14th Irish 2001


A1. Find all solutions to a! + b! + c! = 2ⁿ.
A2. ABC is a triangle. Show that the medians BD and CE are perpendicular iff b² + c² = 5a².
A3. p is an odd prime which can be written as a difference of two fifth powers. Show that √( (4p+1)/5 ) = (n²+1)/2 for some odd integer n.
A4. Show that 2n/(3n+1) ≤ ∑_n<k≤2n 1/k ≤ (3n+1)/(4n+4).
A5. Show that ( a²b²(a+b)²/4 )^1/3 ≤ (a²+10ab+b²)/12 for all reals a, b such that ab > 0. When do we have equality? Find all real numbers a, b for which ( a²b²(a+b)²/4 )^1/3 ≤ (a²+ab+b²)/3.
B1. Find the smallest positive integer m for which 55ⁿ + m32ⁿ is a multiple of 2001 for some odd n.
B2. Three circles each have 10 black beads and 10 white beads randomly arranged on them. Show that we can always rotate the beads around the circles so that in 5 corresponding positions the beads have the same color.
B3. P is a point on the altitude AD of the triangle ABC. The lines BP, CP meet CA, AB at E, F respectively. Show that AD bisects ∠EDF.
B4. Find all non-negative reals for which (13 + √x)^1/3 + (13 - √x)^1/3 is an integer.
B5. Let N be the set of positive integers. Find all functions f : N → N such that f(m + f(n)) = f(m) + n for all m, n.

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