| 11. Prove no three lattice points in the plane form an equilateral triangle. In three dimensions? | |
| 12. All members of the sequence a1, a2, a3, ... are positive and an < (an-2 + an-1)/2 for n > 2. Prove the sequence converges. | |
| 13. x0, x1, x2, ... is a sequence of complex numbers satisfying. xn+1 = (xn + 1/xn)/2 for n > 0. Does it converge? | |
| 14. x0, x1, x2, ... is a real sequence satisfying: xn+1 = (xn + xn-1)/2 for n > 1. What is the limit of the sequence (in terms of x0 and x1)? | |
| 15. x1, x2, ... , xm is a sequence of positive integers satisfying x1 + ... + xm = n. What is the maximum value of ∏ xi? | |
| 16. ∑ an is a convergent series of positive reals. Prove that ∑ (a1a2... an)1/n converges. | |
| 17. n is a given positive integer. p(x) ≡ xm + am-1xm-1 + ... + a1x + a0 is a polynomial with real coefficients ai such that p(r) is an integral multiple of n for all integers r. What is the smallest possible value of m (in terms of n)? | |
| 18. Evaluate √(1 + 2√(1 + 3√(1 + ... ))). | |
| 19. Prove that at any party two people have the same number of friends present. | |
| 20. If n is any integer greater than 1, then n does not divide 2n - 1. |
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
2 Jan 1999