| 91. Can we find a set of positive integers S, such that all sufficiently large integers can be expressed in the same number of ways as a sum a + b, with a ≤ b and a, b ∈ S. | |
| 92. We are interested in sequences of integers a1 < a2 < ... < an, such that the set of n(n - 1)/2 positive differences {ai - aj: i > j} is just {1, 2, 3, ... , n(n - 1)/2}. Such a sequence is possible for n = 4: 1, 3, 6, 7. Is it possible for any n > 4? | |
| 93. Show that the number of partitions of N into odd (positive) integers equals the number of partitions of N into distinct (positive) integers. | |
| 94. Let f(n) = ∫0∞ (1 + x/n)n e-x dx. Estimate f(n) as n → ∞. | |
| 95. Let f(n) = ∫0∞ tn e-t dt. Show that f(n) ~ nne-n√(2πn). [Given functions f(n) and g(n), f(n) ~ g(n) means that f(n)/g(n) → 1 as n → ∞.] | |
| 96. Let f(n) = ∑i=0n ni / i! Show that f(n) ~ en/2. | |
| 97. Estimate ∫01 cosn(x2) dx as n → ∞. [In other words, find suitable upper and lower bounds or an asymptotic expression.] | |
| 98. A and B repeatedly play a fair game of chance with unit stake. In other words, at each turn, either A wins 1 unit from B or A loses 1 unit to B, and each outcome has probability 1/2, independent of all other turns. If A starts with m units and B starts with n units, what is the expected number of turns before one player is wiped out? | |
| 99. A and B repeatedly play a game of chance with unit stake. At each turn, either A wins 1 unit from B with probability 0.51, or A loses 1 unit to B with probability 0.49. Both A and B have unlimited capital. What is A's expected peak cumulative loss? | |
| 100. A plays a game of chance. At each turn he wins an amount k which is uniformly distributed on the interval [0, 1]. The game stops when his cumulative win is at least 1. What is the expected number of turns? |
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