VMO 1974

Find all positive integers $n$ and $b$ with $0 < b < 10$ such that if $a_n$ is the positive integer with $n$ digits, all of them $1$ , then $a_{2n} - b a_n$ is a square.

i) How many integers $n$ are there such that $n$ is divisible by $9$ and $n+1$ is divisible by $25$ ? ii) How many integers $n$ are there such that $n$ is divisible by $21$ and $n+1$ is divisible by $165$ ? iii) How many integers $n$ are there such that $n$ is divisible by $9, n + 1$ is divisible by $25$ , and $n + 2$ is divisible by $4$ ?

Let $ABC$ be a triangle with $A = 90^o, AH$ the altitude, $P,Q$ the feet of the perpendiculars from $H$ to $AB,AC$ respectively. Let $M$ be a variable point on the line $PQ$ . The line through $M$ perpendicular to $MH$ meets the lines $AB,AC$ at $R, S$ respectively. i) Prove that circumcircle of $ARS$ always passes the fixed point $H$ . ii) Let $M_1$ be another position of $M$ with corresponding points $R_1, S_1$ . Prove that the ratio $RR_1/SS_1$ is constant. iii) The point $K$ is symmetric to $H$ with respect to $M$ . The line through $K$ perpendicular to the line $PQ$ meets the line $RS$ at $D$ . Prove that $\angle BHR = \angle DHR, \angle DHS = \angle CHS$ .

$C$ is a cube side $1$ . The $12$ lines containing the sides of the cube meet at plane $p$ in $12$ points. What can you say about the $12$ points?