VMO 1999

Find all real solutions to (1 + $4^{2x-y}$)($5^{y-2x+1}$) = $2^{2x-y+1}$ + 1, $y^3$ + 4x + ln($y^2$ + 2x) + 1 = 0.

let a triangle ABC and A',B',C' be the midpoints of the arcs BC,CA,AB respectively of its circumcircle. A'B',A'C' meets BC at $A_1,A_2$ respectively. Pairs of point $(B_1,B_2),(C_1,C_2)$ are similarly defined. Prove that $A_1A_2 = B_1B_2 = C_1C_2$ if and only if triangle ABC is equilateral.

Let $S = \{0,1,2,\ldots,1999\}$ and $T = \{0,1,2,\ldots \}.$ Find all functions $f: T \mapsto S$ such that

(i) $f(s) = s \forall s \in S.$ (ii) $f(m+n) = f(f(m)+f(n)) \forall m,n \in T.$

Find the maximum value of 2/($x^2$ + 1) - 2/($y^2$ + 1) + 3/($z^2$ + 1) for positive reals x, y, z which satisfy xyz + x + z = y.

OA, OB, OC, OD are 4 rays in space such that the angle between any two is the same. Show that for a variable ray OX, the sum of the cosines of the angles XOA, XOB, XOC, XOD is constant and the sum of the squares of the cosines is also constant.

Find all functions f(n) defined on the non-negative integers with values in the set {0, 1, 2, ... , 2000} such that: (1) f(n) = n for 0 ≤ n ≤ 2000; and (2) f( f(m) + f(n) ) = f(m + n) for all m, n.