(a) For which positive integers $n$ is $\underbrace{11\ldots1}_{2n\text{ digits}}-\underbrace{77\ldots7}_{n\text{ digits}}$ a perfect square?
(b) Determine all digits $b$ for which $\underbrace{11\ldots1}_{2n\text{ digits}}-\underbrace{bb\ldots b}_{n\text{ digits}}$ is a perfect square.
(a) How many consecutive integer pairs $x,x+1$ have $x$ a multiple of $9$ and $x+1$ a multiple of $25$? Find them.
(b) How many pairs $(x,x+1)$ have $x$ a multiple of $21$ and $x+1$ a multiple of $165$?
(c) Find all consecutive integer triples $(x,x+1,x+2)$ such that $x$ is a multiple of $9$, $x+1$ a multiple of $25$, and $x+2$ a multiple of $4$.
A fixed triangle $ABC$ is right-angled at $A$, with altitude $AH$. From $H$ draw perpendiculars $HP,HQ$ to $AB,AC$, respectively. Let $M$ move arbitrarily on the line $PQ$. The line through $M$ perpendicular to $MH$ meets the lines $AB,AC$ at $R,S$.
(a) What observations can you make about the circle through $A,R,S$? Prove them.
(b) Two positions $M_1,M_2$ give $R_1,R_2$ and $S_1,S_2$. Prove that $R_1R_2/S_1S_2$ is constant. Under what condition on $ABC$ do $R,S$ move with equal speeds when $M$ moves on $PQ$?
(c) Let $K$ be the reflection of $H$ in the point $M$. The line through $A$ perpendicular to $PQ$ meets $RS$ at $D$. Prove that $\angle BHR=\angle DHR$. Find another analogous equality.
Editorial scope note: The printed arbitrary-position construction and the printed claim in (c) require scrutiny; the source wording is retained.
In space, consider the $12$ lines containing the $12$ edges of a cube of unit side. Intersect these lines with a plane $P$. Study the intersections of $P$ with the $12$ lines, discussing all possible positions of $P$.