6th Irish 1993


A1. The real numbers x, y satisfy x³ - 3x² + 5x - 17 = 0, y³ - 3y² + 5y + 11 = 0. Find x + y.
A2. Find which positive integers can be written as the sum and product of the same sequence of two or more positive integers. (For example 10 = 5+2+1+1+1 = 5·2·1·1·1).
A3. A triangle ABC has fixed incircle. BC touches the incircle at the fixed point P. B and C are varied so that PB·PC is constant. Find the locus of A.
A4. The polynomial xⁿ + a_n-1x^n-1 + ... + a₀ has real coefficients. All its roots are real and lie in the interval (0, 1). Also f(1) = \|f(0)\|. Show that the product of the roots does not exceed 1/2ⁿ.
A5. The points z₁, z₂, z₃, z₄, z₅ form a convex pentagon P in the complex plane. The origin and the points αz₁, ... , αz₅ all lie inside the pentagon. Show that \|α\| ≤ 1 and Re(α) + Im(α) tan(π/5) ≤ 1.
B1. Given 5 lattice points in the plane, show that at least one pair of points has a distinct lattice point on the segment joining them.
B2. a₁, a₂, ... , a_n are distinct reals. b₁, b₂, ... , b_n are reals. There is a real number α such that ∏_1≤k≤n (a_i + b_k) = α for i = 1, 2, ... , n. Show that there is a real β such that ∏_1≤k≤n (a_k + b_j) = β for j = 1, 2, ... , n.
B3. Given positive integers r ≤ n, show that ∑_d (n-r+1)Cd (r-1)C(d-1) = nCr, where nCr denotes the usual binomial coefficient and the sum is taken over all positive d ≤ n-r+1 and ≤ r.
B4. Show that sin x + (sin 3x)/3 + (sin 5x)/5 + ... + (sin(2n-1)x)/(2n-1) > 0 for all x in (0, π).
B5. An m x n rectangle is divided into unit squares. Show that a diagonal of the rectangle intersects m + n - gcd(m,n) of the squares. An a x b x c box is divided into unit cubes. How many cubes does a long diagonal of the box intersect?

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