MathDB

4

Part of 1991 Irish Math Olympiad

Problems(2)

Eight Politicians Attending Parliament

Source: 1991 IrMO Paper 1 Problem 4

10/1/2017
Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament. They decided on the following rules of attendance:
(a) There should always be at least one person present on each day.
(b) On no two days should the same subset attend.
(c) The members present on day NN should include for each K<NK<N, (K≥1)(K\ge 1) at least one member who was present on day KK.
For how many days can the parliament sit before one of the rules is broken?
combinatorics
Function Equation on Positive Rationals

Source: 1991 IrMO Paper 2 Problem 4

10/1/2017
Let P\mathbb{P} be the set of positive rational numbers and let f:P→Pf:\mathbb{P}\to\mathbb{P} be such that f(x)+f(1x)=1f(x)+f\left(\frac{1}{x}\right)=1 and f(2x)=2f(f(x))f(2x)=2f(f(x)) for all x∈Px\in\mathbb{P}. Find, with proof, an explicit expression for f(x)f(x) for all x∈Px\in \mathbb{P}.
algebrafunctional equationfunctional equation in Q