Advanced Problems


1. Show that for every n > 3 there exist two sets of integers 
   A = {x_1, x_2, ..., x_n}, B = {y_1, y_2, ..., y_n} such that:
    * A and B are disjoint.
    * x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n
    * x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2

2. ABC is an acute angled triangle and L is a line in its plane.
   Three symmetric lines of L with respect to sides of ABC meet
   each other in the points A', B', and C'. 
   Prove that the center of the inscribed circle of A'B'C' lies
   on the circumscribed circle of ABC.

3. In a party there are 12k guests. Every guest knows exactly 
   3k + 6 other guests. Suppose that if x knows y, then y knows
   x too. For every two guests x and y in this party there are 
   exactly n guests who know both x and y. (n is a constant) 
   Determine the number of guests in this party.

4. Let S = {2^m 3^n | m, n are nonnegative integers}.
   Prove that every natural number can be written as a sum of 
   distinct elements of S such that non of them is a multiple 
   of another.

5. Prove that for every integer n > 0 we have:
       [\Sqrt{n} + \Sqrt{n+1} + \Sqrt{n+2}] = [\Sqrt{9n+8}]
   where [x] denotes the smallest integer greater than or equal 
   to x. (ceiling of x)

6. In a tetrahedron ABCD, let A', B', C', and D' be centers of 
   circumscribed circles of BCD, CDA, DAB, and ABC, respectively.
   We denote the plane which passes trough the point X and is 
   perpendicular to the line YZ by S(X, YZ).
   Prove that four planes S(A, C'D'), S(B, D'A'), S(C, A'B'), and 
   S(D, B'C') have a point in common.