1. The contents of a piggy bank consist of quarters, dimes, nickels, and 47 pennies—70 coins in all. If there were exactly 4 times more quarters, 50% fewer dimes, 100% more nickels, and 3 times as many pennies, there would be exactly twice again as much money in the bank. How many of each kind of coin are actually in the bank?

2. The contents of a piggy bank consist of quarters, dimes, nickels, and pennies—17 coins in all. If there were exactly 4 times more quarters, 50% fewer dimes, 100% more nickels, and 8 times as many pennies, there would be exactly twice again as much money in the bank. How many of each kind of coin are actually in the bank?

3. An equilateral triangle is inscribed in a circle. What is the ratio of the area of the triangle to the area of the circle?

4. A square is inscribed in a circle. What is the ratio of the area of the square to the area of the circle?

5. A regular polygon with n sides is inscribed in a circle. What is the ratio of the area of the polygon to the area of the circle?

6. Consider the infinite sequence where the index n > 2 and the term r_n is the ratio of the area of an n-sided regular polygon to the area of its circumscribed circle. Use standard techniques of elementary calculus to prove that lim_{n increases w/o bound} r_n = 1.

7. A merry-go-round floor consists of a circular platform with a concentrically circular hole in it. A painter takes a single 60-foot measurement of the merry-go-round and uses it to figure the area of the floor. What measurement was taken, and what is the area of the floor?

8. Trapezoid ABCD with Bases AB and CD is inscribed in a circle. Chords BD and AC intersect at Point M. The respective lengths of Segments AM and CM are p and q. Chord XY through Point M is parallel to the trapezoid bases. What is the length of Chord XY in terms of p and q?

9. Given that a > 1, x > 0, y > 0, x is not 1, y is not 1, log_xxy = log_yxy, and x + y = 2a, find (x,y).

10. Given that a > 1, prove that a - (a² - 1)^0.5 > 0 but not equal to 1.

11. Indicated below is an infinite sequence of sentences called “Pythagorean runs.”

3² + 4² = 5².
10² + 11² + 12² = 13² + 14².
21² + 22² + 23² + 24² = 25² + 26² + 27².
¼

Each Pythagorean run generates a number sequence:

3, 4, 5;
10, 11, 12, 13, 14;
21, 22, 23, 24, 25, 26, 27;
¼.

Calculate the 1222^nd term of the 1998^th Pythagorean run number sequence.

12. Given in order, the terms a₂, a₃, a₄, a₅, and a₆ of an infinite sequence are 35, 45, 63, 105, and 315. Find a₁, a₇, and the lim_{n increases w/o bound} a_n.

13. Given that x > 0 and y > 0 and that z and n are integers each greater than 1 such that xⁿ + yⁿ = zⁿ, prove independently of Fermat’s Last Theorem that there exists a positive integer p such that for each n > p, x and y are not both integers.

14. Suppose x > 1 and y > 1. For each radius z, the graphs of the equation

x^p + y^p = z^p where p > 1

collectively constitute a region of the x-y plane. What is the greatest lower bound of the set of areas of all such regions?

15. A Triangle ABC is inscribed in a circle. What is the area of the circle in terms of a, b, and c?

16. A circle is inscribed in a Triangle ABC. What is the area of the circle in terms of a, b, and c?

17. Perpendicular to a road that follows a straight shoreline are Pier 1 (240 feet long), then Pier 2 (140 feet long), then an inland path. At the offshore end of Pier 1, one looks directly past the offshore end of Pier 2 to see the point where the road meets the path and directly past the inland end of Pier 2 to see the far end of the path. Find the path length.

18. Will started a 325-mile trip at 8 a.m. going 40 mph in 12 miles of city traffic. Then he went 69 mph on the highway for 1 hour, 40 minutes before a 6-minute break. If he is to move at 55 mph for the rest of the trip, how long can he stop for lunch and still arrive by 2 p.m.?

19. In the figure below, Lines l, m, and n are parallel. Points A, B, and C are collinear; Points A, D, and E are collinear; Points F, D, and C are collinear; and Points F, G, and E are collinear. The respective lengths of Segments AF and BG are 38.5 and 23.1. What is the length of Segment CE?  (Points indicated belong to lines indicated.)

                    A .                                                        . F                        l

                              B .                      .                   . G                           m
                                                      D
                                     C .                             . E                                n

20. At what times between 5:47 and 5:48 are the sweep second hand and the hour hand of a clock perpendicular?

21. On May 10 Pump 213 and Pump 174 were used together for 17 hours to remove 41,480 ft³ of water from flooded land. On May 14 Pumps 213 and 203 removed 33,120 ft³ of water in 12 hours, and on May 25 Pumps 174 and 203 removed 14,040 ft³ of water in 4½ hours. How long would it take all three pumps jointly to remove 45,760 ft³ of water?

22. A minor segment of a circle consists of a chord and its minor arc. Its length l is its chord length, and its width w is the distance between the midpoints of its sides. What is its perimeter in terms of l and w?

23. Find the area of a minor segment of a circle in terms of l and w as described in Puzzle 22.

24. Consider an infinite sequence of finite arithmetic subsequences as exemplified below.

Example 1: 1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5, …
Example 2: 3,5,7,3,5,7,3,5,7,3,5,7,3,5,7, …
Example 3: 0,-3,0,-3,0,-3,0,-3,0,-3, …

Write a general formula for each example and write a general formula for all such infinite sequences.

25. Consider an infinite sequence of finite geometric subsequences as exemplified below.

Example 1: 3,6,12,24,48,3,6,12,24,48,3,6,12,24,48,3,6,12,24,48, …
Example 2: -5,25,-125,-5,25,-125,-5,25,-125,-5,25,-125, …
Example 3: 16,4,1,¼,16,4,1,¼,16,4,1,¼,16,4,1,¼, …

Write a general formula for each example and write a general formula for all such infinite sequences.

26. A circle is inscribed in an equilateral triangle. What is the ratio of the area of the triangle to the area of the circle?

27. A circle is inscribed in a square. What is the ratio of the area of the square to the area of the circle?

28. A circle is inscribed in a regular polygon with n sides. What is the ratio of the area of the polygon to the area of the circle?

29. Consider the infinite sequence where the index n > 2 and the term r_n is the ratio of the area of an n-sided regular polygon to the area of its inscribed circle. Use standard techniques of elementary calculus to prove that lim_{n increases w/o bound} r_n = 1.

30. Write a general formula and a recursive formula for the following sequence:

–1, 1/2, –4, 1/8, –16, 1/32, –64, 1/128, –256, 1/512, ….

31. How many grams of salt must be added to 50 grams of 10% salt solution to achieve the concentration rate that would have resulted from evaporating 8 grams of water from the original solution?

32. Consider an infinite sequence of finite arithmetic subsequences as exemplified below.

Example 1: 1,2,3,4,5,2,3,4,5,6,3,4,5,6,7,4,5,6,7,8,5,6,7,8,9, …
Example 2: 3,5,7,5,7,9,7,9,11,9,11,13, …
Example 3: 0,-3,-3,-6,-6,-9,-9,-12, …

Write a general formula for each example and write a general formula for all such infinite sequences.

33. Consider an infinite sequence of finite geometric subsequences as exemplified below.

Example 1: 3,6,12,24,48,6,12,24,48,96,12,24,48,96,192,24,48,96,192,384, …
Example 2: -5,25,-125,25,-125,625,-125,625,-3125, …
Example 3: 16,4,1,1/4,1/4,1/16,1/64,1/256,1/256,1/1024,1/4096,1/(16,364),
1/(16,364), …

Write a general formula for each example and write a general formula for all such infinite sequences.

34. A. The length of a chord of a circle, with radius length r, is l. In terms of r and l, what is the length of the minor arc made by the chord?
B. The length of a minor arc of a circle, with radius length r, is l. In terms of r and l, what is the length of the longest chord of the arc?

35. There are exactly 7 coins of exactly two kinds in a jar. If there were exactly 3 times as many coins of one kind and 1000% more coins of the other kind, the jar would contain exactly 600% more money. What kinds of coins and how many of each kind are actually in the jar?

36. Consider the infinite sequence where the index n > 2 and the term r_n is the ratio of the area of an n-sided regular polygon inscribed in a circle to the area of an n-sided regular polygon circumscribed about the circle. Write a simple expression for r_n and show that lim_{n increases w/o bound} r_n = 1.

37. The respective formulas for the n^th terms of arithmetic and geometric sequences are

a_n = a₁ + d(n – 1) and a_n = a₁rⁿ^-1.

Describe the relationships among them and the most general formulas given in the solutions to Puzzles 24, 25, 32 and 33.

38. A brace of a trapezoid is a segment that
1) has an endpoint on each leg,
2) is parallel to the bases,
and 3) contains the intersection point of the diagonals.

What is the length of the brace of a trapezoid in terms of the lengths of its bases?

39. If b₁ > 0, b₂ > 0, and b₁ < b₂, prove that

b₁ < 2b₁b₂/(b₁ + b_2⁾< b₂.

40. The probability that a first event will occur is p₁; the probability that an independent second event will occur is p₂. A. Explain the probability that either event will occur. B. Explain the probability that only one of the events will occur.

41. If each of p₁and p₂is between 0 and 1 inclusively, prove that so is

p₁ + (1 – p₁) p_2.

42. Two fair dice are tossed. A. How many times more likely is it that one of them will show six than that both of them will show two? B. How many times as likely is it that only one of them will show two as that both of them will show six?

43. The probabilities that three mutually independent events will occur are p₁,p₂, and p₃. A. Explain the probability that two of the events will occur. B. Explain the probability that exactly two of the events will occur.

44. The probabilities that n mutually independent events will occur are p₁,p₂, …, p_n where n = 2,3,4,5. If m = 1,2,3,4,5 and m < n + 1, display in a table the probabilities that m out of the n events will occur. If necessary, refer to the solutions to Parts A of Puzzles 40 and 43.

45. The rain probabilities in a five-day period are 20% Day 1, 70% Day 2, 30% Day 3, 80% Day 4, and 90% Day 5. Among Days 2, 3, and 4 only, what is the rain probability

one day? B. exactly one day?

C. exactly one day for the entire five-day period?

46. The snow probabilities in a five-day period are 20% Day 1, 70% Day 2, 30% Day 3, 80% Day 4, and 90% Day 5. What is the snow probability

two days? B. exactly two days?

C. two consecutive days?

47. Given in order, the terms a₂, a₃, a₄, a₅, and a₆ of an infinite sequence are 22, 32, 50, 92, and 302. Find a₁, a₇, and the lim_{n increases w/o bound} a_n.

48. Given in order, the first nine terms of an infinite sequence are 26, 16, 2, 44, 254, 376, 166, 124, and 106. Give a general formula for a_n and find the lim_{n increases w/o bound} a_n.

49. Suppose that x > 0, y > 0, z > 0, and x > y. A boat that averages x mph makes a three-legged cruise on a river that averages y mph. The first leg of the cruise ends upstream of the departure dock. The second leg ends z times as far from the dock as it began but downstream of it. The third leg ends at the departure dock. What distance can be traveled in x hours at the average rate of the cruise?

50. Suppose that x > 0, y > 0, z > 0, and x > y. A boat that averages x mph makes a three-legged cruise on a river that averages y mph. The second leg of the cruise ends z times as far from the initial departure dock as it began. The third leg ends at the departure dock. What distance can be traveled in x hours at the average rate of the cruise?

51. Suppose that n and i are integers and that n > 2, 1< i < n, x > 0, y > 0, x > y, and z_i > 0. A boat that averages x mph makes an n-legged cruise on a river that averages y mph. The ith leg of the cruise ends z_i times as far from the initial departure dock as it began. The nth leg ends at the departure dock. What distance can be traveled in x hours at the average rate of the cruise?