29th Swedish 1989

1.  Show that in base n2+1 the numbers n2(n2+2)2 and n4(n2+2)2 have the same digits but in opposite order.
2.  Find all continuous functions f(x) such that f(x) + f(x2) = 2 for all real x.
3.  For which positive integers n is n3 - 18n2 + 115n - 391 a cube?
4.  ABCD is a regular tetrahedron. Find a point P on the edge BD such that the sphere diameter AP touches the edge CD.
5.  The positive reals x1, x2, x3, x4, x5 satisfy x1 < x2 and x2 ≤ each of x3, x4, x5. Also a > 0. Show that 1/(x1+x3)a + 1/(x2+x4)a + 1/(x2+x5)a < 1/(x1+x2)a + 1/(x2+x3)a + 1/(x4+x5)a.
6.  4n points are arranged around a circle. The points are colored alternately yellow and blue. The yellow points are divided into pairs and each pair is joined by a yellow line segment. Similarly for the blue points. At most two segments meet at any point inside the circle. Show that there are at least n points of intersection between a yellow segment and a blue segment.

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

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© John Scholes
8 Oct 2003
Last corrected/updated 8 Oct 03