MathDB

2

Part of 2023 Saint Petersburg Mathematical Olympiad

Problems(3)

Geo with angle 120

Source: St. Petersburg 2023 10.2

8/12/2023
Let ABCABC be a triangle with ∠B=120o\angle B=120^{o}. Let DD be point on the BB-angle bisector, such that ∠ADB=2∠ACB\angle ADB=2\angle ACB. Point EE lies on the segment ABAB, so that AE=ADAE=AD. Show that EC=EDEC=ED.
geometry
Geo ineq with median

Source: St. Petersburg 2023 11.2

8/12/2023
Given is a triangle ABCABC with median BMBM. The point DD lies on the line ACAC after CC, such that BD=2CDBD=2CD. The circle (BMC)(BMC) meets the segment BDBD at NN. Show that AC+BM>2MNAC+BM>2MN.
geometry
Numbers on a circle

Source: St. Petersburg 2023 9.2

8/12/2023
A few (at least 55) integers are put on a circle, such that each of them is divisible by the sum of its neighbors. If the sum of all numbers is positive, what is its minimal value?
number theory