| 81. F: [0, ∞) → (0, ∞) is monotonic increasing. y = f(x) is a solution of y'' + F(x) y = 0. Prove f(x) is bounded as x → ∞. | |
| 82. f is a real-valued twice differentiable function on the reals. f and f '' are bounded. Prove that f ' is also bounded. | |
| 83. For the positive real number a, define x0 = a, xn = axn-1 for n > 0, and define f(a) = limn→∞ xn. Show that f(√2) = 2, but that f(x) = 4 has no solution. What is the supremum of the values k for which f(x) = k has a solution? | |
| 84. f is a continuous real-valued function on the reals, but is not necessarily differentiable. It satisfies limh→0+ ( f(x + 2h) - f(x + h) )/h = 0 for all x. Prove that f is constant. | |
| 85. Show that f:[a, b] → R (the reals) has a continuous derivative iff limh→0 ( f(x+h) - f(x) )/h exists uniformly on [a, b]. | |
| 86. Let f(x) = |sin x sin 2x sin 4x ... sin 2nx|. Prove that f(x) ≤ 2/√3 f(π/3). | |
| 87. x1, x2, x3, ... is a sequence of positive reals such that xn < xn+1 + xn2. Prove that S xn diverges. | |
| 88. Find all possible ways of labeling the faces of two dice with positive integers so that the probability of throwing a total score of N when the two dice are thrown together is the same as with normal dice (so we require p(2) = p(12) = 1/36, p(3) = p(11) = 2/36, p(4) = p(10) = 3/36, p(5) = p(9) = 4/36, p(6) = p(8) = 5/36, p(7) = 6/36, where p(N) is the probability of a total score of N). Note that the two dice do not have to be labeled in the same way. | |
| 89. Find two disjoint sets A and B whose union is the non-negative integers such that every positive integer can be expressed as the sum of two distinct elements of A in the same number of ways as it can be expressed as the sum of two distinct elements of B. | |
| 90. Let N be the set of positive integers. Prove that we cannot find an integer n > 1 and subsets N1, N2, ... , Nn of N, such that (1) the subsets are disjoint, (2) they have union N, (3) each Ni is an arithmetic progression {ai, ai + di, ai + 2di, ... }, and (4) each di is different. |
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