These problems are harder than those in Place Value and Fusing Dots, but they require the same kind of ideas.

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1 These problems are harder than those in Place Value and Fusing Dots, but they require the same kind of ideas. 1. Emphatic Countdown. Find the decimal representation of (10/9!) 8! 7! 6! 5!/4! + 3! 2! + 1!. 2. Evaluate the expression. It describes my behavior in late November Find all integers n for which the sum of the digits is 11 and the product of the digits is For each positive integer n, let G(n) denote the product of the five consecutive integers beginning at n. For example G(1) = = 120. Let R(n) denote the rightmost nonzero digit of n when n is written in decimal notation. For example R(G(1)) = R(120) = 2. (a) Find the smallest n for which R(G(n)) is an odd number. (b) Find the smallest n for which R(G(n)) is 5. (c) Find the smallest n for which R(G(n)) is the digit 9. (d) Find the smallest n for which R(G(n)) is the digit What are the largest and the smallest five-digit numbers the product of whose digits is 2520? 6. What is the value of the base 3 repeating decimal expressed as a base-ten common fraction? 7. The product of two three-digit numbers abc and cba is , where c < a. Find the value of abc. 8. Notice that = Its also true that = , and = What we want to explore here is does the first equation hold for any other pairs of numbers. See the problem on multiplicative palindromes. 9. Suppose a, b and c are three different decimal digits. Then there are six different three digit numbers that can be built using all of the three digits. Suppose the difference between the largest such number and the smallest such number is one of the six numbers. What are the three digits? 1

2 10. Great Numbers. Call a nine-digit number great if it consists of nine different nonzero digits. It is possible that two great numbers can have a great sum. In fact, it is even possible that two great numbers can both have a great sum and also be identical in seven places. Find two such numbers. 11. Finding the unknown digit. Let N = abcde denote the five digit number with digits a, b, c, d, e and a 0. Let N = edcba denote the reverse of N. Suppose that N > N and that N N = 670x3 where x is a digit. What is x? 12. Recall also that the remainder when a decimal integer is divided by 9 is the same as the remainder when the sum of its digits is divided by 9. For example 1234 = 9k + 1 since = 10 = Let a 1 = 1 and for each n 1, define a n+1 as follows: a n+1 = n a n log(n+1). Let s work out the first few values of the sequence. For example, a 2 = 2 + a log(n+1) = log(2) = = = 12 and a 3 is a 3 = 3 + a log(2+1) = log(3) = = = = 123. Using the definition, we can show that a 4 = 1234, etc. This definition is said to be recursive. Recursion is a popular topic in discrete mathematics. We ll see later that the Principle of Mathematical Induction is an important tool in reasoning about recursively defined sequences. (a) How many digits does a 2008 have? (b) Find the smallest n such that a n has at least 2008 digits. 2

3 (c) What is S(a 2008 )? (d) How many 1 s are in the representation of a 2008? (e) How many 0 s are in the representation of a 2008? (f) Find the first 5 multiples of 9 among the a n. (g) Find the remainder when a 2008 is divided by 9. (h) What is the smallest n such that a n is a multiple of 11? 13. Let b 1 = 1 and for each n 1, define b n+1 as follows: b n+1 = (n + 1) 2 + b n log(n+1)2. Let s work out the first few values of the sequence. The function log refers to the common logarithm, ie. log 10. For example, b 2 = b log(n+1)2 = log(4) = = = 14 and b 3 is b 3 = b log(2+1)2 = log(9) = = = = 149. (a) Using the definition and the fact that b 3 = 149, show that b 4 = (b) How many digits does b 9 have? (c) How many digits does b 200 have? (d) Find the smallest n such that b n has at least 2000 digits. (e) Find the first four multiples of 9 among the b n. (f) Find the remainder when b 200 is divided by The third sequence of interest is c n. The first few members are c 1 = 1, c 2 = 13, and c 3 = 135. This sequence is obtained by appending to each c n the decimal representation of the n + 1 st odd positive integer. Thus, for example, c 12 =

4 (a) Use the ideas in the first two problems to define the sequence without using the word append. IE, multiply the number by enough to create some zeros at the right end and then add to this an integer. (b) How many digits does each of the following numbers have? i. c 10 ii. c 100 iii. c 1000 (c) How many times does the digit 1 appear in each of the numbers below? i. c 10 ii. c 100 iii. c 1000 (d) Which of the following numbers are multiples of 9? i. c 9 ii. c 99 iii. c Now for the two numbers. Define another sequence t n as follows: t n = a n /10 1+ log an. Thus t 1 = 0.1, t 2 = 0.12 and t 3 = 123/10 1+ log 123 = 123/1000 = Let a = lim n t n. In other words, a = Prove that a is irrational by showing that there is no repeating block of digits. 16. The other number is defined in the same way, except that we use the sequence b n. Thus, b = This number is also irrational. 17. What is the mean of the list L of nine numbers? L = (9, 99, 999, 9999,..., ) 18. Consider the set of single digit prime numbers, S = {2, 3, 5, 7}. For each nonempty subset D of S, let P (D) denote the product of the members of D. There are 15 such nonempty subsets, A 1, A 2, A 3,..., A 15. What is the sum of the 15 numbers P (A 1 ) + P (A 2 ) + + P (A 15 )? 19. What is the smallest n such that 1/n has a decimal that repeats in blocks of 5. 4

5 20. Let f(n) = n!/(n 5)!. What is the smallest n for which f(n) has a units digit of 5? 21. Choose any positive integer as the starting value, and choose a different positive integer as the ending value. Then, perform any of the following moves on the starting number (a) Add 1. (b) Subtract 1. (c) Multiply by 10. (d) Divide by 10. Continue to perform moves until you ve reached the ending number. For example, you can get from 8 to 71 with the following sequence of moves: Subtract 1: 8 1 = 7. Multiply by 10: 7 10 = 70. Add 1: 70+1 = 71. Find the fewest moves required to get from 23 to 360; from 11 to 85; from 189 to 12. Notice that we have not said whether we allow non-integers during the process. Try this puzzle first by allowing only integers during the process. Then change the rules to allow any numbers during the process. 22. This time we get to add 1, divide by 2, and divide by 3. We want to get from 19 to 1 in as few steps as possible. First argue that we need only consider paths that include only integers. Then find the shortest path. 23. A positive integer equals 11 times the sum of its digits. What is the units digit of this number? 24. It is surprising that each of the following infinite decimals is actually a repeating decimal. Find the pair of integers whose ratio is the given decimal. (a) (b) Find a five digit positive integer in which all the digits are different and are in increasing order from left to right. The number is a multiple of four and its square is a ten-digit number. 26. Is it possible to write the numbers 1, 2,..., 10 in a list so that each pair of adjacent numbers differ by either 3 or 5? (As an example, the list: 1, 4, 7, 10, 5, 8, 3, 6, 9, 2 would not suffice since 9 and 2 differ by 7.) 5

6 27. Consider the equation 10 + N ( N 10 ) log 10 N = 3N N 2. Show that N = is a solution and find all the other positive integer solutions. 28. The integers from 1 to 999 are written on the board. What is the smallest number of these integers that can be wiped off so that the product of the remaining integers ends in 8? 29. Find the sum of all 5-digit numbers that use only the digits 1, 2, 8, 9 when repetition is allowed. Also, what is the sum of all four-digit number that can be built if repetition is not allowed. 30. For positive integer n let a n be the integer consisting of n digits of 9 followed by the digits 488. For example, a 3 = 999, 488 and a 7 = 9, 999, 999, 488. Find the value of n so that a n is divisible by the highest power of Find the least three-digit number that is equal to the sum of its digits plus twice the product of its digits. 32. Define the function f by f(x) = x/ x/10. Compute 99 k=10 f(k). 33. The digits of S = are written from left to right followed by the digits of T = How many digits are written altogether? We can build some notation to simplify the solution and to help us think about the problem. Let x y denote the concatenation of integers x and y. Thus = = Let N = ( ). When N is written in decimal (base 10) notation, what is the sum of the digits of N? 35. The number 2 29 contains nine digits, all of them distinct. Which one is missing? 6

7 36. A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. For example, after traveling one mile the odometer changed from to If the odometer now reads , how many miles has the car actually traveled? 37. You re allowed to perform two operations with numbers. You can either add 1 to the number you have or you can double it. Starting with 0, what is the fewest operations you can perform to get 100? 38. You re allowed to perform three operations with numbers. You can either add 1 or 2 to the number you have or you can triple it. Starting with 0, what is the fewest operations you can perform to get 100? 39. A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. For example, after traveling one mile the odometer changed from to If the odometer now reads , how many miles has the car actually traveled? 40. Let a, b, c, d, e, f, g, and h be distinct elements in the set { 7, 5, 3, 2, 2, 4, 6, 13}. What is the minimum possible value of (a + b + c + d) 2 + (e + f + g + h) 2? 41. place value amc10b 2005 Let x and y be two-digit integers such that y is obtained by reversing the digits of x. The integers x and y satisfy x 2 y 2 = m 2 for some positive integer m. What is x + y + m? 42. Find all three digit numbers abc such that abc 2 ends in the three digits abc. 7