A triangle with sides $a,b,c$ is given. Denote its semiperimeter by $s=(a+b+c)/2$. Construct a triangle with sides $s-a,s-b,s-c$. Repeat this process until a triangle can no longer be constructed with the given side lengths. For which original triangles can this process be repeated indefinitely?
In a circle $C$ with centre $O$ and radius $r$, let $C_1,C_2$ have centres $O_1,O_2$ and radii $r_1,r_2$. Each $C_i$ is internally tangent to $C$ at $A_i$, and $C_1,C_2$ are externally tangent at $A$. Prove that the lines $OA,O_1A_2,O_2A_1$ are concurrent.
Editorial qualification (AI): assume $A\ne O$, so the line $OA$ is defined. The printed problem omits this qualification; the exceptional configuration is explained below.
Let $n\gt 3$ be an integer. Choose three numbers from $\{1,2,\ldots,n\}$. Using each chosen number once and using addition, multiplication and parentheses, form all possible combinations.
(a) Show that if all three numbers exceed $n/2$, the values of these combinations are all distinct.
(b) Let $p\le\sqrt n$ be prime. Show that the number of ways to choose three numbers with smallest number $p$ for which the combination values are not all distinct equals the number of positive divisors of $p-1$.
Editorial clarification (AI): expressions differing only by commutativity or associativity are counted as the same combination.
Determine all pairs $(h,s)$ of positive integers with the following property. If one draws $h$ horizontal lines and another $s$ lines such that (i) they are not horizontal, (ii) no two of them are parallel, and (iii) no three of the $h+s$ lines are concurrent, then the number of regions formed is $1992$.
The diagram is a small schematic example of the line conditions.
Find a sequence of maximal length consisting of nonzero integers in which the sum of any seven consecutive terms is positive and the sum of any eleven consecutive terms is negative.
