34th Swedish 1994

1.  x√8 + 1/(x√8) = √8 has two real solutions x1, x2. The decimal expansion of x1 has the digit 6 in place 1994. What digit does x2 have in place 1994?
2.  In the triangle ABC the medians from B and C are perpendicular. Show that cot B + cot C ≥ 2/3.
3.  The vertex B of the triangle ABC lies in the plane P. The plane of the triangle meets the plane in a line L. The angle between L and AB is a, and the angle between L and BC is b. The angle between the two planes is c. Angle ABC is 90o. Show that sin2c = sin2a + sin2b.
4.  Find all integers m, n such that 2n3 - m3 = m n2 + 11.
5.  The polynomial xk + a1xk-1 + a2xk-2 + ... + ak has k distinct real roots. Show that a12 > 2ka2/(k-1).
6.  Let N be the set of non-negative integers. The function f:N→N satisfies f(a+b) = f(f(a)+b) for all a, b and f(a+b) = f(a)+f(b) for a+b < 10. Also f(10) = 1. How many three digit numbers n satisfy f(n) = f(N), where N is the "tower" 2, 3, 4, 5, in other words, it is 2a, where a = 3b, where b = 45?

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

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© John Scholes
9 Oct 2003
Last corrected/updated 9 Oct 03