13th Irish 2000


A1. Let S be the set of all numbers of the form n²+n+1. Show that the product of n²+n+1 and (n+1)²+(n+1)+1 is in S, but give an example of a, b ∈ S with ab ∉ S.
A2. ABCDE is a regular pentagon side 1. F is the midpoint of AB. G, H are points on DC, DE respectively such that ∠DFG = ∠DFH = 30^o. Show that FGH is equilateral and GH = 2 cos 18^o cos²36^o/cos 6^o. A square is inscribed in FGH with one side on GH. Show that its side has length GH√3/(2+√3).
A3. Let f(n) = 5n¹³ + 13n⁵ + 9an. Find the smallest positive integer a such that f(n) is divisible by 65 for every integer n.
A4. A strictly increasing sequence a₁ < a₂ < ... < a_M is called a weak AP if we can find an arithmetic progression x₀, x₁, ... , x_M such that x_n-1 ≤ a_n < x_n for n = 1, 2, ... , M. Show that any strictly increasing sequence of length 3 is a weak AP. Show that any subset of {0, 1, 2, ... , 999} with 730 members has a weak AP of length 10.
A5. Let y = x² + 2px + q be a parabola which meets the x- and y-axes in three distinct points. Let C_pq be the circle through these points. Show that all circles C_pq pass through a common point.
B1. Show that x²y²(x²+y²) ≤ 2 for positive reals x, y such that x+y = 2.
B2. ABCD is a cyclic quadrilateral with circumradius R, side lengths a, b, c, d and area S. Show that 16R²S² = (ab+cd)(ac+bd)(ad+bc). Deduce that RS√2 ≥ (abcd)^3/4 with equality iff ABCD is a square.
B3. For each positive integer n, find all positive integers m which can be written as 1/a₁ + 2/a₂ + ... + n/a_n for some positive integers a₁ < a₂ < ... < a_n.
B4. Show that in any set of 10 consecutive integers there is one which is relatively prime to each of the others.
B5. p(x) is a plynomial with non-negative real coefficients such that p(4) = 2, p(16) = 8. Show that p(8) ≤ 4 and find all polynomials where equality holds.

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