounts larger. Here s why: Suppose you can get the amounts 3,,5,6,7, then by the addition of the box of size 5, you can also get 8,9,30,3,3, and another addition of the box of size 5 produces 33,3,35,36,37 and so on. This would also be true of seven consecutive amounts and ten consecutive amounts, but five consecutive amounts would occur first. So look for amounts smaller than the first five consecutive amounts.} b) Do the same problem, except the cherries come in boxes of 6, 9, and 0. 6. If x x Two Squares Don t Get Along A Difference Of Squares! 0,000 0,000 5,000 5,000,500,500 y 5, x y, x y 3, x y, and 0,000 0,000 y,500,500, then find the value of x Hint: a ba b a b. y. 0,000 0,000 I Hate This Problem To The N th Degree. 7. Use the following properties of exponents to find the exact value of the given expressions. n m n m x x x xy, n n n n x y, x m x nm a) 3 6 9 90,000 90,000 89,999 90,000 b) 3 6 9 00,000 0,000 00,000 5000
8. Notice that Getting Solutions Without Actually Solving. x a x b x a b x ab 3 x a x b x c x a b c x ab ac bc x abc abc abd acd bcd x abcd. x a x b x c x d x a b c d x ab ac ad bc bd cd x 3 n a) Use inductive reasoning to determine the value of the coefficient of x and the constant x a x a x a. term in the expansion of the following product: n b) Use the previous result to determine the sum of the seventeenth powers of the 7 7 solutions of the equation x 3x 0. {Hint: The Fundamental Theorem of Algebra guarantees that the equation 7 x 3x 0 has seventeen solutions(counting duplicates). The seventeen 7 solutions of x a x a x a7 x 3x 0 are a, a,, a 7. So adding the seventeen equations together yields: 7 a 3a 0 a 3a 0 7 a 3a 0 7 7 7 7 7 7 a a a7 a a a7 3 7 0. Now use the previous result.} Even So, It s Odd. 9. Show that for every positive integer n, n 3n 8 is even. {Hint: n is either even or odd, so take it from here.} An Odd Product. 0. Find the value of the following product: 3 5 7 9 0. 3 5 07 {Hint: What factors can you cancel out?}
. Consider the following figure: Call It Like You See It. a) What fraction of the large square is shaded? b) What fraction of the large square is shaded? c) What fraction of the large square is shaded? Lucky 3, I Repeat, Lucky 3. Lucky 7, I Repeat.. a) Show that every six-digit number of the form abc,abc (for example 8,8 or 35,35) is divisible by 3. b) Show that every six-digit number of the form abc,abc (for example 8,8 or 35,35) is divisible by 7. c) Show that every six-digit number of the form abc,abc (for example 8,8 or 35,35) is divisible by. A Whole Lotta Eights; A Whole Lotta 3. a) What is the smallest whole number that when multiplied by 9 gives a number whose digits are all 8 s? b) What is the smallest whole number that when multiplied by 9 gives a number whose digits are all 5 s? c) What is the smallest whole number that when multiplied by 9 gives a number whose digits are all 3 s?
. A man born in the year he born? You re A Real Square, Man. x died, on his 87 th birthday, in the year x. In what year was It s As Easy As 3. 5. If the digit 9 is written to the right of a certain number, that number is increased by,,. Find the number. {Hint: If the original number is x, then 0x 9 is the new number.} See How Everything Lines Up. 6. Given the following incomplete distance chart for points in a plane, find the distance from A to B. A B C D A 0? 9 B? 0 5 7 C 5 0 D 9 7 0 Stick puzzles involve rearranging, removing, or adding sticks in order to accomplish the requirements of a problem. In the following problems, you might want to use the following suggestions: This arrangement of three sticks can be used to represent a square root: A stick representing an over bar can be used to multiply a Roman numeral by,000: represents 5,000
7. a) Move one stick to make a true equation. Just Stick It. b) Remove two sticks to make a true equation. c) Move one stick to make a square. d) Move one stick to make a true equation. e) Move one stick to make a true equation.
What An Intersecting Little Problem. 8. Find the area of the following shaded region formed by the two perpendicular intersecting rectangles. Who Said Holes Have To Be Round? 9. Find the area of the square hole in the middle of the square. 3 6 {Hint: Find the areas of the four right triangles, and subtract it from the area of the large square.} The Arc Of Triangle. 30. In right triangle ABC with legs of 5 and, arcs of circles are drawn, one with center A and radius, the other with center B and radius 5. What is the length of MN? N M B A C
I Go Cuckoo For Coconuts. 3. Five sailors were stranded on a desert island, and their only food was coconuts. One day they gathered all the coconuts on the island together, and the next day they would divide them evenly. The first sailor woke up early and gave one coconut to a monkey and hid his fifth of the remaining coconuts. Then the second sailor woke up and gave one coconut to the same monkey and hid his fifth of the remaining coconuts. The third, fourth, and fifth sailors all did the same. Upon arising the next day, one coconut was given to the monkey, and the remaining coconuts were divided equally among the five sailors. What is the smallest starting number of coconuts possible? Sailor Remaining coconuts N 5 N 3 N 5 5 555 5 5 55 5 5 5 5 5 N 5 N Sailor Remaining coconuts R R 5 5 5 5 6 R 6 R 6 5 5 R 5 R 9 3 5 5 5 5 65 369 56 6 6 5 R 5 R N is the starting number of coconuts, and R is the remaining coconuts after the five sailors have done their secret removals. Or start with a smaller problem: If one sailor gives a coconut to the monkey, takes one-fifth of the remaining coconuts, and then another coconut is given to the monkey and the rest are divided among the five sailors then, the number of remaining coconuts would have to be a 55k multiple of 5: 5 N 5k. So N, which means that 5k must be a multiple of, and 6 is the smallest possible multiple of that works. This means that the smallest number of coconuts in this case would be. If one sailor gives a coconut to the monkey, takes one-fifth of the remaining coconuts, and another sailor gives a coconut to the monkey, takes one-fifth of the remaining coconuts, and
another coconut is given to the monkey, and the rest are divided among the five sailors then, the number of remaining coconuts would have to be a multiple of 5: 5k 6 55 N 5k. So N, which means that 5k 6 must be a 6 multiple of 6, and 936 is the smallest possible multiple of 6 that works. This means that the smallest number of coconuts in this case would be. Keep going! I Don t Give A Square s S! 3. A right triangle with leg measurements of a and b has an inscribed square with side measurement s as shown in the figure. Find the value of s. b s s s s {Hint: The areas of the square and two little right triangles must equal the area of the big right triangle.} a The Age Of Man. 33. A man lived one-sixth of his life in childhood, one-twelfth in youth, and one-seventh as a bachelor. Five years after his marriage, a son was born who died four years before his father at half his father s final age. What was the man s final age? Hint: 6 M M 7 M 5 M is the age of the man up to the death of his son, and this is years less than the man s final age. This Game s Just A Pile Of Shi-llings. 3. A game involves a pile of coins and two players who alternately take turns removing,, 3,, or 5 coins from the pile. The player who removes the last coin(s) wins the game. How many coins should the first player remove in order to guarantee that he can win on his next turn?
Crazy 8. 35. Show that the difference of the squares of two odd numbers is divisible by 8. Hint: Suppose the two odd numbers are xn and ym. Then x y n m n n m m n mn m. If you can show that n mn m must be divisible by, then you re done. n m n m n m n mn m even even even odd? odd odd even odd? even odd odd even? odd even odd even? Where There s A Will. 36. A father in his will left all his money to his children in the following manner: $000 to the first born and /0 of what then remains, then $000 to the second born and /0 of what then remains, then $3000 to the third born and /0 of what then remains, and so on. When this was done each child had the same amount. a) How much money did the father leave in his will? b) How many children were there? {Hint: If S is the amount in the will, then the first born gets,000,000 0 S, and the 9 second born gets,000,000,000 0 0 S. Set them equal to each other, and solve for S.} Finally, A Light A The End Of The Tunnel. 37. A train which is mile long is traveling at a steady speed of 0 miles per hour. It enters a tunnel mile long at noon. At what time does the rear of the train emerge from the tunnel? (Both ends of the tunnel are in the same time zone!) {Hint: How far does the front of the train have to travel?}
If The Glove Don t Fit, Then I Quit! 38. There are 0 gloves in a drawer: 5 pairs of black gloves(5 left and 5 right), 3 pairs of brown gloves(3 left and 3 right), and pairs of grey gloves( left and right). You will select gloves from the drawer in the dark, and you may check them only after the selection has been made. What is the fewest number of gloves you need to select in order to be guaranteed of selecting at least a) one matching pair of gloves?(left and right of the same color) b) one matching pair of each color? Making The Most And The Least Of Them. 39. Using the four digits 5, 6,, and 9, a) Make two -digit numbers that have the largest possible sum. b) Make two -digit numbers that have the smallest possible positive difference. c) Make two -digit numbers that have the largest possible product. d) Make two -digit numbers that have the smallest possible product. It s More Magical! 0. The numbers in each row, column, and diagonal add up to 3. All the numbers from to 6 appear, but only once each. Fill in the missing numbers. 6 6 0 8