MathDB

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Part of 2022 Oral Moscow Geometry Olympiad

Problems(2)

circumcenter of BLD lies on circumcircle of trapezoid ABCD

Source: 2022 Oral Moscow Geometry Olympiad grades 8-9 p1

4/17/2022
Given an isosceles trapezoid ABCDABCD. The bisector of angle BB intersects the base ADAD at point LL. Prove that the center of the circle circumscribed around triangle BLDBLD lies on the circle circumscribed around the trapezoid.
(Yu. Blinkov)
geometrytrapezoidcircumcircleConcyclic
one of XB_1, OA_1 and C_1Y is equal to sum of other two

Source: 2022 Oral Moscow Geometry Olympiad grades 10-11 p1

4/18/2022
In a circle with center OO, chords ABAB and ACAC are drawn, both equal to the radius. Points A1A_1, B1B_1 and C1C_1 are projections of points A,BA, B and CC, respectively, onto an arbitrary diameter XYXY. Prove that one of the segments XB1XB_1, OA1OA_1 and C1YC_1Y is equal to the sum of the other two.
(A. Shklover)
geometryequal segments