Miscellaneous Problems

Problem 44. Show that every positive real number is a sum (possibly infinite) of a subset of the numbers {1, 1/2, 1/3, 1/4, . . .}.

Problem 43. What is the prime factorization of 1,005,010,010,005,001?

Problem 42. Find the volume of a torus (doughnut) of inner radius b whose cross-section by a plane through the axis is a semicircle of radius a, with its straight boundary parallel to the axis and curved boundary away from the axis.

Problem 41. Consider an isosceles right triangle with legs of fixed length a. Inscribe a rectangle and a circle inside the triangle as indicated in the figure below. Find the dimensions of the rectangle (and the radius of the circle) which make the total area of the rectangle and circle a maximum.

Problem 40. Consider a balance that is used to measure loads of integral weights. The balance has two scales, a load scale and a weight scale. On the weight scale one can place only certain measuring weights. On the load scale one can place the load to be measured and any desired subset of the measuring weights. Show that with four suitably chosen weights one can measure the weight of any load whose weight is an integer between 1 and 40 kg.

Problem 39. Before steers are introduced to a pasture, there is a given amount of grass per acre, and the grass keeps growing at a constant rate. If 12 steers take 16 weeks to deplete the grass on 10 acres, and if 18 steers take 8 weeks to deplete the grass on 10 acres, how many steers does it take to deplete the grass on 40 acres in 6 weeks?

Problem 38. A ball of radius 1 is in a corner touching all three walls. Find the radius of the largest ball that can be fitted into the corner behind the given ball.

Problem 37. A destroyer is hunting a submarine in dense fog. The fog lifts for a moment, disclosing the submarine on the surface three miles away, upon which the submarine immediately descends. The speed of the destroyer is twice that of the submarine, and it is known that the latter will depart at once at full speed on a straight course of unknown direction. The wily captain of the destroyer sails straight to the point 2/3 of the way to the spot where the submarine was sighted and then sets out on a spiral course that is bound to make him pass directly over the submarine. What is the equation of this spiral?

Problem 36. Given finitely many points in the plane situated so that any three of them are the vertices of a triangle of area ≤ 1. Show that all the points can be enclosed in a rectangle of area ≤ 4.

Problem 35. The winning team of the World Series must win four games out of seven. Assuming that teams are equally matched, find the probabilities that the Series lasts

• (a) exactly four games,
• (b) exactly five games,
• (c) exactly six games,
• (d) exactly seven games.  

Problem 34. The probability that the square of a positive integer (in decimal notation) ends with the digit 1 is 2/10 because out of every 10 numbers those and only those ending with the digits 1 or 9 have squares ending with

Problem 33. What is the probability that the cube of a positive integer chosen at random ends with the digits II? Prove your answer.

Problem 32. A fair coin is tossed ten times. Find the probability that two tails do not appear in succession.

Problem 31. Find all real functions f such that, for all real x, f(x + 2) = f(x) and fʹ(x) = f(x + 1) – 2.