1

Part of 1991 Romania Team Selection Test

Problems(2)

Source: Romanian TST for BMO 1991

a,b

are positive integers for which A\equal{}\frac{a\plus{}1}{b}\plus{}\frac{b}{a} is an integer.Prove that A\equal{}3.

quadraticsnumber theory proposednumber theory

Source: Laurentiu Panaitopol,Romania, TST 1991

M=\{A_{1},A_{2},\ldots,A_{5}\}

be a set of five points in the plane such that the area of each triangle

A_{i}A_{j}A_{k}

, is greater than 3. Prove that there exists a triangle with vertices in

M

and having the area greater than 4. Laurentiu Panaitopol

geometrygeometry proposed