Asian Pacific Mathematics Olympiad · 1992

Problems

Problem 1A triangle with sides a,b,c is given. Let s=(a+b+c)/2 be its semiperimeter. Construct a triangle with sides s-a,s-b,s-c, and repeat this process while possible. For which original triangles can the process be repeated indefinitely?Solutions: 1
Problem 2In 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 $OA$ , $O_1A_2$ , and $O_2A_1$ are concurrent.Solutions: 1
Problem 3Let $n>3$ be an integer. Choose three numbers from $\{1,2,\ldots,n\}$ . Using each once, together with addition, multiplication, and parentheses, form all possible combinations. (a) Show that if all three chosen numbers are greater than $n/2$ , their values are all distinct. (b) Let $p$ be a prime with $p\le \sqrt{n}$ . Show that the number of choices whose smallest number is $p$ and whose combination values are not all distinct is exactly the number of positive divisors of $p-1$ .Solutions: 1
Problem 4Determine all pairs $(h,s)$ of positive integers with the following property: if one draws $h$ horizontal lines and another $s$ lines satisfying (i) they are not horizontal, (ii) no two are parallel, and (iii) no three of the $h+s$ lines are concurrent, then the number of regions formed is $1992$ .Solutions: 1
Problem 5Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.Solutions: 1