17th Vietnam 1979 problems

A1.  Show that for all x > 1 there is a triangle with sides, x4 + x3 + 2x2 + x + 1, 2x3 + x2 + 2x + 1, x4 - 1.
A2.  Find all real numbers a, b, c such that x3 + ax2 + bx + c has three real roots α,β,γ (not necessarily all distinct) and the equation x3 + a3x2 + b3x + c3 has roots α3, β3, γ3.
A3.  ABC is a triangle. Find a point X on BC such that area ABX/area ACX = perimeter ABX/perimeter ACX.
B1.  For each integer n > 0 show that there is a polynomial p(x) such that p(2 cos x) = 2 cos nx.
B2.  Find all real numbers k such that x2 - 2 x [x] + x - k = 0 has at least two non-negative roots.
B3.  ABCD is a rectangle with BC/AB = √2. ABEF is a congruent rectangle in a different plane. Find the angle DAF such that the lines CA and BF are perpendicular. In this configuration, find two points on the line CA and two points on the line BF so that the four points form a regular tetrahedron.

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