MathDB

1

Part of 2006 Turkey MO (2nd round)

Problems(2)

Show that D, C, K, L lie on a circle

Source: Turkey National Olympiad 2006 - D1 - P1

1/6/2007
Points PP and QQ on side ABAB of a convex quadrilateral ABCDABCD are given such that AP=BQ.AP = BQ. The circumcircles of triangles APDAPD and BQDBQD meet again at KK and those of APCAPC and BQCBQC meet again at LL. Show that the points D,C,K,LD,C,K,L lie on a circle.
geometrycircumcirclepower of a pointradical axiscyclic quadrilateralgeometry unsolved
turkey nmo 2006 q4

Source:

1/6/2007
x1,...,xnx_{1},...,x_{n} are positive reals such that their sum and their squares' sum are equal to tt. Prove that ∑i≠jxixj≥(n−1)2⋅tt−1\sum_{i\neq{j}}\frac{x_{i}}{x_{j}}\ge\frac{(n-1)^{2}\cdot{t}}{t-1}
inequalities unsolvedinequalities