MathDB

3

Part of 2000 All-Russian Olympiad

Problems(3)

K is the circumcenter of triangle AOC

Source: All-Russian MO 2000

12/30/2012
Let OO be the center of the circumcircle ω\omega of an acute-angle triangle ABCABC. A circle ω1\omega_1 with center KK passes through AA, OO, CC and intersects ABAB at MM and BCBC at NN. Point LL is symmetric to KK with respect to line NMNM. Prove that BL⊥ACBL \perp AC.
geometrycircumcircleparallelogramAsymptoteperpendicular bisector
P,B,Q,R lie on a circle

Source: Russia 2000 10.3

12/25/2009
In an acute scalene triangle ABCABC the bisector of the acute angle between the altitudes AA1AA_1 and CC1CC_1 meets the sides ABAB and BCBC at PP and QQ respectively. The bisector of the angle BB intersects the segment joining the orthocenter of ABCABC and the midpoint of ACAC at point RR. Prove that PP, BB, QQ, RR lie on a circle.
geometryparallelogramtrigonometrygeometric transformationreflectioncircumcircleangle bisector
Convex Pentagon in a lattice

Source: All-Russian MO 2000

12/30/2012
A convex pentagon ABCDEABCDE is given in the coordinate plane with all vertices in lattice points. Prove that there must be at least one lattice point in the pentagon determined by the diagonals ACAC, BDBD, CECE, DADA, EBEB or on its boundary.
analytic geometrygeometry unsolvedgeometry