MathDB

7

Part of 2018 Saint Petersburg Mathematical Olympiad

Problems(3)

Tangent lines

Source: St Petersburg Olympiad 2018, Grade 11, P7

6/21/2018
Points A,BA,B lies on the circle SS. Tangent lines to SS at AA and BB intersects at CC. MM -midpoint of ABAB. Circle S1S_1 goes through M,CM,C and intersects ABAB at DD and SS at KK and LL. Prove, that tangent lines to SS at KK and LL intersects at point on the segment CDCD.
geometry
Moving checker

Source: St Petersburg Olympiad 2018, Grade 10, P7

6/22/2018
The checker moves from the lower left corner of the board 100×100100 \times 100 to the right top corner, moving at each step one cell to the right or one cell up. Let aa be the number of paths in which exactly 7070 steps the checker take under the diagonal going from the lower left corner to the upper right corner, and bb is the number of paths in which such steps are exactly 110110. What is more: aa or bb?
combinatoricscatalan numbersSaint Petersburg
Arrows in cells

Source: St Petersburg Olympiad 2018, Grade 9, P7

7/13/2018
In 10×1010\times 10 square we choose nn cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or the distance between their ends is at least 2. What is the maximum possible value of nn?
combinatorics