MathDB

6

Part of 2012 Saint Petersburg Mathematical Olympiad

Problems(3)

Light beam and mirrors

Source: St Petersburg Olympiad 2012, Grade 11, P6

9/29/2017
On the coordinate plane in the first quarter there are 100100 non-intersecting single unit segments parallel to the coordinate axes. These segments aremirrors (on both sides), they reflect the light according to the rule. "The angle of incidence is equal to the angle of reflection." (If you hit the edge of the mirror, the beam of light does not change its direction.) From the point lying in the unit circle with the center at the origin, a ray of light in the direction of the bisector of the first coordinate angle. Prove that, that this initial point can be chosen so that the ray is reflected from the mirrors not more than 150150 times.
geometrycombinatorics
Interesting geometry

Source: St Petersburg Olympiad 2012, Grade 10, P6

9/29/2017
ABCDABCD is parallelogram. Line ll is perpendicular to BCBC at BB. Two circles passes through D,CD,C, such that ll is tangent in points PP and QQ. MM - midpoint ABAB. Prove that ∠DMP=∠DMQ\angle DMP=\angle DMQ
geometryparallelogram
Some geometry

Source: St Petersburg Olympiad 2012, Grade 9, P6

9/29/2017
ABCABC is triangle. Point LL is inside ABCABC and lies on bisector of ∠B\angle B. KK is on BLBL. ∠KAB=∠LCB=α\angle KAB=\angle LCB= \alpha. Point PP inside triangle is such, that AP=PCAP=PC and ∠APC=2∠AKL\angle APC=2\angle AKL. Prove that ∠KPL=2α\angle KPL=2\alpha
geometry