| 51. Define the sequence an as follows: a0 = 0, a1 = 1, an = an-1 + an-2/2 for n > 1. Prove that an is not an integer, except for n = 0, 1, 2, 4, 8. | |
| 52. Let In = {1, 2, 3, ... , n}. We define the density of a set T of positive integers as inf (In ∩ T)/n. Let a1, a2, ... , ak be any positive integers. Let S be the set of all positive integers not divisible by any ai. Prove that the density of S is at least ∏ (1 - 1/ai). | |
| 53. Given n distinct real numbers x1, x2, ... , xn, and n arbitary real numbers y1, y2, ... , yn. Prove that we can find a polynomial p(x) all of whose zeros are real such that p(xi) = yi for 1 ≤ i ≤ n. | |
| 54. Is there a non-trivial continuous real-valued function defined on the real numbers such that f(x) + f(2x) + f(3x) = 0 for all x? | |
| 55. Exhibit: (A) an infinite group with no infinite proper subgroups; (B) a field isomorphic to a proper subfield; (C) a ring with no maximal ideals. | |
| 56. a1, a2, a3, ... is a sequence of positive integers. It satisfies an+1 > aan for n > 0. Prove that an = n. | |
| 57. Let ABC be a triangle with angle BAC = 90. P1, P2, ... , Pn is a finite set of points inside ABC. Prove that we can relabel the points Q1, ... , Qn so that ∑0<i<n QiQi+12 ≤ BC2. | |
| 58. In some game a player's batting average for a given period is defined as his total score for the period divided by his total at-bats. [Scores must be non-negative integers, and at-bats must be positive integers.] Player A has a higher batting average than player B for both the first half of the season and the second half of the season. Does he necessarily have a higher batting average for the season as a whole? | |
| 59. Let eex = ∑ anxn. Estimate an. [In other words, find upper and lower bounds for an.] | |
| 60. Define x0, x1, x2, ... by x0 = 1, xn = xn-1 + 1/xn-1 (for n > 0). Prove xn → ∞. How fast? [In other words, find upper and lower bounds for xn.] |
Donald J Newman, A problem seminar (Springer, problem books in mathematics, 1982). Highly recommended. It comes with short hints, giving the key idea and solutions which are concise but motivated (in other words, they often explain how you might have found the solution yourself).
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John Scholes
jscholes@kalva.demon.co.uk
25 Sep 1999