4th Irish 1991


A1. Given three points X, Y, Z, show how to construct a triangle ABC which has circumcenter X, Y the midpoint of BC and BZ an altitude.
A2. Find all polynomials p(x) of degree ≤ n which satisfy p(x²) = p(x)² for all real x.
A3. For any positive integer n, define f(n) = 10n, g(n) = 10n+4, and for any even positive integer n, define h(n) = n/2. Show that starting from 4 we can reach any positive integer by some finite sequence of the operations f, g, h.
A4. 8 people decide to hold daily meetings subject to the following rules. At least one person must attend each day. A different set of people must attend on different days. On day N for each 1 ≤ k < N, at least one person must attend who was present on day k. How many days can the meetings be held?
A5. Find all polynomials xⁿ + a₁x^n-1 + ... + a_n such that each of the coefficients a₁, a₂, ... , a_n is ±1 and all the roots are real.
B1. Prove that the sum of m consecutive squares cannot be a square for m = 3, 4, 5, 6. Give an example of 11 consecutive squares whose sum is a square.
B2. Define a_n = (n² + 1)/√(n⁴ + 4) for n = 1, 2, 3, ... , and let b_n = a₁a₂ ... a_n. Show that b_n = (√2 √(n²+1) )/√(n²+2n+2), and hence that 1/(n+1)³ < b_n/√2 - n/(n+1) < 1/n³.
B3. ABC is a triangle and L is the line through C parallel to AB. The angle bisector of A meets BC at D and L at E. The angle bisector of B meets AC at F and L at G. If DE = FG show that CA = CB.
B4. P is the set of positive rationals. Find all functions f:P→P such that f(x) + f(1/x) = 1 and f(2x) = 2 f( f(x) ) for all x.
B5. A non-empty subset S of the rationals satisfies: (1) 0 ∉ S; (2) if a, b ∈ S, then a/b ∈ S; (3) there is a non-zero rational q not in S such that if s is a non-zero rational not in S, then s = qt for some t ∈ S. Show that every element of S is a sum of two elements of S.

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