MathDB

6

Part of 2023 Novosibirsk Oral Olympiad in Geometry

Problems(3)

<ADB+<BEC+<CFA=? BA=BC, AD=AC, BE=BA , CF=CB (2023 Novosibirsk Oral Geo Oly 7.6)

Source:

9/4/2023
An isosceles triangle ABCABC with base ACAC is given. On the rays CACA, ABAB and BCBC, the points D,ED, E and FF were marked, respectively, in such a way that AD=ACAD = AC, BE=BABE = BA and CF=CBCF = CB. Find the sum of the angles ∠ADB\angle ADB, ∠BEC\angle BEC and ∠CFA\angle CFA.
geometryequal segmentsanglesisosceles
area bisector of mushroom-gon (2023 Novosibirsk Oral Geo Oly 8.6)

Source:

9/4/2023
Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half. https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png
geometryareasareaconstructiongeometric construction
angle wanted, 2 quarter-circles (2023 Novosibirsk Oral Geo Oly 9.6)

Source:

9/5/2023
Two quarter-circles touch as shown. Find the angle xx. https://cdn.artofproblemsolving.com/attachments/b/4/e70d5d69e46d6d40368f143cb83cf10b7d6d98.png
geometryangle