MathDB

2

Part of 2000 Federal Competition For Advanced Students, Part 2

Problems(2)

Show that points P,Q,X, Y lie on a circle

Source: Austrian Mathematical Olympiad 2000, Part 2, D1, P2

6/28/2011
A trapezoid ABCDABCD with AB∥CDAB \parallel CD is inscribed in a circle kk. Points PP and QQ are chose on the arc ADCBADCB in the order A−P−Q−BA-P -Q-B. Lines CPCP and AQAQ meet at XX, and lines BPBP and DQDQ meet at YY. Show that points P,Q,X,YP,Q,X, Y lie on a circle.
geometrytrapezoidgeometry proposed
On the equation |(m^2+2000m+999999) - (3n^3+9n^2+27n) |=1

Source: Austrian Mathematical Olympiad 2000, Part 2, D2, P2

6/28/2011
Find all pairs of integers (m,n)(m, n) such that \left| (m^2 + 2000m+ 999999)- (3n^3 + 9n^2 + 27n) \right|= 1.
number theory proposednumber theory