| 1. P is a point inside a sphere. Three chords through P are mutually perpendicular. Show that the sum of the squares of their lengths is independent of their directions. | |
| 2. n is a square such that if a prime p divides n, then p has an even number of digits. Show that if the rationals x, y satisfy xn - 1987x = yn - 1987y, then x = y. | |
| 3. f is a real-valued function on the reals such that f(x+1) = f(x) + 1. The sequence x0, x1, x2, ... satisfies xn = f(xn-1) for all positive n. For some n > 0, xn - x0 = k, an integer. Show that lim xn/n exists and find it. | |
| 4. Is there a subset of {1, 2, ... , 3000} with 2000 elements such that n and 2n do not both belong to the subset for any n? | |
| 5. Space is partitioned into three disjoint sets. Show that for each d > 0 we can find two points a distance d apart in one of the sets. |
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| 6. C is a circle radius 1 and n is a fixed positive integer. Let F be the set of all sets S of n points on C and let D be the set of all diameters of C. Given any element S of F and any element d of D, let f(S, d) be the shortest (perpendicular) distance from a member of S to d. Find g(n) = minFmaxDf(S, d), and find all sets S for which maxDf(S, d) = g(n). | |
| 7. A palindrome is a number which is the same read backwards (for example, 43534). Show that there are infinitely many palindromes whose digit sum is 1 more than one-third of the product of their digits. Show that only finitely many of these have all digits > 1 and find them. | |
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8. A closed contour is made up of translates of the four quarters of a circle. The translates are fitted together smoothly (so that the tangents are the same at the join). Show that the number of translates is a multiple of 4. An example is shown below:
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| 9. Let X be the set of points { (x, y): x = 1, 2, ... , 12; y = 1, 2, ... , 13}. Show that every subset of X with 49 elements has 4 points which are the vertices of a rectangle with sides parallel to the axes. Show that there is a subset of X with 48 elements which does not contain such points. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
17 Dec 2002
Last corrected/updated 17 Dec 2002