MathDB

3

Part of 2012 Czech-Polish-Slovak Junior Match

Problems(2)

projections of point on circumcirle on radii form equilateral of fixed area

Source: Czech-Polish-Slovak Junior Match 2012, Individual p3 CPSJ

3/11/2020
Different points A,B,C,DA, B, C, D lie on a circle with a center at the point OO at such way that ∠AOB\angle AOB =∠BOC== \angle BOC = ∠COD=\angle COD = 60o60^o. Point PP lies on the shorter arc BCBC of this circle. Points K,L,MK, L, M are projections of PP on lines AO,BO,COAO, BO, CO respectively . Show that (a) the triangle KLMKLM is equilateral, (b) the area of triangle KLMKLM does not depend on the choice of the position of point PP on the shorter arc BCBC
circumcircleEquilateralfixedarea of a trianglegeometryprojections
2(n^2 +1)-n not a perfect square

Source: Czech-Polish-Slovak Junior Match 2012, Team p3 CPSJ

3/12/2020
Prove that if nn is a positive integer then 2(n2+1)−n2 (n^2 + 1) - n is not a square of an integer.
Perfect Squarenumber theory