MathDB

5.

Part of 2020 Saint Petersburg Mathematical Olympiad

Problems(3)

Geometric inequality again

Source: Saint Petersburg MO 2020 Grade 11 Problem 5

5/7/2020
The altitudes BB1BB_1 and CC1CC_1 of the acute triangle △ABC\triangle ABC intersect at HH. The circle centered at ObO_b passes through points A,C1A,C_1, and the midpoint of BHBH. The circle centered at OcO_c passes through A,B1A,B_1 and the midpoint of CHCH. Prove that B1Ob+C1Oc>BC4B_1 O_b +C_1O_c > \frac{BC}{4}
geometric inequalityinequalities
Rays passing through a point

Source: Saint Petersburg MO 2020 Grade 10 Problem 5

5/7/2020
Rays ℓ,ℓ1,ℓ2\ell, \ell_1, \ell_2 have the same starting point OO, such that the angle between ℓ\ell and ℓ2\ell_2 is acute and the ray ℓ1\ell_1 lies inside this angle. The ray ℓ\ell contains a fixed point of FF and an arbitrary point LL. Circles passing through FF and LL and tangent to ℓ1\ell_1 at L1L_1, and passing through FF and LL and tangent to ℓ2\ell_2 at L2L_2. Prove that the circumcircle of △FL1L2\triangle FL_1L_2 passes through a fixed point other than FF independent on LL.
geometrycircumcircle
an excircle and 4 points

Source: Saint Petersburg MO 2020 Grade 9 Problem 5

5/7/2020
Point IaI_a is the AA-excircle center of △ABC\triangle ABC which is tangent to BCBC at XX. Let A′A' be diametrically opposite point of AA with respect to the circumcircle of △ABC\triangle ABC. On the segments IaX,BA′I_aX, BA' and CA′CA' are chosen respectively points Y,ZY,Z and TT such that IaY=BZ=CT=rI_aY=BZ=CT=r where rr is the inradius of △ABC\triangle ABC. Prove that the points X,Y,ZX,Y,Z and TT are concyclic.
geometrycircumcircleinradius