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#### Challenging Problems in Algebra

**By ALFRED S. POSAMENTIER, CHARLES T. SALKIND**

**Dover Publications, Inc.**

**Copyright © 1988 Alfred S. Posamentier**

All rights reserved.

ISBN: 978-0-486-13154-2

All rights reserved.

ISBN: 978-0-486-13154-2

CHAPTER 1

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 *a*1 and *b*1 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 *a*4 and *b*4 has property *P*, and that at least one of *a*5 and *b*5 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 *x*25. 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 *x*2. 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 3*n*/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 2**Arithmetic: 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 + 1*n*, 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, 4*5TD*4. 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 2*k* + 1 positive even integers, or the number of partitions of *N* into 2*k* + 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 *x*3 + *a* is divided by *x* + 2, the remainder is known to be -15. Find the numerical value of *a*.

*(Continues...)*

Excerpted fromChallenging Problems in AlgebrabyALFRED S. POSAMENTIER, CHARLES T. SALKIND. Copyright © 1988 Alfred S. Posamentier. Excerpted by permission of Dover Publications, Inc..

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