IrMO 1991

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 $N$ should include for each $K<N$ , $(K\ge 1)$ at least one member who was present on day $K$ .

For how many days can the parliament sit before one of the rules is broken?

Let $$a_n=\frac{n^2+1}{\sqrt{n^4+4}}, n=1,2,3,\dots$$ and let $b_n$ be the product of $a_1,a_2,a_3,\dots ,a_n$ . Prove that $$\frac{b_n}{\sqrt{2}}=\frac{\sqrt{n^2+1}}{\sqrt{n^2+2n+2}},$$ and deduce that $$\frac{1}{n^3+1}<\frac{b_n}{\sqrt{2}}-\frac{n}{n+1}<\frac{1}{n^3}$$ for all positive integers $n$ .

Let $\mathbb{P}$ be the set of positive rational numbers and let $f:\mathbb{P}\to\mathbb{P}$ be such that $$f(x)+f\left(\frac{1}{x}\right)=1$$ and $$f(2x)=2f(f(x))$$ for all $x\in\mathbb{P}$ . Find, with proof, an explicit expression for $f(x)$ for all $x\in \mathbb{P}$ .

Let $\mathbb{Q}$ denote the set of rational numbers. A nonempty subset $S$ of $\mathbb{Q}$ has the following properties:

(a) $0$ is not in $S$ ;

(b) for each $s_1,s_2$ in $S$ , the rational number $s_1/s_2$ is in $S$ ;

(c) there exists a nonzero number $q\in \mathbb{Q} \backslash S$ that has the property that every nonzero number in $\mathbb{Q} \backslash S$ is of the form $qs$ for some $s$ in $S$ .

Prove that if $x$ belongs to $S$ , then there exists elements $y,z$ in $S$ such that $x=y+z$ .