34th Vietnamese Mathematical Olympiad 1996 Problems

34th Vietnamese Mathematical Olympiad 1996 Problems

A1.  Find all real x, y such that √(3x) (1 + 1/(x + y) ) = 2 and √(7y) (1 - 1/(x + y) ) = 4√2.

A2.  SABC is a tetrahedron. DAB, EBC, FCA are triangles in the plane of ABC congruent to SAB, SBC, SCA respectively. O is the circumcenter of DEF. Let K be the exsphere of SABC opposite O (which touches the planes SAB, SBC, SCA, ABC, lies on the opposite side of ABC to S, but on the same side of SAB as C, the same side of SBC as A, and the same side of SCA as B). Show that K touches the plane ABC at O.

A3.  Let n be a positive integer and k a positive integer not greater than n. Find the number of ordered k-tuples (a₁, a₂, ... , a_n) such that: (1) all a_i are different, but all belong to {1, 2, ... , n}; (2) a_r > a_s for some r < s; (3) a_r has the opposite parity to r for some r.

B1.  Find all functions f(n) on the positive integers with positive integer values, such that f(n) + f(n+1) = f(n+2) f(n+3) - 1996 for all n.

B2.  The triangle ABC has BC = 1 and angle A = x. Let f(x) be the shortest possible distance between its incenter and its centroid. Find f(x). What is the largest value of f(x) for 60^o < x < 180^o?

B3.  Let w, x, y, z be non-negative reals such that 2(wx + wy + wz + xy + xz + yz) + wxy + xyz + yzw + zwx = 16. Show that 3(w + x + y + z) ≥ 2(wx + wy + wz + xy + xz + yz).