MathDB

2

Part of 2013 Dutch IMO TST

Problems(2)

(4n-2)/(n+5) is squre of a rational

Source: Dutch IMO TST 2013 day 1 p2

8/29/2019
Determine all integers nn for which 4n−2n+5\frac{4n-2}{n+5} is the square of a rational number.
Perfect Squaresnumber theoryrational
Convex quadrilaterals and equal ratios

Source: Netherlands IMO Team Selection Test 2013

10/21/2014
Let PP be the point of intersection of the diagonals of a convex quadrilateral ABCDABCD.Let X,Y,ZX,Y,Z be points on the interior of AB,BC,CDAB,BC,CD respectively such that AXXB=BYYC=CZZD=2\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2. Suppose that XYXY is tangent to the circumcircle of △CYZ\triangle CYZ and that YZY Z is tangent to the circumcircle of △BXY\triangle BXY.Show that ∠APD=∠XYZ\angle APD=\angle XYZ.
ratiogeometrycircumcirclegeometry unsolved