28th Vietnam 1990 problems

A1.  -1 < a < 1. The sequence x1, x2, x3, ... is defined by x1 = a, xn+1 = ( √(3 - 3xn2) - xn)/2. Find necessary and sufficient conditions on a for all members of the sequence to be positive. Show that the sequence is always periodic.
A2.  n-1 or more numbers are removed from {1, 2, ... , 2n-1} so that if a is removed, so is 2a and if a and b are removed, so is a + b. What is the largest possible sum for the remaining numbers?
A3.  ABCD is a tetrahedron with volume V. We wish to make three plane cuts to give a parallelepiped three of whose faces and all of whose vertices belong to the surface of the tetrahedron. Find the intersection of the three planes if the volume of the parallelepiped is 11V/50. Can it be done so that the volume of the parallelepiped is 9V/40?
B1.  ABC is a triangle. P is a variable point. The feet of the perpendiculars from P to the lines BC, CA, AB are A', B', C' respectively. Find the locus of P such that PA PA' = PB PB' = PC PC'.
B2.  The polynomial p(x) with degree at least 1 satisfies p(x) p(2x2) = p(3x3 + x). Show that it does not have any real roots.
B3.  Some children are sitting in a circle. Each has an even number of tokens (possibly zero). A child gives half his tokens to the child on his right. Then the child on his right does the same and so on. If a child about to give tokens has an odd number, then the teacher gives him an extra token. Show that after several steps, all the children will have the same number of tokens, except one who has twice the number.

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

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© John Scholes
23 July 2002