7th Irish 1994


A1. m, n are positive integers with n > 3 and m² + n⁴ = 2(m-6)² + 2(n+1)². Prove that m² + n⁴ = 1994.
A2. B is an arbitrary point on the segment AC. Equilateral triangles are drawn as shown. Show that their centers form an equilateral triangle whose center lies on AC.
A3. Find all real polynomials p(x) satisfying p(x²) = p(x)p(x-1) for all x.
A4. An n x n array of integers has each entry 0 or 1. Find the number of arrays with an even number of 1s in every row and column.
A5. The sequence a₁, a₂, a₃, ... is defined by a₁ = 2, a_n+1 = a_n² - a_n + 1. Show that 1/a₁ + 1/a₂ + ... + 1/a_n lies in the interval (1-½^N, 1-½^2N), where N = 2^n-1.
B1. The sequence x₁, x₂, x₃, ... is defined by x₁ = 2, nx_n = 2(2n-1)x_n-1. Show that every term is integral.
B2. p, q, r are distinct reals such that q = p(4-p), r = q(4-q), p = r(4-r). Find all possible values of p+q+r.
B3. Prove that n( (n+1)^2/n - 1) < ∑₁ⁿ (2i+1)/i² < n(1 - 1/n^2/(n-1) ) + 4.
B4. w, a, b, c are distinct real numbers such that the equations: x + y + z = 1 xa² + yb² + zc² = w² xa³ + yb³ + zc³ = w³ xa⁴ + yb⁴ + zc⁴ = w⁴ have a real solution x, y, z. Express w in terms of a, b, c.
B5. A square is partitioned into n convex polygons. Find the maximum number of edges in the resulting figure. You may assume Euler's formula for a polyhedron: V + F = E + 2, where V is the no. of vertices, F the no. of faces and E the no. of edges.

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