15th Swedish 1975

1.  A is the point (1, 0), L is the line y = kx (where k > 0). For which points P (t, 0) can we find a point Q on L such that AQ and QP are perpendicular?
2.  Is there a positive integer n such that the fractional part of (3 + √5)n > 0.99?
3.  Show that an + bn + cn ≥ abn-1 + bcn-1 + can-1 for real a, b, c ≥ 0 and n a positive integer.
4.  P1, P2, P3, Q1, Q2, Q3 are distinct points in the plane. The distances P1Q1, P2Q2, P3Q3 are equal. P1P2 and Q2Q1 are parallel (not antiparallel), similarly P1P3 and Q3Q1, and P2P3 and Q3Q2. Show that P1Q1, P2Q2 and P3Q3 intersect in a point.
5.  Show that n divides 2n + 1 for infinitely many positive integers n.
6.  f(x) is defined for 0 ≤ x ≤ 1 and has a continuous derivative satisfying |f '(x)| ≤ C |f(x)| for some positive constant C. Show that if f(0) = 0, then f(x) = 0 for the entire interval.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
26 September 2003
Last corrected/updated 30 Dec 03