

1. x√8 + 1/(x√8) = √8 has two real solutions x_{1}, x_{2}. The decimal expansion of x_{1} has the digit 6 in place 1994. What digit does x_{2} 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 90^{o}. Show that sin^{2}c = sin^{2}a + sin^{2}b.


4. Find all integers m, n such that 2n^{3}  m^{3} = m n^{2} + 11.


5. The polynomial x^{k} + a_{1}x^{k1} + a_{2}x^{k2} + ... + a_{k} has k distinct real roots. Show that a_{1}^{2} > 2ka_{2}/(k1).


6. Let N be the set of nonnegative 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 2^{a}, where a = 3^{b}, where b = 4^{5}?

