Classwork ProblemsDo you understand the meaning of the words in the problem?

1. How much dirt is there in a hole which is 5 feet long, 2 feet deep, and 3 feet wide?

2. Can a man marry his widow’s sister?

3. Which word, if read right is wrong, but if read wrong is right?

4. From the bottom of a 30 foot well, a worm climbs up 4 feet each day, but by the end of theday slips back 3 feet. On which day can we be certain that the worm escapes from the well?(We will say that in order to escape he must be at least 1/4 inch out of the well not just atthe top.)

5. If a ↑ b means ab and a ↓ b means b√

a, what is the value of [(2 ↑ 6) ↓ 3] ↑ 2?

6. If 3! · 5! · 7! = n! find n.

What simple facts and basic principles can you bring to bear on the problem?

7. If 333 + 333 + 333 = 3x, find x.

8. What is the real value of x which 3

√

x√

x = 4?

9. The sum of two prime numbers is 999. What is their product?

10. Two congruent circles are externally tangent at point P . (See Figure 1.) A segment is drawnfrom the center of one circle to a point on the other circle where it is tangent. The length ofthis segment is 12. What is the length of the radius of one of the circles?

P

Figure 1: The two congruent circles in Classwork Problem 10.

11. What is the degree measure of the least positive angle x for which log2(cos x) = −12?

12. If (logx 2x)(log10 x) = 3, find x.

13. In a right triangle ABC, what is the numerical value of sin2 A + sin2 B + sin2 C?

14. What is the simplified numerical value of

(sin 10)(cos 10)(tan 10)(cot 10)(sec 10)(csc 10)

(sin 20)(cos 20)(tan 20)(cot 20)(sec 20)(csc 20)?

Angles here are in degrees.

15. What are all values of x for which sin x >√

1 − sin2 x?

1

What simple observations can you make?

16. A nonstop train leaves Moscow for Leningrad, traveling at a constant rate of 60 miles perhour. Another nonstop train leaves Leningrad for Moscow, at a constant rate of 40 miles perhour both traveling on the same straight track. How far apart are the fronts of the trains inmiles 1 hour before they meet?

17. What is the difference between the sum of the first 20 positive even integers and the sum ofthe first 20 positive odd integers.

18. Find all values of x which satisfy x2 − 8 ≤ 2x and x2 − 2x ≥ 8.

19. If

(

1 − 1

3

)(

1 − 1

4

)(

1 − 1

5

)

· · ·(

1 − 1

126

)

=1

n, then find n.

20. Let 10101 − 1 be written as an integer in standard form. Find the sum of the digits of thisinteger.

21. (The numbers game) Here are the rules of the game.

(a) Only two players play at a time.

(b) Player A mentions any positive integer that is not less than 2 nor greater than 10. PlayerB mentions any number that is not less than 2 nor greater than 10, and that is addedto Player A’s number. This process continues until 100 is reached, each player addingto the growing total. Whoever is first to reach 100 wins. Let’s play! What patterns doyou notice? What is a winning strategy?

Will drawing a picture help?

22. Two congruent circles are externally tangent at P . The line joining their centers has length10. A line segment is drawn tangent to the two circles such that it does not pass through Pand it does not go beyond the points where it is tangent. How long is this segment?

23. At a certain hospital there are 100 patients, all of whom have at least one of the followingailments: a cold, the flu, or an ear ache. Thirty-eight have a cold, forty have the flu, andsixty-five have ear aches. If seventeen have both colds and the flu, ten have colds and earaches, twenty-three have flu and ear aches, and seven have all three, how many people haveonly an ear ache? How many have the flu, but not a cold?

24. A circle is inscribed in a quadrilateral, ABCD, with sides AB = 5, BC = 6, CD = 7. Findthe length of DA.

Can you bring some algebra to bear on the problem?

25. Nine congruent rectangles are placed as shown in Figure 2 to form a large rectangle whosearea is 180 square inches. What is the perimeter of the large rectangle?

26. My father is 24 years older than I am. In two years, he will be three times as old as I amthen. How old is each of us, now?

27. Mr. Smith traveled from Detroit to Chicago, via a certain route, and returned via the exactsame route, so his distance traveled in each direction was the same. On the trip to Chicago,he averaged 30 mph. At what rate must he make the return trip, if his goal is to average 60mph for the entire trip?

2

Figure 2: The nine congruent rectangles in Classwork Problem 25.

28. Serial number trick. (Attributed to Royal Heath.)

Before getting bogged down in heavy computations, can you observe something that

might make the problem simpler?

29. In Figure 3, the rectangle is inscribed in the quarter circle. Find the length of the rectangle’sdiagonal AB.

4’6’ A

B

Figure 3: The rectangle inscribed in the quarter circle of Classwork Problem 29.

30. When 3x3 − 8x2 + 7 is written in the form a(x− 2)3 + b(x− 2)2 + c(x− 2) + d, determine thenumerical value of the sum a + b + c + d.

31. What is the volume of a rectangular solid whose faces have areas 6, 8, and 12?

Can you introduce something new that might make the problem easier or reduce it to

a simpler one?

32. For which values of x is the following equation true?

(

2x + 3

3x + 2

)2

+

(

2x + 3

3x + 2

)

= 6.

33. What are all values of x for which x2 − cos x + 1 = 0?

34. A cube is illustrated in Figure 4; what is the measure of angle ABC?

35. If sin θ = 2cos θ, then what is the numerical value of cos2 θ?

36. Express the following product as a single fraction:(

1 +1

3

)(

1 +1

9

)(

1 +1

81

)

· · ·(

1 +1

3(2n)

)

3

C

A

B

Figure 4: The cube in Classwork Problem 34.

37. In an isosceles triangle whose base is 30, the altitude to the base is 20. What is the length ofthe altitude drawn to one of the legs?

Can you take a smaller problem of a similar nature or do you see any patterns that

might give you the answer?

38. When the first 1995 odd primes are multiplied, what is the units digit of the result?

39. What is the final digit in

(

· · ·(

(

(77)7)7

)7

· · ·)7

, where the 7th power is taken 100 times?

40. What is the value of i256 + i257 + i258 + i259? (Here, i =√−1).)

41. The lengths of the sides of a rectangular parallelepiped are 3, 4, and 12. Find the length ofthe diagonal of the parallelepiped.

42. Suppose that the counting numbers are arranged in columns as shown below:

A B C D E F G1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19

In which column will we find the number 1001?

43. Five hundred students arrived for the first day of school and found 500 numbered lockers,all of which were open. The first student switched every locker (switching a locker in thisproblem means changing it from closed to open, or from open to closed). The second studentswitched every other locker (lockers 2, 4, 6, . . .). The third student switched every third

locker, and so on, so that the nth student switched every nth locker (lockers n, 2n, 3n, . . .).

If this pattern continues up to and including the 500th student, what lockers will remain openat the end of this process?

If a problem seems to have no solution, try thinking outside the box.

44. A large basket has 5 apples. How do you divide the apples among 5 girls, so that each getsan apple, but one apple remains in the basket.

4

45. In Figure 5, you see 12 sticks, forming the incorrect equation, 6−4 = 9 using Roman numerals.Move only one stick to make the statement a true statement.

Figure 5: Figure for Classwork Problem 45

Some problems are easiest solved by trial and error.

46. There are two different ways to put the digits 1, 2, 3, 4, and 5 into the blank parentheses,one digit per parentheses, so that

( ) × ( )

( )= ( ) + ( )

is a true statement. What are these two ways? Note: Changes in order only are not considereddifferent.

47. What is the least positive integer n > 1 for which the expression√

1 + 2 + 3 + · · · + n sim-plifies to an integer?

48. Can one find three different single digits such that any two of them, written in any order,serve as the digits of a two digit prime? If so, find them; if not, explain why not.

Sometimes just listing the cases is the easiest way to go

49. If every side of a nondegenerate triangle has a different integral length, what is the smallestpossible length of the shortest side?

50. “Give me change for a dollar please,” said the customer.

“I’m sorry.” said Mrs. Jones, the cashier, “but I can’t do that with the coins I have.”

“Can you change a half dollar then?”

Mrs. Jones shook her head. In fact, she said she couldn’t make change for a quarter, dime,or nickel!

“Do you have any coins at all?” asked the customer.

“Oh yes,” said Mrs. Jones. “I have $1.15 in coins.”

Exactly what coins were in the cash register? (Assume there are no dollar coins.)

Do we have enough information to solve the problem?

51. If 3 cats can catch 3 rats in 3 minutes, exactly how many cats will catch 100 rats in 100minutes?

52. An island was inhabited by two tribes. Members of one tribe always tell the truth, membersof the other tribe always lie.

A missionary met two of these natives, one tall, and the other short. “Are you a truth teller?”he asked the taller one.

“Oopf,” the tall native answered.

5

The missionary recognized this native word as either meaning yes or no, but he couldn’t recallwhich. The short native spoke English, so the missionary asked him what his companion hadsaid.

“He say, ’yes’,” replied the short native “but him big liar!”

What tribe did each native belong to?

53. A man walked a total of 5 hours. He walked along a level road, then up a hill, then heimmediately turned around losing no time and walked back to his starting point, along thesame route. He walks at a constant rate of 4 miles per hour on level ground, 3 miles per houruphill, and 6 miles per hour down hill. Can we find the distance he traveled?

54. A cylindrical hole, 6 inches long, has been drilled through the center of a sphere. (Parts abovethe cylinder in the picture are drilled out also as a result.) Can we find the volume of theremaining sphere? (See Figure 6.)

Figure 6: A cylindrical hole drilled in a sphere. (See Classwork Problem 54.)

Sometimes it pays to try to simplify something before proceeding.

55. What are both values of x which satisfy

(

1

25

)x

(125)x2

= (125)x(

1

25

)

56. For how many ordered triples of unequal positive integers, (x, y, z) does the expression

x

(x − y)(x − z)+

y

(y − x)(y − z)+

z

(z − x)(z − y)

take on positive values?

Are you assuming something that you shouldn’t be?

57. What is the least integer x for which 12x+1 is an integer?

58. An ordinary die is tossed, and 9 times in a row, a 6 comes up. True or False: The probabilitythat the die will be come up 6 on the next roll is less than 1/6.

59. Arrange a total of 10 trees in exactly 5 rows of 4 trees each.

6

Can you reduce the problem to a simpler problem?

60. Show that for any positive numbers, a, b, c, d, e, and f , that

(a2 + b2)(c2 + d2)(e2 + f2) ≥ 8abcdef.

61. In Figure 7, each circle has radius of 1, and the circles are externally tangent. What is thetotal area of the region between the circles?

Figure 7: The externally tangent circles in Classwork Problem 61.

62. How many zeros does 500! end with?

Avoid common fallacies.

63. “But I don’t have time for school,” Eddie complained to the truant officer. “I sleep 8 hoursa day, which adds up to 122 days a year, there is no school on Saturdays and Sundays, whichamounts to 104 days a year. We have 60 days of summer vacation. I need three hours a dayfor meals—that is more than 45 days a year. And I need at least two hours of recreation,which amounts to 30 days a year. This totals 361 days. This leaves only 4 sick days, andthat doesn’t even take into account the 7 school holidays. So I have no time for school.” Thetruant officer scratched his head and looked baffled. What is wrong here?

64. Here is a proof that the weight, x, of an elephant, equals the weight, y, of a mosquito. Beginby calling the sum of the weights 2w. (So if the sum of the weights were 1001 pounds, thenw = 500.5). So, as we said, x + y = 2w. Now from this equation we can, by moving thingsaround, obtain two more equations, x − 2w = −y and x = −y + 2w. When you multiplythese two equations, you get x2 − 2wx = y2 − 2wy, and then we add w2 to both sides of theequation, to get x2 − 2wx + w2 = y2 − 2wy + w2. This simplifies to (x−w)2 = (y −w)2, andthen taking the square root of both sides, we get x−w = y−w, and finally adding w to bothsides gives x = y. That is, the elephant’s weight is the same as the mosquito’s weight! Whatis wrong here, if anything?

65. Consider the infinite series, 1 − 2 + 4 − 8 + 16 − 32 + · · · = S. Rewrite the series as

S = 1 − (2 − 4 + 8 − 16 + 32 − · · ·= 1 − 2(1 − 2 + 4 − 8 + 16 − 32 + · · ·)= 1 − 2S.

In summary, S = 1 − 2S. So S = 13 . But all the terms are integers. How could this be?

7

66. We give a proof here that all triangles are isosceles. We begin with triangle ABC in Figure 8.

CA

B

D

Figure 8: The isosceles triangle in Classwork Problem 66.

Let D be the intersection of the angle bisector of angle B and the perpendicular bisector ofside AC. Now draw DE perpendicular to AB, DF perpendicular to BC, and draw AD andDC to give you Figure 9.

F

A

B

E

D

C

Figure 9: The isosceles triangle in Classwork Problem 66.

Now 6 DEB = 6 DFB (Both are 90 degrees) and since 6 EBD = 6 FBD (Why?) and BD iscommon to triangles DEB and DFB, triangles DEB and DFB are congruent by angle-angle-side. Hence DE = DF . Since D is on the perpendicular bisector of AC, we have AD = DC.[Theorem: Any point on the perpendicular bisector of a line segment is equidistant from the

endpoints.] Hence, triangles ADE and CDF are congruent by hypotenuse leg. Thus

AE = FC. (1)

From the congruent triangles DEB and DFB, we have that

BE = BF. (2)

Adding (1) and (2), we get AB = BC. That is, the triangle is isosceles. Since we startedwith an arbitrary triangle and showed it was isosceles, all triangles are isosceles. What, ifanything, is wrong?

8

Logic.

67. Each of the four cards shown in Figure 10 has a letter on one side and a digit on the other.

3A F 2

Figure 10: The four cards in Classwork Problem 67.

Identify every card you MUST turn over to determine if the following sentence is true or falsefor these four cards:

“Whenever there is a vowel on one side of a card,there is an even number on the other side of that card.”

68. Three logicians, A, B, and C, are wearing hats which they know are either black or white,but not all are white. A can see the hats of B and C; B can see the hats of A and C; andC is blind. Each is asked in turn if they know the color of their own hat. Their answers, inturn are A: “No.” B: “No” and C: “Yes”. What color is C’s hat and how does she know?

69. In a certain village we have a Mr. Carpenter, a Mr. Smith and a Mr. Machinist. One is acarpenter, one is a smith and one is a machinist. None follows the vocation of his name. IfMr. Machinist is not a carpenter, what profession does each person have? Now, suppose thateach person is assisted in his work by the son of another, and that none of the sons, followsthe trade of his name. Then what is the occupation of young Mr. Smith?

70. In certain village, each person either lies all the time, or tells the truth all the time. The liarsare from one race, and the truthtellers from another. A, B, and C are three villagers wereasked the following questions and they gave the following answers.

Q: Mr. A, is Mr. B a truthteller?A: Yes.

Q. Mr. B, do Mr. A and Mr. C belong to the same race?A: No.

Q: Mr. C, what do you say about Mr. B? Is he is a truthteller?A: Yes.

Tell which race each villager belonged to.

71. Here is a list of words: HOE OAR PAD TOE VAT

In this problem, we have the following facts:

(a) Each of three logicians was told one letter of a certain word, so that each logician knewonly one of the letters, and no two logicians knew the same letter.

(b) The logicians were then told their three letters could be arranged to spell one of thewords in the list above.

(c) When each logician was asked in turn, “Do you know the word the letters spell?” Firstone logician answered “Yes”, then a second logician answered “Yes”, and finally, thethird logician answered “Yes”.

Which word did the letters spell?

9

72. Five students, Adele, Betty, Carol, Doris, and Eileen, answered five questions on an examconsisting of two multiple choice (a, b, or c) questions and three true or false questions; hereis a table of their answers:

I II III IV VAdele a a t t tBetty b b t f tCarol a b t t fDoris b c t t fEllen c a f t t

If no two students got the same number of answers correct, who had the most correct answers?

One of the most difficult things is proving things are correct.

73. Eight chickens are seated at a circular table in such a way that the number of feathers of eachchicken at the table is the average of the number of feather of the chicken immediately to theright and immediately to the left of that chicken. If the first chicken has 500 feathers, willthe eighth chicken have more less feathers than the first, or can you tell?

74. For what positive integral values of n is 1n + 2n + 3n + 4n divisible by 5?

75. What are all real values of x for which 5x + 12x = 13x?

76. Show that x2 − y2 = a3 always has integral solutions for x and y when a is a positive integer.

77. The equation (x+2)4+x4 = 82 has solutions x = 1 and x = −3 as is easily seen by inspection.Prove that there are no other real solutions.

78. Prove by induction that 1 + 2 + 3 + · · · + n = n(n+1)2 .

79. Find a formula for the sum of the first n odd numbers, and then prove it.

80. Using the fact that |x + y| ≤ |x| + |y| , give a proof by induction that

|x1 + x2 + · · · + xn| ≤ |x1| + |x2| + · · · + |xn| .

(This is known as the triangle inequality.)

81. Consider a 2n by 2n square that has a 2n−1 by 2n−1 square removed from its upper rightcorner. Use induction to show that this L-shaped region can be completely covered by non-overlapping small L-shaped tiles in the shape of a 2 by 2 square missing an upper-right corner.[You are allowed to rotate these small tiles to piece them together to cover the large region.]

82. Prove by induction that the number of subsets of a set with n elements is 2n.

Specialized types of problems:

Place value.

83. When a two digit number is subtracted from the number with the digits reversed, the resultis one less than the number. If 3 times the tens digit is added to 4 times the units digit, theresult is the number. Find the number.

84. If 2x3 + 4x2 + 6x + 8 = 2468, and x is a positive real number, what is the value of x3 + 9x2 +8x + 8?

10

85. Ask a fellow student you know to write down on a piece of paper the number of brothers heor she has, (if this number is less than 10) and multiply the result by two. Now add three tothis result and multiply the new result by 5. Add to this the number of sisters this personhas (if this is less than 10). Multiply the total by 10. Finally tell him/her to add the numberof courses he or she is taking this semester and then tell you the result. You will be able tellyou how many brothers, sisters and courses he has. How do you do it?

Number theory.

86. For which ordered pairs of positive integers, (x, y), is 36 × 5x = 225 × 4y? How do you knowthere is only one solution?

87. Show that the only integral solutions for a and b in the equation a2 − b2 = 16, are (a = ±4,b = 0), and (a = ±5, b = ±3).

88. A grid has 100 rows and 100 columns, numbered from 1 to 100, as shown in Figure 11. In row2 every second box is shaded. In row 3, every third box is shaded etc. The shaded boxes havethe letter s written in them. What is the only column to contain 5 (vertically) consecutiveshaded boxes?

1 2 3 4 5 6 7 8

12345

s s

s s

s s

s

· · ·

...

Figure 11: The grid in Classwork Problem 88.

89. A person buys 3 cents stamps and 6 cent stamps. He pays for them with a $5.00 bill andreceives 75 cents in change? Does he receive the right change?

90. What is the smallest positive integer N for which all the digits in the product 9N are 1’s?

91. Take any number with more than 3 digits and write it down. Scramble the digits, and subtractthe smaller from the larger. Circle any nonzero digit, in the answer, and tell me the sum ofthe remaining digits. I will tell you the number you circled. How do I do it?

92. The number 909 has 1900 different positive integral divisors. How many of these are squaresof integers?

93. Show that if a 6= b, and a and b are positive, thena

b+

b

ais never an integer.

94. Suppose f(x) is a polynomial with integral coefficients. If f(x) = 2 for three different integersa, b, and c, prove that there is no integer x such that f(x) = 3.

11

Problems involving an ± bn and its factorization.

95. What is the sum of the series 1002 − 992 + 982 − 972 + · · ·+ 42 − 32 + 22 − 12, where the signsalternate between squares of consecutive integers?

96. If x2 − y2 = 100 and x + y = 10, find x and y without using the method of substitution.

97. What is the ordered pair of real numbers (x, y) for which 16x − 16y = 192 and 4x − 4y = 8?

98. (a) If x3 + y3 = 400 and x2y + xy2 = 200, find the value of x + y.

(b) Show that if a and b are integers, then the equation a3 + b3 = 1 has only two solutions.What are they?

(c) What are all ordered pairs of integers (x, y) which satisfy

x3 + y3 − 3x2 + 6y2 + 3x + 12y + 6 = 0?

99. For which values of n is 11 × 14n + 1 a prime?

100. What is the simplest form of3

√

7 + 5√

2 +3

√

7 − 5√

2? Is this number rational?

101. For which real values of x ≥ 1 is√

x + 2√

x − 1 +√

x − 2√

x − 1 = 2?

102. Two cubes with integral sides have their combined volumes equal to the combined lengths ofall their edges. What are the dimensions of the cubes?

103. What are all ordered pairs of numbers (x, y) which satisfy

x2 − xy + y2 = 13 and x − xy + y = −5?

104. Prove that if the sides of a triangle satisfy a2 + b2 + c2 = ab + bc + ac, then the triangle isequilateral.

Pigeonhole principle.

105. (a) Suppose there are 13 people in a room. Why is it true that two people have a birthdayin the same month?

(b) Now suppose that there are 30 people in a room. Why must at least two of them havethe same first initial?

(c) Suppose that in (b) there were 60 people. What conclusion could you make?

106. Suppose that there are 6 pairs of blue socks all alike and 6 pairs of black socks all alikescrambled in a drawer. Suppose you wake up to a power outage in the middle of the nightand need to find a matching pair in complete darkness. How many socks will you need totake out of the drawer to be certain to have a matching pair? Answer the same question ifthere are 3 pairs of black socks, 7 pairs of green socks, and 4 pairs of blue socks.

107. Prove that there exist two different powers of 3 whose difference is divisible by 1997.

108. Prove that if I pick any 5 points inside of a 2 × 2 square, at least two of them are within adistance of

√2 from each other.

12

Arithmetic and geometric progressions.

109. In an arithmetic progression, the 25th term is 552 and the 52nd term is 5279. What is the79th term?

110. What is the product of the first 10 terms of a geometric series whose first term is 1 and whose10th term is 2?

111. Consider the geometric series

x + x2 + x3 + · · · =x

1 − x

and the geometric series

1 +1

x+

1

x2+ · · · =

1x

1 − 1x

=x

x − 1.

When we add the two series, we get

· · · + 1

x2+

1

x+ 1 + x + x2 + · · · =

x

1 − x+

x

x − 1=

x

1 − x− x

1 − x= 0.

When we consider any positive value for x in this doubly infinite series, all terms are positive;either the terms to the left of 1 are positive and growing while the terms to the left of 1 arepositive and shrink to zero, or vice versa. What is wrong here?

112. Find the sum of the series 1 + 2x + 3x2 + 4x3 + · · · where |x| < 1

Some problems are done most easily by working backwards.

113. When Mr. Spendmore counted the money in his pocket, the “morning after”, he found asingle crumpled one dollar bill. Ruefully he recalled the hectic evening he had spent on thetown, weaving a trail from one night spot to another.

He tried to remember exactly how much money he had with him when he started the evening.But all he could remember was that he had spent half his money at the top hat, and then ashe left, he tipped the hat check girl one dollar. At the Golden Eagle club, the second clubhe visited, he spent half his remaining money and again tipped the hat check girl $1. Herepeated the same performance at the Glass Slipper and then again at the Pirate Ship beforehe staggered home.

How much did Mr. Spendmore have when he started out?

Probability.

114. Two people, Mrs. A and Mr. B, are about to play a round of 5 games of pool. Mr. B says,“The way I see it, your chances of winning 3 games with me are exactly the same as yourchance of winning four games.” If Mr. B is right, what is Mrs. A’s probability of winning all5 games?

115. True or False: In a room of 40 people, there is almost a 90% chance that two people will havethe same birthday.

116. Six playing cards are lying face down on a table. You have been told that exactly two of themare kings, but you do not know the positions of the kings. You pick two cards at random andturn them face up. Which is the most likely?

(a) There will be at least one king among the two cards.

(b) There will be no kings among the two cards.

13

Supplementary Problems1. The sum of the angles of a triangle is 180 degrees. Figure 12 gives an example of what is

called a pentagram. Is it true that in any pentagram, the sum of angles A, B, C, D, and Eis constant? If so, what is it? If not, show why not.

A

C

B

E D

Figure 12: An example of a pentagram. (See Supplementary Problem 1.)

2. Find the sum of the reciprocals of the divisors of 360. Then find a formula which gives thesum of the reciprocals of the divisors of any number N and show why it is true.

3. Find all solutions to the system of equations:

xyz3 = 24xy3z = 54x3yz = 6

4. Figure 13 shows two parallelograms ABCD and AEFG, where E is a point that is 1/3 ofthe way from B to C. Is the area of AEFG greater than or less than or equal to the area ofABCD? How do you know?

F

A

B C

D

E

G

Figure 13: The two intersecting parallelograms in Supplementary Problem 4.

5. From a point inside an equilateral triangle, perpendiculars are drawn to the sides of thetriangle. Find the sum of the lengths of these perpendiculars.

6. Suppose that a one-mile-wide river runs East-West between two towns, A and B. A and Bare situated as follows. If you walk three miles south from town A, you hit the northern bankof the river at point N ; If you walk five miles north from town B hit the southern bank of theriver at a point S, fifteen miles upstream from N . Find the length of the shortest path fromtown A to town B, ff crossing the riverbanks can only be done at right angles to the banks.

7. A fourth degree polynomial P has integer coefficients and its lead coefficient is equal to 1. IfP (

√2 +

√7) = 0, what is the value of P (1)?

14

8. In triangle ABC, AC = 18, and D is the point on AC for which AD = 5. Perpendicularsdrawn from D to AB and CB have lengths 4 and 5, respectively. What is the area of triangleABC?

9. What are all pairs of real numbers (x, y) for which 17x2 − 10xy + 2y2 − 6x + 2 = 0?

10. On a straight road, an inspecting officer traveled from the rear to the front of an army columnand back, while the column marched forward its own length. If the officer and the columnmaintained steady (but different) speeds, what was the ratio of their speeds, faster to slower?

11. Suppose you have twelve identical coins, one of which is counterfeit and so weighs slightlydifferent than the other eleven. You have at your disposal a two-pan balance so that you candetermine whether two piles of coins weigh the same. How can you determine the counterfeitcoin in only three weighings?

12. The value of

(

7 +√

x +1

5 −√x

)

is rational for only one positive integer x that is not a

square. What is x?

13. In triangle ABC, what is the ordered pair of real numbers (x, y) for which sin A : sinB :sin C = 4 : 5 : 6 and cos A : cos B : cos C = x : y : 2?

15

Homework Problems1. Mary bought a dozen apples and ate all but 4. How many apples were left?

2. What eleven letter English word does everyone pronounce incorrectly?

3. What is the fewest number of elephants you can have in a straight line, if there are twoelephants directly in front of an elephant, two elephants directly in back of an elephant, withan elephant in the middle?

4. We know a young lady in Dundee,whose age has its last digit “three”.The square of the firstis her whole age reversed,so what must this lady’s age be?

5. A girl bought a dog for $10, sold it for $15, bought it back for $20, and finally sold it for $25.Did the girl make or lose money, and if so, how much?

6. “I seem to have overdrawn my account,” said Mr. Green to the bank president, “though Ican’t for the life of me understand how it could have happened. You see, I originally had 100dollars in the bank. Then I made 6 withdrawals. These withdrawals add up to $100. Butaccording to my records, there was only $99 in the bank to draw from.” Mr. Green handedthe bank president a sheet of paper on which was written,

Withdrawals Amount Left on Deposit$50 $50$25 $25$10 $15$8 $7$5 $2$2 $0

Total: $100 $99

“As you see,” said Mr. Green, “I seem to owe the bank $1”. The bank president looked overthe figures, and smiled. “I appreciate your honesty, Mr. Green, but you owe us nothing.”

“Then there is a mistake in the figures?”

“No, your figures are correct.”

Can you explain where the error lies?

7. In terms of x, what is x% of x% of x% of 1,000,000?

8. For which integral values of x is∣

∣x2 − 9∣

∣ a prime number?

9. When 270 is divided by the number x, the quotient is a positive prime, and the remainder is0. What is the smallest possible positive value of x?

10. What is the probability that a point which is interior to a circle of radius of two is furtherthan one unit from the center?

11. A palindrome is a number that reads the same backwards and forwards. For example, both121 and 3443 are palindromes. An American family was driving along a highway at a constantlegal speed, which in America is less than 85 miles per hour. The driver noted that the carsodometer reading was 45954 miles, a palindrome. Two hours later the odometer displayedyet another palindrome! Determine the car’s speed in miles per hour.

16

12. What is the value of n for which n! = 225 · 313 · 56 · 74 · 112 · 132 · 17 · 19 · 23?

13. If a ↑ b means ab, what is the value of x which satisfies

4 ↑ (3 ↑ 2) ÷ (4 ↑ 3) ↑ 2 = 4 ↑ (3 ↑ x)?

14. In simplest form, what is the numerical value of (√

1985)( 3√

1985)( 6√

1985)?

15. For which values of x, if any, is (xx)1986 = 1.

16. If a, b, and c are the smallest positive integers for which 3a = 4b = 5c, calculate the suma + b + c.

17. Solve the equation x2 +1 = y for all pairs (x, y) where x and y are primes. How do you knowyou have all pairs?

18. For how many of the first 100 positive integers x does

(6x2 − 13x + 6)(4x + 3) = (8x2 − 6x − 9)(3x − 2)?

19. Solve the following equation for x and y:

|x + y + 7| + |2x − y + 2| = 0.

20. If sin2(π9 )+sin2(2π

9 )+sin2(3π9 )+sin2(4π

9 ) = 94 , evaluate cos2(π

9 )+cos2(2π9 )+cos2(3π

9 )+cos2(4π9 ).

21. What is the simplified value of (tan 15◦)(tan 30◦)(tan 45◦)(tan 60◦)(tan 75◦)?

22. In degrees, what is the measure of the least positive angle x for which

(2sin2 x)(2cos2 x)(2tan2 x) = 22?

23. A train leaves New York for Fort Lauderdale traveling at 125 miles per hour. An hour later,another train leaves Fort Lauderdale traveling to New York at a rate of 140 miles per hour.When the two trains meet each other, which is closer to New York? (The distance means thedistance from the front of the train to the city.)

24. Consider the sum 1 + 2 + 3 + · · ·+ 101. Is it possible to change some of the plus signs so thatthe sum is zero? How do you know?

25. Prove that if 1 is added to the sum of the squares of 3 consecutive odd numbers, the result isalways divisible by 3.

26. There is only one triple of numbers, (x, y, z) which makes x + y + z = 9, 1x + 1

y + 1z = 1,

and xy + yz + xz = 27. What is it? (Proving that there is only one triple is an interestingexercise. Try it on your own time and show me what you get.)

27. Although four students tried to find the sum of the first 21 positive primes, only one got thecorrect answer. Pat got 709, Lee got 711, Sandy got 712, and Dale got 723. Can you tell whowas correct?

28. What is the probability that the dates, April 4th, June 6th, August 8th, October 10th, andDecember 12th all fall on the same day of the week?

29. What is the positive value of k which makes log 1 + log 9 + log 8 + log 8 = 2 log k? (Here logmeans log10.)?

17

30. Iflog a

log b= 1000, then what is the numerical value of

log(ab )

log b?

31. If b = log3 x, what value of x satisfies logb(log3 x2) = 3?

32. The sum of the squares of the lengths of the three sides of a right triangle is 200. What isthe length of the hypotenuse?

33. Solve the following equation for x:

32√

3x + 3

4= 3

√3x

34. Show that1

log2 N+

1

log3 N+

1

log4 N+ · · · + 1

log100 N=

1

log100! N.

35. What is the only ordered pair of numbers (x, y) which satisfies

xayb =

(

3

4

)a−b

and xbya =

(

3

4

)b−a

, for all a and b?

36. What are the values of k, if any, which make bothk − 1

k + 1and

k + 1

k − 1integers?

37. Two circles with radii 8 and 10 are externally tangent. A line segment is drawn tangent tothe two circles, but not at their common point of tangency. How long is this segment?

38. Consider an isosceles triangle ABC with vertex angle C = 80 degrees. Insert bisectors ofvertex angles A and B so that they meet sides a and b at D and E, respectively. Thesebisectors meet at F . Find the measure of angle AFB.

39. In a convex quadrilateral ABCD, AB = AD = 10, BC = CD = 17, and BD = 16. What isthe area of the quadrilateral, ABCD?

40. The length of a given rectangle is 1 more than twice the width. If the length is increased by1, then the area of the new rectangle is 2 more than that of the original rectangle. Find thedimensions of the original rectangle.

41. In a class of 42 students, the following data were collected about their studying during thepast weekend. Nine had studied on Friday, eighteen had studied on Saturday, thirty hadstudied on Sunday, three had studied on both Friday and Saturday, ten had studied on bothSaturday and Sunday, six had studied on both Friday and Sunday, two had studied on all ofFriday, Saturday and Sunday. Assuming that the data are accurate, answer the following:

(a) How many students studied on Sunday, but on neither Friday nor Saturday?

(b) How many students did all their studying on only one day?

(c) How many students did not study at all over the weekend?

42. The Men of Delta Omicron Gamma (DOG) fraternity house, did a survey on the playinghabits of its members. They found that 18 play basketball, 20 play baseball, and 23 playfootball. Only 3 play all three sports. Nine play baseball and basketball, eight play baseballand football, and ten play basekball and football. The other fourteen members are onlyinvolved in track, tennis, or swimming. How many men are in the DOG house?

18

43. During a vacation it rained on 13 days. But when it rained in the morning, the afternoon wasfine, and every rainy afternoon was preceeded by a fine morning. There were 11 fine morningsand 12 fine afternoons. How long was the vacation?

44. You are stranded in the desert and you come upon a circular pit of diameter 10 feet thatseems to extend infinitely deep. Magically hanging above the exact center of this pit at aheight reachable on your tiptoes is a bottle of water. You look around and see that there aretwo boards of length 8 feet that support your weight. How can you use these boards to geta drink of water without falling to your death?

45. Is it possible to inscribe a circle in a quadrilateral with consecutive sides 13, 14, 15, and 16?Explain.

46. Here is an amusing trick. With your back turned away from your friend, have her write ina column, any two numbers. Then let her add them to obtain a third number, which shewrites below the other two. Then add the third number to the previous number to generatea fourth, and so on until she has 10 numbers in a vertical column. (For example, if she writes2 and 4 as her first two numbers, then the remaining numbers will be 6, 10, 16, 26, 42, 68,110, and 178.) You can then turn around and quickly give her the sum of the numbers in thecolumn as follows: Look at the fourth number from the bottom, multiply it by 11, and thatis the sum. Use algebra to explain this trick. [Hint: Call the first two numbers in the column

a and b.]

47. A boy has as many sisters as brothers, but each sister has only half as many sisters as brothers.How many brothers and sisters are there in the family?

48. Fred came in with a shopping bag. “They didn’t have a large capon,” he said, “So insteadI bought 2 birds weighing a total of 9 pounds. I paid $5.28 for the bigger one and $3.64 forthe other.” The small bird was 8 cents a pound more. How much did each bird weigh?

49. Baby Jake was born on January 1st 2000, while his father big Jake was born on the samedate, but in the year 1980. In 2002, the baby will be two, and the father twenty two, andthus the father will be 11 times as old as the son. In 2004, the father will be 24 and the son4, and the father will be 6 times as old as his son. Finally, in 2005, the son will be 5 and thefather 25. So the father will be 5 times as old as the son.

(a) When will the father be 3 times as old as the son?

(b) When will they be the same age?

50. Doug was yawning when he came down for breakfast at 8 AM. “What time did you comehome last night?” asked Amy, “I didn’t hear you. I went to bed at midnight.” “Not too late,Mom,” Doug said. “I checked with your clock and the minute hand was exactly on a minutemark seventeen minutes ahead of the hour hand.” What time was it when Doug got home?

51. A horse and a mule, both heavily loaded, were walking side by side. When the horse com-plained of its load, the mule cried, “What are you complaining about? If I take one sack fromyour back, and place it on my back, my load will be twice yours, but if you take one sackfrom my back and place it on yours, your load will be equal to mine.” How many sacks wasthe horse carrying? (Assume all sacks have the same weight.)

52. A train traveling at 30 miles per hour reaches a tunnel which is 9 times as long as the train.If the train takes two minutes to completely clear the tunnel, how long is the train? (A mileis 5280 feet.)

19

53. A farmer has hens and rabbits. These animals have 50 heads and 140 feet. How many hensand rabbits has the farmer?

54. Every week Jack purchases 50 cents worth of apples. This week he was pleased to see thatfor the same 50 cents he got 5 apples more. “Hmm,” he said. “I see the price of apples hasgone down by 10 cents per dozen.” What was the new price per dozen?

55. Suppose that it is now noon.

(a) Exactly when will the hour and minute hands of the clock be together again? Expressyour answer accurately to the fraction of a second.

(b) How many times between noon and midnight do the hour hand and the minute hand ofa clock point the same direction?

(c) What theorem from calculus applies to this problem?

56. When 21990 is multiplied by 51991, the product has 1991 digits. What is the sum of thesedigits?

57. In the guard tower, six hundred guards are preparing for battle. Five percent of the guardsare armed with one weapon, while of the remaining ninety-five percent, half are armed withtwo weapons each and the other half have no weapons. How many weapons are there totalin the guard tower? What is the easy way to calculate this total?

58. (This problem will blow your mind!) Imagine a sphere with radius three yards and imagineputting a steel band tightly around the equator. (See picture below.) Now add one yard tothe length of the steel band, and imagine placing the new band around the equator in such away that it is the same distance x off the sphere all the way around. (See Figure 14.)

(a) What is the distance x that the band is from the sphere?

(b) Now imagine a sphere the size of the sun, and let us do the same thing, namely put asteel band tightly around the equator, and then add only one yard. Then place the newband around the sun. True or false? You will be able to slip a baseball under the bandwith no trouble.

x

Figure 14: The figure on the left represents a band placed tightly around the equator. The figureon the right represents a slightly longer band placed around the equator at an equal distance xabove the ground. (See Homework Problem 58.)

59. (Another surprise!) On a hot day, a hypothetical railroad, originally 1 mile in length (5280feet), expands by one foot. Because the ends of the railroad track are anchored, the rail bowsupward along an arc (not a semicircle). (See Figure 15.) Without using a calculator, estimatehow far off the ground the midpoint of the bowed rail is and answer: Will it be a few inches,a few centimeters, or a few feet?

20

?

Figure 15: A mile-long railroad track that expands by a foot. (From Homework Problem 59.)

60. What is the value of x which satisfies

17

85+

19

95+

21

105+

23

115+

25

125+

x

135= 1

61. Express the following as a fraction in simplest form.

3 + 6 + 9 + 12 + · · · + 291 + 294

4 + 8 + 12 + 16 + · · · + 388 + 392

62. What are the only four integer values of x for which xx−2 is an integer?

63. Find the smallest positive number x, in radians, for which tan2 x + sec2 x = 1?

64. On an 8× 15 rectangle, congruent triangles ABC and A′B′C ′ are drawn with correspondingsides parallel, as illustrated in Figure 16. If AB = 7, AC = 8, and BC = 9, then what is thevalue of AA′?

A

C’

B’

B

C

A’

Figure 16: The rectangle and two congruent triangles in Homework Problem 64.

65. A set with 1 element, say {a}, has two subsets: the empty set and {a} itself. (The empty setis considered to be a subset of every set.)

(a) List all the subsets of {a,b}(b) List all the subsets of {a,b,c}(c) List all the subsets of {a,b,c,d}(d) Guess at a formula for the number of subsets of a set with n elements in it.

(e) Do you believe the following? Given a nonempty set A, the number of subsets havingan even number of elements is the same as the number of subsets having an odd numberof elements. Why do you think that way?

21

66. In Figure 17, the two circles are concentric with O as their common center. The radius of thesmaller circle is 1 and the radius of the larger circle is 2. Both AO and BC are perpendicularto OC. What is the measure of angle 6 BAO?

A

O C

B

Figure 17: The two concentric circles in Homework Problem 66.

67. Only four integers between 100 and 1000 are equal to the sum of the cubes of its digits. Threeof these are 370, 153, and 407. What is the fourth?

68. What is the units digit of 1 + 9 + 92 + · · · + 91989?

69. How many integers are strictly between 1996! + 19 and 1996! + 96?

70. What are the final three digits of 5100?

71. Take any positive integer n with fewer than 10 digits. Then perform the following two steps.

(a) Let x be the number of even digits in n and let y be the number of odd digits in n, andlet z = x + y.

(b) Replace n with the three digit number whose hundreds digit is x, (which may be zero),whose tens digit is y, and whose units digit is z.

Repeat steps (a) and (b) 1995 times. What is the final value of n? How do you know?

72. What is the remainder when 661986 is divided by 12?

73. In the set of perpendicular segments shown in Figure Figure 18, AB = 10, and every othersegment is 1 unit shorter than the previous one (so BC = 9, CD = 8, DE = 7, etc.). Whatis the straight line distance from A to K?

I

A

B C

DE

F G

H

KJ

Figure 18: Spiral of perpendicular segments in Homework Problem 73.

22

74. Make a square of nine dots as shown in Figure 19. Draw exactly four straight line segmentsthat visit all nine dots without picking your pencil up off the paper and without retracingany line segment.

• • •

• • •

• • •

Figure 19: A square of nine dots. (See Homework Problem 74.)

75. At one time the Russian postal system was very corrupt. Any letter, package or box, whichwas open, or easily openable, would be opened in the sorting office, and anything inside wouldbe removed, whether or not it had any value. However, since the pickings were so rich, thesorters never bothered to open anything that was locked, even if they suspected it containedvaluables.

Boris, in Moscow, bought a beautiful gem for his girlfriend Natascha, who lived in SaintPetersburg. Neither he, nor Natascha, could travel to the other’s city, so what was he to do?He had a strongbox, with a hasp to which a number of padlocks could be attached. If hebought a padlock, and key, he could put the gem in the box, lock the padlock, and send thebox through the postal system, knowing that it would not be pried open and that it wouldbe delivered to his beloved. But what good would that do? Natascha would not have thekey to open the padlock. Boris couldn’t send the key separately by letter, as it would beopened and the key removed. Nor did he know anyone traveling to St Petersburg to bring itto her. However, Boris phoned Natascha, and between them they hatched a clever plan bywhich they could get the precious stone from Moscow to St. Petersburg in safety, despite thecorrupt postal system. How did they do this? (Note: They cannot use combination locks,nor can they break boxes or locks.)

76. On morning, early at sunrise, a Buddhist monk began to climb a tall mountain. The narrowpath, no more than a foot or two wide, spiraled around the mountain to a glittering templeat the summit.

The monk ascended the path at varying rates of speed, stopping many times along the way torest, and to eat the dried fruit which he carried with him. He reached the temple shortly beforesunset. After several days of fasting and meditation, he began his journey back, walking alongthe same path, starting at the same time he did when he ascended the mountain. He againwalked at variable speeds with many pauses along the way. His average speed descendingwas, of course, greater than his average speed climbing. Show that there is a spot on thepath that the monk will occupy on both trips, at precisely the same time of day. (You neednot give a specific time of day, just show that this occurs at SOME time. Also, assume thatsunrise and sunset occur at the same time on both the journey up and the journey down.)

77. Cross out nine letters from the following in such a way that the remaining letters form asingle word.

NAISNIENLGELTETWEORRSD

78. Place three matches on a table, parallel to one another. Add two more matches to makeeight.

23

79. A prisoner is about to be brought before the executioner. She is ordered to make make a finalstatement. If the statement is true, she will be hanged; if the statement is false she will beburned at the stake. What is the statement that she should make to avoid either punishment?

80. Nearing the end of a tournament, two horsemen are tied. The organizer tells the two horsementhat the horseman whose horse arrives at the tower on the horizon the first will lose thetournament. Upon hearing this, the two horsemen run into the stable, and each ride off on ahorse as fast as possible towards the tower. How can you explain this seemingly contradictorybehavior?

81. You are in a dark room. You have 12 coins on the table of which 5 are heads up, and 7 aretails up. You want to form two piles with the same number of heads, but you can’t see thecoins nor can you tell by feel which are heads up. Can you do it? You are allowed to turncoins over if you wish.

82. Take one end of an unknotted cord in each of your two hands. Show that it is possible tocreate a knot in this cord without ever letting go of either end!

83. A king has taken his worst enemy prisoner. In the public square the following morning, theking will give the prisoner one last chance to be pardoned. He will place two marbles intoan urn—a white marble to represent freedom and a black marble to represent death—thenin front of everyone, the prisoner will choose one marble from the urn to decide his own fate.That night, a spy learns that the king will place two black marbles in the urn and relays thisinformation to the prisoner. With this information, how can the prisoner save himself?

84. Suppose you are competing in an archery tournament where the archer with the largest oddscore wins. You are given six arrows to shoot at the target in Figure 20. Discuss your optimalstrategy.

3 115 11 7 5

Figure 20: Figure for Homework Problem 84.

85. A nonstop train, A, leaves Moscow for Leningrad, traveling at a constant rate of 60 miles perhour. Another nonstop train, B, leaves Leningrad for Moscow, traveling on the same track,at 40 miles per hour. Exactly one hour before they collide, a fly places himself between thetwo trains exactly in front of A. He flies towards B, then when he meets B immediately turns,losing no time and flies towards A. When he reaches A he immediately turns, losing no timeand flies until he reaches B. He keeps going back and forth until the two trains collide, andthen fly becomes history. If the fly travels at 100 miles per hour, what is the total distancehe traveled back and forth on this final trip of his?

24

86. In Figure 21, move exactly two matchsticks to make the equation true.(Do not create a “does not equal” sign.)

Figure 21: An equation made out of matchsticks for Homework Problem 86.

87. I have ten bags of coins, each bag having ten coins. One bag contains only counterfeit coins.Each legitimate coin weighs 1 ounce, while each counterfeit coin weighs 0.9 ounces. You havea bathroom type scale in front of you that will weigh objects up to 100 ounces. What is theminimum number of weighings needed to determine which bag has the counterfeit coins? Youare allowed to take coins out of the bags if you wish.

88. Given a five liter jar, a three liter jar, and an unlimited supply of water, how can you measureout exactly four liters of water?

89. Rufus and Dufus each took some money from a piggy bank to buy an ice cream cone, butRufus was 24 cents short, and Dufus was 2 cents short of the price of the cone. They decidedto pool their resources, but found they still could not afford to buy the cone. How many centsdid the ice cream cone cost?

90. Alice has three pennies, three nickels, and three dimes. How many different amounts of moneycan Alice make using one or more of these nine coins?

91. Points A, B, C, and D lie on a straight line, but not necessarily in that order. If AB = 3,BC = 4, and CD = 5, what is the smallest possible value of AD?

92. On his birthday, Brian was 14 years old and his father was 41. Brian noticed that his agewas the reverse of his fathers age. How old will Brian be the next time his age is the reverseof his father’s age?

93. My street address is a three digit number. If the product of the digits is 140, and the digitsappear in increasing order, from left to right, what is my street address?

94. In how many ways can you enter two numbers into the following sentence to make it true?

“This sentence has consonants and vowels.”

95. Augustus Demorgan (1806–1871) was x years old in the year x2 (x is an integer). Find x.

96. In the addition problem below, the ∗ and # symbols both represent unknown digits. Thevarious ∗ symbols may represent different digits. What digit is represented by the # symbol?

∗ 3∗ ∗∗ ∗∗ ∗

3 9 #

25

97. With the knowledge that black dots move to the right and white dots move to the left, and amove consists of a dot moving either to an adjacent empty space or jumping over one otherdot to reach an empty space, determine a sequence of moves which switch the positions ofthe black and the white dots in Figure 22.

• • • ◦ ◦ ◦

Figure 22: Initial dot positions in Homework Problem 97.

98. Pat said, “I am thinking of two numbers. One number is three times the other, and their sumis 8 more than twice the smaller number.” Can you tell Pat exactly what the two numbersare? Explain.

99. A woman has 3 daughters. Her friend wants to know how old the daughters are. The womangives them a hint. She says, the product of their ages is 72. (All ages are in whole numberof years.) The sum of their ages is equal the house number of the house directly across thestreet. The friend looks at the number of house directly across the street and says, “I stilldon’t have enough information to determine their ages. Can you give me one more hint.”Reluctantly, the mother agrees and says, “My only oldest daughter has blond hair.” Thefriend says, “Aaah, yes! Now I know their ages.” Do you?

100. What is the real value of x which satisfies

(212 + 2−12)(212 − 2−12) = 8x − 8−x?

Try to prove that only one value of real value of x solves this equation.

101. Simplify completely:

1√1 +

√3

+1√

3 +√

5+

1√5 +

√7

+1√

7 +√

9

102. What is the coefficient of x14 in (x− a)(x− b) · · · (x− z) in terms of the variables a, b, . . . , z?

103. The sum of two positive numbers equals the sum of the reciprocals of the same two numbers.What is the product of the two numbers?

104. On a shelf in Danny’s basement there are three boxes. One contains only nickels, the sec-ond, only dimes, and the third a mixture of nickels and dimes. The three labels, “Nickels”,“Dimes”, and “Mixed”, all fell off, and were all put back on the wrong boxes. Without lookinginside any of the boxes, Danny can select only one coin from one of the mislabeled boxes, andthen correctly label all three boxes. Which box should he pick from, and once he picks, howdoes one tell which box contains which coins?

105. Determine which of 2.23.3 and 3.32.2 is larger without the use of a calculator.

106. A cup of coffee and a cup of tea each contain the same amount of liquid. Jack takes ateaspoon of liquid from the cup of tea and places it in the cup of coffee and the mixture ismixed thouroughly. Then a teaspoon is taken from the mixture and placed back into the teacup. Can you determine if there more coffee in the coffee cup than there is tea in the tea cupor vice versa? If so, which is true?

26

107. You are offered a chance to play a game. The rules are simple. There are 100 cards facedown. Of these, 55 say WIN and 45 say LOSE. You begin with 10,000 dollars. You must bet1/2 of your money on each card turned over, and you either win or lose that amount basedon what the card says. At the end of the game all the cards have been turned over. Are youahead or behind at the end of the game? How much do you have at the end of the game?What arrangement of cards will earn you the most money?

108. If the lengths of the sides of a right triangle are 3 and 4, what is the least possible value ofthe length of third side?

109. Plant a total of 13 trees, in exactly 12 rows of 3 trees per row.

110. Four men sat down to play, and played all night till the break of day. The played for cashand not for fun, with separate scores for everyone. Yet, when they came to square accounts,they had all made quite fair amounts. Can you explain how? How could everyone gain?

111. Show that for all positive values of p, q, r, and s

(p2 + p + 1)(q2 + q + 1)(r2 + r + 1)(s2 + s + 1)

pqrs≥ 81

112. Consider the 8 by 8 grid of squares in Figure 23.

(a) Find the number of squares in the grid.

(b) Find the number of rectangles in the grid. (Remember, a square is a rectangle also.)

Figure 23: Grid for Homework Problem 112.

113. In Figure 24, you see a schematic of some of the streets in a certain town. Determine howmany paths exist from A to B that travel only to the right and up. Two such pathsare given in the figure, one using a dashed line and one using a dotted line (they overlap inmultiple places).

114. Can you find the error in the following? We begin with a = b. We multiply both sides bya to get a2 = ab. We subtract b2 from both sides to get a2 − b2 = ab − b2. Factoring, weget (a − b)(a + b) = b(a − b). Finally, dividing by (a − b), we get a + b = b. Now, if you seta = b = 1, then this tells us that 2 = 1.

115. Starting with the statement that (−1)/1 = 1/(−1), take the square root of both sides, to get√

(−1)/1 =√

1/(−1). This is equivalent to√−1/

√1 =

√1/

√−1. That is, i/1 = 1/i. Now

cross multiply to get −1 = 1. What is wrong?

27

B

A

Figure 24: Street layout in Homework Problem 113.

116. Here is another one. The series 1 − 1 + 1 − 1 + 1 − 1 + · · · has sum 0 as is easily seen byrewriting it as (1 − 1) + (1 − 1) + (1 − 1) + · · ·. Its sum is also 1 as is seen by rewriting it as1− (1− 1)− (1− 1)− (1− 1)− · · · = 1 + 0 + 0 + 0 + · · · = 1. How could this be? (Interestinghistorical note: This was given as one of the proofs of God. If something, 1 can be formedfrom nothing, 0, then there must be a God!)

117. This next one is a bit different. It can be shown that the series

1 − 1

2+

1

3− 1

4+ · · ·

does converge, and has the sum ln 2. Call the sum of this series, S (which we have said isln 2). Now look at the following computations:

S = 1 − 1

2+

1

3− 1

4+

1

5− 1

6+

1

7− 1

8· · · Multiply by 1/2 to get

(1/2)S =1

2− 1

4+

1

6− 1

8+ · · · Add the series, we get

(3/2)S = 1 + 0 +1

3− 1

2+

1

5+ 0 +

1

7− 1

4+ · · · When more terms of this

series are taken and the series rearranged, we get the series

1 − 1

2+

1

3− 1

4+

1

5− 1

6+ · · · which we know is S.

In summary, 32S = S. So S = 0, not ln 2! This one really gave mathematicians heartburn.

118. You may recall from high school that an angle inscribed in a semicircle is a right angle. Usingthis fact, we prove that there is a triangle with two right angles! Referring to Figure 25, wesuppose that two circles intersect at A and B, and that BC and BD are diameters. Nowdraw CD, intersecting the first circle at F and the other at E. Since angle CEB is inscribedin a semicircle, it is a right angle. Since angle BFD is inscribed in a semicircle, it is also aright angle. Thus triangle BEF is a triangle with two right angles. What is wrong?

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E

B

A

C

D

F

Figure 25: The two intersecting circles in Homework Problem 118.

119. In Figure 26, we see a 13×5 rectangle and an 8×8 square made up of the same two trapezoidsand two triangles. However, the area of the rectangle is 65 and the area of the square is 64.What is wrong? Building a model of this to show the class would be greatly appreciated.

8

8 5

85

5

3

35

3

3

35

5

5

53

Figure 26: Figure for Homework Problem 119.

120. A Sultan wanted to increase the number of women in his country, as compared to the numberof men, so that men could have larger harems. (Sorry ladies!) To accomplish this, he proposedthe following law: As soon as a mother gave birth to her first son, she would be forbiddento have any more children. In this way, the Sultan argued, some families would have severalgirls, and only one boy, but no family would have more than one boy. It would not be longbefore the females greatly outnumbered the males. Do you think the Sultan’s law wouldwork?

121. Find all solutions of x2 − y2 = 63 where x and y are positve integers. Then solve thefollowing problem. Henry, Eli, and Cornelius and their wives, Gertrude, Katherine andAnna (not necessarily in that order), each purchase at least one animal from a farm auction.Coincidentally, each of the 6 people spent exactly the amount for each animal as the numberof animals purchased. (So one could have purchased 4 animals at $4 a piece, or 5 animals at$5 a piece, and so on.) Henry purchased 23 more animals than Katherine did, and Eli spent$11 more per animal than Gertrude. Also each husband spent $63 more than his wife. Whois married to whom?

122. Mr. A, Mr. B, Mr. C, Mr. D, and Mr. E as well as their wives were seated at a round table.The seats were arranged so that the men and women alternated, and each woman was threeplaces distant from her husband. Mrs. C, sat directly on Mr. A’s right. Mr. E sat 2 placesto the left of Mr. C, while Mrs. E sat two places to the right of Mrs. B. Who sat directly onMr. A’s left?

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123. We can determine the ages of Ambrose, Brandon, and Chester (all positive integers) using thediagram in Figure 27, where just one digit is written in each box, and the following statementshold.

a

b

Figure 27: Age diagram for Homework Problem 123.

(a) starting at a and reading across is Ambrose’s age in years

(b) starting at a and reading down is the sum of Ambrose’s age and Brandon’s age in years.

(c) starting at b and reading across across is the sum of Ambrose’s age, Brandon’s age, andChester’s age in years.

(d) Two of Ambrose, Brandon and Chester, are the same age in years.

Which of the people has a different age in years from the other two? (Note: In this problem,single digit ages, like 7, cannot be written as 07.)

124. A, B, and C, each fired six shots each and got 71 points. The results of the shots are shownin Figure 28. A’s first two shots earned 22 points and B’s first shot was worth 3 points. Whohit the bull’s eye?

3 12550 25 20 10

Figure 28: Target and shots fired in Homework Problem 124.

125. Vera, one of the performers in a play, was murdered in her dressing room. Determine whokilled Vera using the following facts and the layout of the dressing rooms shown in Figure 29.Each of the 5 performers in the play—Vera, Adam, Babe, Clay, and Dawn—had his or herown dressing room. Rooms 2, 3, and 5 are the same size. Rooms 1 and 4 are the same size.

(a) Both the killer’s and Vera’s dressing rooms border on the same number of rooms.

(b) Vera’s dressing room borders on Adam’s dressing room and Babe’s dressing room.

(c) Clay’s dressing room and Dawn’s dressing room are the same size.

(d) Babe’s dressing room does not border on Clay’s dressing room.

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5

1 2

3 4

Figure 29: Dressing room layout in Homework Problem 125.

126. One of Mr. Horton, his wife, their son, and Mr. Horton’s mother is a doctor and another isa lawyer.

(a) If the doctor is younger than the lawyer, then the doctor and the lawyer are not bloodrelatives.

(b) If the doctor is a woman, then the doctor and the lawyer are blood relatives.

(c) If the lawyer is a man, then the doctor is a man.

Whose occupation do you know?

127. Avery, Blake, Clark, and Doyle each live in an apartment building. Their apartments arearranged as in Figure 30; use the following information to determine which of the four is thelandlord.

da b c

Figure 30: Apartment arrangement in Homework Problem 127.

(a) One of the four is the landlord.

(b) If Clark’s apartment is not next to Blake’s, then the landord is Avery and lives inapartment a.

(c) If Avery’s apartment is east of Clark’s apartment, then the landord is Doyle, and livesin apartment d.

(d) If Blake’s apartment is not next to Doyle’s apartment, then the landlord is Clark andlives in apartment c.

(e) If Doyle’s apartment is east of Avery’s apartment, then the landlord is Blake, and livesin apartment b.

Who is the landlord?

128. Prove that multiplying a four-digit number by 2 can never reverse the digits.

129. Using all the digits 1 through 9 and using each only once, form numbers whose sum is 100.If you think it can’t be done, then try to prove it. Of course, if you find the numbers, thereis nothing to prove.

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130. The sum and difference of two squares may be a prime. For example 4− 1 = 3 and 4+1 = 5;9 − 4 = 5 and 9 + 4 = 13, and so on. Let’s reverse the question. Can we pick 2 primes p andq such that the sum and difference of these two primes is a square? If so, find all such pairsof numbers (p, q). If not, explain why not.

131. What is the least positive integer that can be added to the product of any 4 consecutiveintegers so that the result is always the square of some integer?

132. Consider this 15 by 15 matrix:

1 2 3 4 · · · 1516 17 18 19 · · · 3031 32 33 34 · · · 45...

......

.... . .

...211 212 213 214 · · · 225

.

Suppose we pick a set of numbers, one number from each row and one from each column.True or False: The sums we get are always the same. If it is true, prove it. If it is false, findthe largest possible sum we can get.

133. Prove that 1 · 2 + 2 · 3 + 3 · 4 + · · · + (n − 1) · n = (n−1)(n)(n+1)3 . for n ≥ 2.

134. Prove that 1n+1 + 1

n+2 + · · · + 12n > 13

24 when n ≥ 2.

135. Using the fact that A ∩ (X ∪ Y ) = (A ∩ X) ∪ (A ∩ Y ) for any sets A, X, and Y , prove thatA ∩ (B1 ∪ B2 ∪ · · · ∪ Bn) = (A ∩ B1) ∪ (A ∩ B2) ∪ · · · ∪ (A ∩ Bn).

136. Show that if x1, x2, . . . , xn are positive numbers between 0 and 1, then

(1 − x1)(1 − x2) · · · (1 − xn) ≥ 1 − x1 − x2 − · · · − xn.

137. The Fibonacci sequence begins with the numbers F1 = 1, F2 = 1, and then each subsequentnumber is the sum of the two previous numbers. Thus the sequence is 1, 1, 2, 3, 5, 8, 13, . . .. In

symbols, the nth Fibonacci number, Fn, is the sum of the two previous Fibonacci numbers,

Fn−1 + Fn−2. Prove that every 5th Fibonacci number is divisible by 5.

138. Prove that 32n+2 + 8n − 9 is divisible by 16 for each positive integer n.

139. Find a formula which gives the sum of the first n positive integers that give remainder twomodulo three. (For example, 2, 5, 8, 11, etc.)

140. Theorem: All horses are of the same color.

Proof: (by induction) Consider a set with one horse. It clearly has only a horse of one color.Now assume that any set of k horses has the same color, and take a set of k + 1 horses. Lookat the subset consisting of the first k horses. By the inductive assumption, they all have thesame color. Now consider the subset consisting of the horses starting from the second andgoing to the last horse. That set also has k horses, and by the inductive assumption, all thehorses in that set have the same color. These two subsets overlap, and therefore all the horsesin the set have the same color. Since the truth of the statement for k horses implies the truthof the statement for k + 1 horses, by the principle of mathematical induction, all horses havethe same color.

Now, how is that for a horse of a different color???

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141. Suppose that 100 soldiers are lined up in 10 rows of 10 soldiers each. Suppose that in each rowthe soldiers are lined up from shortest to tallest from front to back. Then the commandingofficer orders that all soldiers that are the same depth in each row (for example all 10 soldiersthat are in the front of the row, second-back in the row, etc.) to rearrange themselves sothat they are lined up from shortest to tallest from left to right. Prove that even in this newassignment of positions, each row is still lined up from shortest to tallest from front to back.

142. Give a friend a coin with a value in cents which is even. Since a dime is worth 10 cents and10 is even, you can give him a dime. Give him another coin, whose value in cents is odd, saya nickel. Tell him to place one coin in his right hand and the other in his left hand. Nowask him to triple the value of the coin in his right hand, and double the value of the coin inhis left hand, and then add the two numbers obtained. If the sum obtained is even, the dimeis in his right hand, if the sum is odd, the dime is in his left hand. Explain why this trickworks. Can you think up some variations of this problem?

143. Take two dice. Throw one and note the number which shows uppermost on the first die.Multiply that number by 2 and add 1. Now multiply the result by 5. Throw the second die,and add to the above total, the number which shows uppermost on the second die. Announcethe result. I will tell you what numbers came up on the first and second die. How do I dothis?

144. Find all two digit numbers with the property that if I sum the digits and add this sum to theproduct of the digits, I get the number.

145. Have a person take out three objects (all spelled with a different number of letters or elsethe trick will not work.) Ask the person to think of one of the objects he took out. Ask himto take the number of letters in the object, and then perform the following three operations:Multiply this number by 5, add 3, and then double the total. To make is even more interesting,have someone whisper a digit from 1 to 9, into his ear which he then adds to the total above.When he gives you the answer, you immediately tell him what object he was thinking of andwhat number was whispered into his ear. How do you do it?

146. There is an interesting 5 digit number N with the property that with a 1 after it, it is 3 timesas large as it is with a 1 before it. Find N .

147. What is the largest integral factor of 111,111,111,111 which is less than 111,111,111,111?

148. Can you replace the missing digits in the number 789XY Z, so that the resulting number isdivisible by 7, 8, and 9? The only restriction is that you cannot use a 7, 8 or 9.

149. Tell someone to pick a positive integer, double it, add 9, multiply by 5, and then subtractthe number he started with. Tell him to remove any nonzero digit in the answer, and tell youthe remaining digits in any order he or she wishes. You then tell him the digit he removed.Make up some examples, and see if you can explain how this is done.

150. Two triangles share a common side. The lengths of the sides of the first triangle are in theratio of 5 : 6 : 7, while those of the second are in the ratio of 4 : 9 : 11. If the lengths of allsix sides are integral, what is the smallest possible length of this common side?

151. A positive integer such as 4334 is a palindrome if it reads the same backwards and forwards.How many prime palindromes are there with an even number of digits?

152. If 1, 2, and 4 are three of the digits of a four digit number N , and N is divisible by 4, whatis the greatest possible value of N?

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153. One solution of the equation (x − a)(x − b)(x − c)(x − d) = 9 is x = 2. If a, b, c, and d aredifferent integers, find the value of a + b + c + d.

154. A certain miser, before he starved to death, hoarded a quantity of five, ten, and twenty dollargold pieces. He kept them in 5 bags that were exactly alike, in that each bag contained thesame number of five dollar pieces, the same number of 10 dollar pieces, and the same numberof twenty dollar pieces.

The miser counted his treasure by pouring it all on the table, then dividing it into four pilesthat were also exactly alike in containing the same amounts of each type of coin. His finalstep was to take any two of these piles, put them together, then divide their coins into threepiles which were exactly alike in the sense already explained. What was the least amount ofmoney this poor old man could have had?

155. Without using a calculator, tell me the difference between the smallest perfect square largerthan one million, and the largest perfect square smaller than 1 million?

156. What are all ordered pairs of real numbers (x, y) for which√

x +√

y = 17 and x − y = 85?(Hint: Don’t use the method of substitution.)

157. Let x > 12 . What is the simplest form of the expression

1 +√

2x − 1√

x +√

2x − 1?

158. If x3 + y3 = 400 and x2y + xy2 = 200, find all values of x and y which solve this equation.

159. If x + 1x = 5, find, in simplest form, the value of x3 + 1

x3 .

160. Simplfiy the following fraction as much as possible.

(8 + 2√

15)3/2 + (8 − 2√

15)3/2

(12 + 2√

35)3/2 − (12 − 2√

35)3/2

161. No person in a certain city has more than one million hairs on his head. If this city has atleast fifteen million people, show there are at least five people in city with the same numberof hairs on their heads. Are there at least ten people with the same number of hairs on theirheads? How do you know?

162. Seven people have first names Alfie, Ben, and Cissi, and last names Dumont and Elm. Showthat at least two of the people have the same first and last names.

163. Seven points are placed inside of a regular hexagon with sides of length 1. Show that at leasttwo of the points are a distance 1 or less from each other.

164. Fifty-one points are placed in a square 100 centimeters by 100 centimeters. Show that somethree points can be covered by a square with side 20 centimeters.

165. True or False: Suppose that there are 12 people in a room, Then there will be two people inthe room with the same number of friends in the room. (We are assuming here that if A andB are people in the room and A is friendly with B, then B is friendly with A.)

166. Consider the following two sets of consecutive integers: {10, 11, . . . , 20} and {21, 22, . . . , 30}.Each element of the first set is multiplied, in turn, by each element in the second set. Findthe sum of all these products.

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167. Find the value of the infinite product, (3)(1/3)(9)(1/9)(27)(1/27) · · ·.

168. Suppose your internet connection is absurdly slow and you hear about the latest and greatestsong that you just HAVE to download from iTunes. You find the file and when the download-ing dialog pops up it says that you have 60 minutes remaining. You return 60 minutes laterand the dialog now says 30 minutes remaining. Your frustration builds as when you return 30minutes later, an estimated 15 minutes remain. If the downloading continues in this fashion,how long will the whole download take in all? Also, how many times will you return to lookat your screen?

169. You are at the center of a circle of radius 9. You begin to hop towards the circumference ofthe circle in a straight line. Your first hop is 41

2 feet, your next hop is 214 feet, and so on,

where each successive hop is half the amount of the previous hop. How many hops will ittake you to escape the circle?

170. Together, Al, Barb, Cal, Di and Ed earned a total of $150 but in unequal amounts. In orderto equalize their earnings exactly, first Barb gave half her earnings to Al. Next Cal gave 1/3his earnings to Barb. Then Di gave 1/4 of her earnings to Cal. Finally, Ed gave 1/6 hisearnings to Di. How many dollars did Al have before equalization?

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