10th Irish 1997


A1. Find all integer solutions to 1 + 1996m + 1998n = mn.
A2. ABC is an equilateral triangle. M is a point inside the triangle. D, E, F are the feet of the perpendiculars from M to BC, CA, AB. Find the locus of M such that ∠FDE = 90^o.
A3. Find all polynomials p(x) such that (x-16) p(2x) = (16x-16) p(x).
A4. a, b, c are non-negative reals such that a + b + c ≥ abc. Show that a² + b² + c² ≥ abc.
A5. Let S denote the set of odd integers > 1. For x∈S, define f(x) to be the largest integer such that 2^f(x) < x. For a, b ∈ S define a * b = a + 2^f(a)-1(b-3). For example, f(5) = 2, so 5 * 7 = 5 + 2(7-3) = 13. Similarly, f(7) = 2, so 7 * 5 = 7 + 2(5-3) = 11. Show that a * b is always an odd integer > 1 and that the operation * is associative.
B1. Let σ(n) denote the sum of the positive divisors of n. Show that if σ(n) > 2n, then σ(mn) > 2mn for any m.
B2. The quadrilateral ABCD has an inscribed circle. ∠A = ∠B = 120^o, ∠C = 30^o and BC = 1. Find AD.
B3. A subset of {0, 1, 2, ... , 1997} has more than 1000 elements. Show that it must contain a power of 2 or two distinct elements whose sum is a power of 2.
B4. How many 1000-digit positive integers have all digits odd, and are such that any two adjacent digits differ by 2?
B5. p is an odd prime. We say n satisfies K_p if the set {1, 2, ... , n} can be partitioned into p disjoint parts, such that the sum of the elements in each part is the same. For example, 5 satisfies K₃ because {1, 2, 3, 4, 5} = {1, 4} &cup {2, 3} ∪ {5}. Show that if n satisfies K_p, then n or n+1 is a multiple of p. Show that if n is a multiple of 2p, then n satisfies K_p.

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