MathDB

4

Part of 2020 Tuymaada Olympiad

Problems(2)

Ratio of perimeter and area with weird angle condition

Source: Tuymaada 2020 Senior, Problem 3 (Junior, P4)

10/6/2020
Points DD and EE lie on the lines BCBC and ACAC respectively so that BB is between CC and DD, CC is between AA and EE, BC=BDBC = BD and ∠BAD=∠CDE\angle BAD = \angle CDE. It is known that the ratio of the perimeters of the triangles ABCABC and ADEADE is 22. Find the ratio of the areas of these triangles.
ratiogeometryperimeter
Give me a bound of $g(k)$

Source: Tuymaada 2020 Senior, Problem 4

10/6/2020
For each positive integer kk, let g(k)g(k) be the maximum possible number of points in the plane such that pairwise distances between these points have only kk different values. Prove that there exists kk such that g(k)>2k+2020g(k) > 2k + 2020.
combinatoricsfreedistance