9th Irish 1996


A1. Find gcd(n!+1, (n+1)!).
A2. Let s(n) denote the sum of the digits of n. Show that s(2n) ≤ 2s(n) ≤ 10s(2n) and that there is a k such that s(k) = 1996 s(3k).
A3. R denotes the reals. f : [0,1] → R satisfies f(1) = 1, f(x) ≥ 0 for all x∈[0,1], and if x, y, x+y all ∈[0,1], then f(x+y) ≥ f(x) + f(y). Show that f(x) ≤ 2x for all x∈[0,1].
A4. ABC is any triangle. D, E are constructed as shown so that ABD and ACE are right-angled isosceles triangles, and F is the midpoint of BC. Show that DEF is a right-angled isosceles triangle.
A5. Show how to dissect a square into at most 5 pieces so that the pieces can be reassembled to form three squares all of different size.
B1. The Fibonacci sequence F₀, F₁, F₂, ... is defined by F₀ = 0, F₁ = 1, F_n+2 = F_n+1 + F_n. Show that F_n+60 - F_n is divisible by 10 for all n, but for any 1 ≤ k < 60 there is some n such that F_n+k - F_n is not divisible by 10. Similarly, show that F_n+300 - F_n is divisible by 100 for all n, but for any 1 ≤ k < 300 there is some n such that F_n+k - F_n is not divisible by 100.
B2. Show that 2^1/24^1/48^1/8 ... (2ⁿ)^1/2ⁿ < 4.
B3. If p is a prime, show that 2^p + 3^p cannot be an nth power (for n > 1).
B4. ABC is an acute-angled triangle. The altitudes are AD, BE, CF. The feet of the perpendiculars from A, B, C to EF, FD, DE respectively are P, Q, R. Show that AP, BQ, CR are concurrent.
B5. 33 disks are placed on a 5 x 9 board, at most one disk per square. At each step every disk is moved once so that after the step there is at most one disk per square. Each disk is moved alternately one square up/down and one square left/right. So a particular disk might be moved L,U,L,D,L,D,R,U ... in successive steps. Prove that only finitely many steps are possible. Show that with 32 disks it is possible to have infinitely many steps.

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