11th Irish 1998


A1. Show that x⁸ - x⁵ - 1/x + 1/x⁴ ≥ 0 for all x ≠ 0.
A2. P is a point inside an equilateral triangle. Its distances from the vertices are 3, 4, 5. Find the area of the triangle.
A3. Show that the 4 digit number mnmn cannot be a cube in base 10. Find the smallest base b > 1 for which it can be a cube.
A4. Show that 7 disks radius 1 can be arranged to cover a disk radius 2.
A5. x is real and xⁿ - x is an integer for n = 2 and some n > 2. Show that x must be an integer.
B1. Find all positive integers n with exactly 16 positive divisors 1 = d₁ < d₂ < ... < d₁₆ = n such that d₆ = 18 and d₉ - d₈ = 17.
B2. Show that for positive reals a, b, c we have 9/(a+b+c) ≤ 2/(a+b) + 2/(b+c) + 2/(c+a) ≤ 1/a + 1/b + 1/c.
B3. Let N be the set of positive integers. Show that we can partition N into three disjoint parts such that if \|m-n\| = 2 or 5, then m and n are in different parts. Show that we can partition N into four disjoint parts such that if \|m-n\| = 2, 3 or 5, then m and n are in different parts, but that this is not possible with only three disjoint parts.
B4. The sequence x₀, x₁, x₂, ... is defined by x₀ = a, x₁ = b, x_n+2 = (1+x_n+1)/x_n. Find x₁₉₉₈.
B5. Find the smallest possible perimeter for a triangle ABC with integer sides such that ∠A = 2∠B and ∠C > 90^o.

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