MathDB

10

Part of 2019 India PRMO

Problems(2)

Geometry Angle Bisectors Problem

Source:

8/11/2019
Let ABCABC be a triangle and let Ω\Omega be its circumcircle. The internal bisectors of angles A,BA, B and CC intersect Ω\Omega at A1,B1A_1, B_1 and C1C_1, respectively, and the internal bisectors of angles A1,B1A_1, B_1 and C1C_1 of the triangles A1A2A3A_1 A_2 A_ 3 intersect Ω\Omega at A2,B2A_2, B_2 and C2C_2, respectively. If the smallest angle of the triangle ABCABC is 40∘40^{\circ}, what is the magnitude of the smallest angle of the triangle A2B2C2A_2 B_2 C_2 in degrees?
geometryangle bisectorcircumcircle
A weird walk timing

Source: PRMO 2019 leg 2 P10

8/25/2019
One day I went for a walk in the morning at xx minutes past 5′O5'O clock, where xx is a 2 digit number. When I returned, it was yy minutes past 6′O6'O clock, and I noticed that (i) I walked for exactly xx minutes and (ii) yy was a 2 digit number obtained by reversing the digits of xx. How many minutes did I walk?
algebraClocktimePRMO