22nd Swedish 1982

1.  How many solutions does x2 - [x2] = (x - [x])2 have satisfying 1 ≤ x ≤ n?
2.  Show that abc ≥ (a+b-c)(b+c-a)(c+a-b) for positive reals a, b, c.
3.  Show that there is a point P inside the quadrilateral ABCD such that the triangles PAB, PBC, PCD, PDA have equal area. Show that P must lie on one of the diagonals.
4.  ABC is a triangle with AB = 33, AC = 21 and BC = m, an integer. There are points D, E on the sides AB, AC respectively such that AD = DE = EC = n, an integer. Find m.
5.  Each point in a 12 x 12 array is colored red, white or blue. Show that it is always possible to find 4 points of the same color forming a rectangle with sides parallel to the sides of the array.
6.  Show that (2a-1) sin x + (1-a) sin(1-a)x ≥ 0 for 0 ≤ a ≤ 1 and 0 ≤ x ≤ π

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

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© John Scholes
8 October 2003
Last corrected/updated 27 Feb 04