IrMO 1990

The real number $x$ satisfies all the inequalities $$2^k<x^k+x^{k+1}<2^{k+1}$$ for $k=1,2,\dots ,n$ . What is the greatest possible value of $n$ ?

Let $ABC$ be a right-angled triangle with right-angle at $A$ . Let $X$ be the foot of the perpendicular from $A$ to $BC$ , and $Y$ the mid-point of $XC$ . Let $AB$ be extended to $D$ so that $|AB|=|BD|$ . Prove that $DX$ is perpendicular to $AY$ .

Suppose that $p_1<p_2<\dots <p_{15}$ are prime numbers in arithmetic progression, with common difference $d$ . Prove that $d$ is divisible by $2,3,5,7,11$ and $13$ .

Let $t$ be a real number, and let $$a_n=2\cos \left(\frac{t}{2^n}\right)-1, n=1,2,3,\dots$$ Let $b_n$ be the product $a_1a_2a_3\cdots a_n$ . Find a formula for $b_n$ that does not involve a product of $n$ terms, and deduce that $$\lim_{n\to \infty}b_n=\frac{2\cos t+1}{3}$$

Let $n=2k-1$ , where $k\ge 6$ is an integer. Let $T$ be the set of all $n$ -tuples $$\textbf{x}=(x_1,x_2,\dots ,x_n), \text{ where, for } i=1,2,\dots ,n, \text{ } x_i \text{ is } 0 \text{ or } 1.$$ For $\textbf{x}=(x_1,x_2,\dots ,x_n)$ and $\textbf{y}=(y_1,y_2,\dots ,y_n)$ in $T$ , let $d(\textbf{x},\textbf{y})$ denote the number of integers $j$ with $1\le j\le n$ such that $x_j\neq x_y$ . $($ In particular, $d(\textbf{x},\textbf{x})=0)$ .

Suppose that there exists a subset $S$ of $T$ with $2^k$ elements which has the following property: given any element $\textbf{x}$ in $T$ , there is a unique $\textbf{y}$ in $S$ with $d(\textbf{x},\textbf{y})\le 3$ .

Prove that $n=23$ .