MathDB

5

Part of 2021 Saint Petersburg Mathematical Olympiad

Problems(3)

Sum of reciprocals of primes in a range is < 2

Source: St Petersburg 2021 11.5

12/23/2021
A natural number nn is given. Prove that ∑n≤p≤n21p<2\sum_{n \le p \le n^2} \frac{1}{p} < 2 where the sum is across all primes pp in the range [n,n2][n, n^2]
number theory
Dania splits big polygon and wants special quadrilateral

Source: St Petersburg 2021 10.5

12/23/2021
The vertices of a convex 25502550-gon are colored black and white as follows: black, white, two black, two white, three black, three white, ..., 50 black, 50 white. Dania divides the polygon into quadrilaterals with diagonals that have no common points. Prove that there exists a quadrilateral among these, in which two adjacent vertices are black and the other two are white.
D. Rudenko
combinatoricscombinatorial geometryHi
Isosceles trapezoid, parralel lines and a tangency

Source: St Petersburg 2021 9.5

12/23/2021
Given is an isosceles trapezoid ABCDABCD, such that ADAD and BCBC are bases and AD=2ABAD=2AB, and it is inscribed in a circle cc. Points EE and FF are selected on a circle cc so that ACAC || DEDE and BDBD || AFAF. The line BEBE intersects lines ACAC and AFAF at points XX and YY, respectively. Prove that the circumcircles of triangles BCXBCX and EFYEFY are tangent to each other.
geometry