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Posers: Innocent and Sophisticated
The number of hairs on a human head, a castaway on a desert island trying to conserve his supply of water, a stubborn watch that stops for fifteen minutes every hour — these are some of the fanciful settings for the problems in this opening section. A variety of mathematical ideas are encountered in the problems, with the Pigeon Hole Principle and the mathematics of uniform motion receiving the greatest share of attention.
1-1 Suppose there are 6 pairs of blue socks all alike, and 6 pairs of black socks all alike, scrambled in a drawer. How many socks must be drawn out, all at once (in the dark), to be certain of getting a matching pair?
Challenge 1
Suppose the drawer contains 3 black pairs of socks, 7 green pairs, and 4 blue pairs, scrambled. How many socks must be drawn out, all at once (in the dark), to be certain of getting a matching pair?
Challenge 2
Suppose there are 6 different pairs of cuff links scrambled in a box. How many links must be drawn out, all at once (in the dark), to be certain of getting a matching pair?
1-2 Find five positive whole numbers a, b, c, d, e such that there is no subset with a sum divisible by 5.
1-3 A multiple dwelling has 50 letter boxes. If 101 pieces of mail are correctly delivered and boxed, show that there is at least one letter box with 3 or more pieces of mail.
Challenge 1
What conclusion follows if there are 102 pieces of mail?
Challenge 2
What conclusion follows if there are 150 pieces of mail?
Challenge 3
What conclusion follows if there are 151 pieces of mail?
Challenge 4
If no human being has more than 300,000 hairs on his head, and the population of Megalopolis is 8,000,000 persons, what is the least value of n in the assertion that there are n persons in Megalopolis with the same number of hairs on their heads? (Assume that all people have at least one hair on their head.)
1-4 Assume that at least one of a1 and b1 has property P, and at least one of a2 and b2 has property P, and at least one of a3 and b3 has property P. Prove that at least two of a1, a2, a3, or at least two of b1, b2, b3 have property P.
Challenge 1
Add to the information in Problem 1-4 that at least one of a4 and b4 has property P, and that at least one of a5 and b5 has property P. Prove that at least 3 of the a's, or at least 3 of the b's have property P.
Challenge 2
Assume that property Q is possessed by at least one of a1, b1, c1, by at least one of a2, b2, c2, ..., by at least one of a10, b10, c10. Find the largest value of k in the assertion that at least k of the a's, or at least k of the b's, or at least k of the c's have property Q.
Challenge 3
Assume that property R is possessed by at least two of a1, b1, c1, by at least two of a2, b2, c2, ..., by at least two of a5, b5, c5. Find the largest value of m for which it can be said that at least m of the a's, or of the b's, or of the c's have property R.
1-5 An airplane flies round trip a distance of L miles each way. The velocity with head wind is 160 m.p.h., while the velocity with tail wind is 240 m.p.h. What is the average speed for the round trip?
1-6 Assume that the trains between New York and Washington leave each city every hour on the hour. On its run from Washington to New York, a train will meet n trains going in the opposite direction. If the one-way trip in either direction requires four hours exactly, what is the value of n?
Challenge
Change "four hours exactly" to "three and one-half hours exactly" and solve the problem.
1-7 A freight train one mile long is traveling at a steady speed of 20 miles per hour. It enters a tunnel one mile long at 1 P.M. At what time does the rear of the train emerge from the tunnel?
1-8 A watch is stopped for 15 minutes every hour on the hour. How many actual hours elapse during the interval the watch shows 12 noon to 12 midnight?
Challenge 1
A watch is stopped for 15 minutes every hour on the hour. According to this watch, how many hours elapse between 12 noon and 12 midnight (actual time)?
Challenge 2
Between 12 noon and 12 midnight, a watch is stopped for 1 minute at the end of the first full hour, for 2 minutes at the end of the second full hour, for 3 minutes at the end of the third full hour, and so forth for the remaining full hours. What is the true time when this watch shows 12 midnight?
1-9 The last three digits of a number N are x25. For how many values of x can N be the square of an integer?
1-10 A man born in the eighteenth century was x years old in the year x2. How old was he in 1776? (Make no correction for calendric changes.)
Challenge 1
Is there a corresponding puzzle for the nineteenth century? If so, find the man's age in 1876.
Challenge 2
Show that there is no corresponding puzzle for the twentieth century.
1-11 To conserve the contents of a 16 oz. bottle of tonic, a castaway adopts the following procedure. On the first day he drinks 1 oz. of tonic and then refills the bottle with water; on the second day he drinks 2 oz. of the mixture and then refills the bottle with water; on the third day he drinks 3 oz. of the mixture and again refills the bottle with water. The procedure is continued for succeeding days until the bottle is empty. How many ounces of water does he thus drink?
Challenge
Assume that the castaway drinks only 1/2 oz. the first day, 1 oz. the second day, 1[1/2] oz. the third day, and so forth. Find the total consumption of water.
1-12 Which yields a larger amount with the same starting salary: Plan I, with four annual increases of $100 each, or Plan II, with two biennial increases of $200 each?
Challenge
How does the result change if the increase under Plan II is $250?
1-13 Assuming that in a group of n people any acquaintances are mutual, prove that there are two persons with the same number of acquaintances.
1-14 The smallest of n consecutive integers is j. Represent in terms of j (a) the largest integer L (b) the middle integer M.
Challenge 1
Let j be the largest of n consecutive integers. Represent in terms of j (a) the smallest integer S (b) the middle integer M.
Challenge 2
Let j be the smallest of n consecutive even integers. Represent in terms of j (a) the largest integer L (b) the middle integer M.
Challenge 3
Let j be the smallest of n consecutive odd integers. Represent in terms of j (a) the largest integer L (b) the middle integer M.
Challenge 4
If n is a multiple of 4, find the integer in position 3n/4 for the original problem and each of Challenges 1, 2, and 3.
1-15 We define the symbol |x| to mean the value x if x ≥ 0, and the value -x if x < 0. Express |(x - y)| in terms of max(x, y) and min(x, y) where max(x, y) means x if x > y, and y if x < y, and min(x, y) means x if x < y, and y if x > y.
Challenge 1
Does the result cover the case when x = y?
Challenge 2
Prove that max(x, y) = x + y/2 + [absolute value of (x - y)]/2, and find the corresponding expression for min(x, j).
1-16 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Express:
(a) x in terms of x+ and x-
(b) |x| in terms of x+ and x-
(c) x+ in terms of |x| and x
(d) x- in terms of |x| and x.
(See problem 1-15 for the meaning of |x|.)
1-17 We define the symbol [x] to mean the greatest integer which is not greater than x itself. Find the value of [y] + [1 - y].
Challenge 1
Find the value of (a) [y] - [1 - y] (b) [1 - y] - [y].
Challenge 2
Evaluate F = [x]/x when (a) x is an integer (x ≠ 0) (b) x is a positive non-integer (c) x is a negative non-integer.
Challenge 3
Evaluate D = [x2] - [x]2 when (a) 0 < x < 1 (b) 1 < x < 2 (c) 2 < x < 3.
Challenge 4
Find an x satisfying the equation [x]x = 11.
Challenge 5
Let (x) = x - [x]; express (x + y) in terms of (x) and (y).
1-18 At what time after 4:00 will the minute hand overtake the hour hand?
Challenge 1
At what time after 7:30 will the hands of a clock be perpendicular?
Challenge 2
Between 3:00 and 4:00 Noreen looked at her watch and noticed that the minute hand was between 5 and 6. Later, Noreen looked again and noticed that the hour hand and the minute hand had exchanged places. What time was it in the second case?
Challenge 3
The hands of Ernie's clock overlap exactly every 65 minutes. If, according to Ernie's clock, he begins working at 9 A.M. and finishes at 5 P.M., how long does Ernie work, according to an accurate clock?
CHAPTER 2Arithmetic: Mean and Otherwise
"Mathematics is the queen of the sciences, and arithmetic is her crown," said the great mathematician Carl Friedrich Gauss. School arithmetic eventually grows up and turns into the branch of mathematics called number theory, which has fascinated mathematicians and amateurs alike through the ages. In this section, you will find problems from number theory involving such topics as means, factorization, primes, divisibility, partitions, and remainders.
2-1 The arithmetic mean (A.M.), or ordinary average, of a set of 50 numbers is 32. The A.M. of a second set of 70 numbers is 53. Find the A.M. of the numbers in the sets combined.
Challenge 1
Change the A.M. of the second set to - 53, and solve.
Challenge 2
Change the number of elements in each set to 1, and solve.
Challenge 3
Find the point-average of a student with A in mathematics, A in physics, B in chemistry, B in English, and C in history — using the scale: A, 5 points; B, 4 points; C, 3 points; D, 1 point — when (a) the credits for the courses are equal (b) the credits for the courses are mathematics, 4; physics, 4; chemistry, 3; English, 3; and history, 3.
Challenge 4
(a) Given n numbers each equal to , and two numbers each equal to 1; find their A.M. (b) Given n numbers each equal to 1 + 1n, and one number 1; find their A.M. Which of (a) and (b) is larger?
Challenge 5
Given n numbers each equal to 1 - 1/n, and one number 2; find their A.M.
Challenge 6
In order to find the A.M. of 8 numbers a1, a2, ..., a8, Carl takes one-half of [1/4]s1 + [1/4]s2 where s1 = a1 + a2 + a3 + a4, and s2 = a5 + a6 + a7 +a8; and Caroline takes one-half of [1/4]s3 + [1/4]s4 where s3 = a1 + a3 + a5 + a7, and s4 = a2 +a4 +a6 +a8. Explain why both obtain the correct A.M.
Challenge 7
Estimate the approximate A.M. of the set {61, 62, 63, 65, 68, 73, 81, 94}.
2-2 Express the difference of the squares of two consecutive even integers in terms of their arithmetic mean (A.M.).
Challenge
How does the result change if two consecutive odd integers are used?
2-3 It is a fundamental theorem in arithmetic that a natural number can be factored into prime factors in only one way — if the order in which the factors are written is ignored. This is known as the Unique Factorization Theorem. For example, 12 is uniquely factored into the primes 2, 2, 3. Consider the set S1 = {4, 7, 10, ..., 3k + 1, ...}, in which k = 1, 2, ..., n,... Does S1 have unique factorization?
Challenge
Is factorization unique in S2 = {3, 4, 5, ..., k ...}?
2-4 What is the smallest positive value of n for which n2 + n + 41 is not a prime number?
Challenge
Examine the expression n2 - n + 41 for primes.
2-5 Given the positive integers a, b, c, d with a/b < c/d < 1; arrange in order of increasing magnitude the five quantities: b/a, d/c, bd/ac, b + d/ a + c, 1.
2-6 It can be proved (see Appendix I) that, for any natural number n, the terminal digit of n5 is the same as that of n itself; that is, n5TDn, where the symbol TD means "has the same terminal digit." For example, 45TD4. Find the terminal digit of (a) 212 (b) 230 (c) 7 7 (d) 810 (e) 810 · 711
Challenge
Find the terminating digit of (a) (5/8)5 (b) (4/7)5.
2-7 If N = 1 · 2 · 3 ··· 100 (more conveniently written 100!), find the number of terminating zeros when the multiplications are carried out.
Challenge
Find the number of terminating zeros in D = 36! - 24!
2-8 Find the maximum value of x such that 2x divides 21!
Challenge 1
Find the highest power of 3 in 21!
Challenge 2
Find the highest power of 2 in 21! excluding factors also divisible by 3.
2-9 The number 1234 is not divisible by 11, but the number 1243, obtained by rearranging the digits, is divisible by 11. Find all the rearrangements that are divisible by 11.
Challenge
Solve the problem for 12034.
2-10 Let k be the number of positive integers that leave a remainder of 24 when divided into 4049. Find k.
Challenge 1
Find the largest integer that divides 364, 414, and 539 with the same remainder in each case.
Challenge 2
A somewhat harder problem is this: find the largest integer that divides 364, 414, and 541 with remainders R1, R2, and R3, respectively, such that R2 = R1 + 1. and R3 = R2 + 1.
Challenge 3
A committee of three students, A, B, and C, meets and agrees that A report back every 10 days, B, every 12 days, and C, every 15 days. Find the least number of days before C again meets both A and B.
2-11 List all the possible remainders when an even integer square is divided by 8.
Challenge
List all the possible remainders when an odd integer square is divided by 8.
2-12 Which is larger: the number of partitions of the integer N = k* 102 into 2k + 1 positive even integers, or the number of partitions of N into 2k + 1 positive odd integers, where k = 1, 2, 3, ...? To partition a positive integer is to represent the integer as a sum of positive integers.
2-13 Given the three-digit number N = a1 a2 a3, written in base 10, find the least absolute values of m1, m2, m3 such that N is divisible by 7 if m1a1 + m2a2 + 22 m3a3 is divisible by 7.
Challenge 1
Solve the problem for the six-digit number N = a1a2a3a4a5a6.
NOTE: Only |m1|, |m2|, and |m3| are needed.
Challenge 2
Solve the problem for the four-digit number N = a1a2a3a4.
2-14 When x3 + a is divided by x + 2, the remainder is known to be -15. Find the numerical value of a.
Excerpted from Challenging Problems in Algebra by ALFRED S. POSAMENTIER, CHARLES T. SALKIND. Copyright © 1988 Alfred S. Posamentier. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
The non-routine algebra problems in this text provide a stimulating intellectual workout. By non-routine, I mean that the problems require insight and, in some cases, ingenuity to solve. Rather than teaching you a skill and asking you to practice it, the authors assume that you have already developed those skills and ask you to apply them to unfamiliar and difficult problems. The problems draw upon topics taught in elementary, intermediate, and advanced algebra classes. Those topics include equations and inequalities; systems of linear equations; arithmetic, geometric, and harmonic means; relations and functions; maxima and minima; the relationship between algebra and geometry; sequences and series; combinatorics and probability; number theory; and Diophantine equations.
Answers to all the problems in which a numerical answer or an algebraic expression is sought are given in an answer key, which gives you a chance to check your answer before reading the authors' solutions. However, not all those answers are correct.
Solutions to all of the principal problems are given in a solution key, which is more reliable than the answer key. However, no solution is given to some of the problems that are variations on or extensions to the principal problems. While the authors label these variations and extensions "Challenges," they are generally no more challenging than the principal problems. The solution key is worth reading even if you have solved a problem correctly. The authors often solve not only the problem at hand but show you solve an entire class of related problems. Reading the solutions is also useful since the techniques developed there can sometimes be applied to subsequent problems in the text.
Reading the appendices before commencing work on the problems is advisable since the relationships and techniques discussed in the appendices are useful in solving the problems. The appendices address terminating digits; the remainder and factor theorems; maximum product, minimum sum problems; arithmetic, geometric, and harmonic means; divisibility tests; the binomial theorem; some useful algebraic relationships; and how to write a proof by mathematical induction.
Working through this text will enhance your problem-solving skills and extend your knowledge of algebra. The level of difficulty of the problems is similar to those in the American Mathematics Competition (AMC), which is not surprising since Charles T. Salkind was the editor of the American High School Mathematics Examination (AHSME), as the AMC was then known, from its inception in 1950 until his death in 1968. Unlike those problems, these problems are not multiple choice. While in many problems a numerical answer or an algebraic expression is sought, these problems also include proofs and investigations of algebraic relationships. While some editing errors detract from the quality of this text, the quality of the problems makes working through it worthwhile.
Overview
Designed for high school students and teachers with an interest in mathematical problem-solving, this stimulating collection includes more than 300 problems that are "off the beaten path" — i.e., problems that give a new twist to familiar topics that introduce unfamiliar topics. With few exceptions, their solution requires little more than some knowledge of elementary algebra, though a dash of ingenuity may help.
Readers will find here thought-provoking posers involving ...