MathDB

1

Part of 1989 Irish Math Olympiad

Problems(2)

Inequality on a Quadrilateral

Source: 1989 IrMO Paper 1 Problem 1

9/29/2017
A quadrilateral ABCDABCD is inscribed, as shown, in a square of area one unit. Prove that 2≤∣AB∣2+∣BC∣2+∣CD∣2+∣DA∣2≤42\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4 [asy] size(6cm); draw((0,0)--(10,0)); draw((10,0)--(10,10)); draw((0,10)--(10,10)); draw((0,0)--(0,10));
dot((0,8.5)); dot((3.5,10)); dot((10,3.5)); dot((3.5,0)); label("DD",(0,8.5),W); label("AA",(3.5,10),NE); label("BB",(10,3.5),E); label("CC",(3.5,0),S); draw((0,8.5)--(3.5,10)); draw((3.5,10)--(10,3.5)); draw((10,3.5)--(3.5,0)); draw((3.5,0)--(0,8.5)); [/asy]
geometryinequalities
Locus of Q as P Varies Along L

Source: 1989 IrMO Paper 2 Problem 1

9/29/2017
Suppose LL is a fixed line, and AA is a fixed point not on LL. Let kk be a fixed nonzero real number. For PP a point on LL, let QQ be a point on the line APAP with ∣AP∣⋅∣AQ∣=k2|AP|\cdot |AQ|=k^2. Determine the locus of QQ as PP varies along the line LL.
geometry