MathDB

8

Part of 2019 Sharygin Geometry Olympiad

Problems(4)

Sharygin CR 2019 P8

Source:

3/6/2019
The circle ω1\omega_1 passes through the vertex AA of the parallelogram ABCDABCD and touches the rays CB,CDCB, CD. The circle ω2\omega_2 touches the rays AB,ADAB, AD and touches ω1\omega_1 externally at point TT. Prove that TT lies on the diagonal ACAC
geometry
Diagonal length-sum

Source: Sharygin 2019 Finals Day 2 Grade 8 P8

7/31/2019
What is the least positive integer kk such that, in every convex 1001-gon, the sum of any k diagonals is greater than or equal to the sum of the remaining diagonals?
geometrySharygin Geometry Olympiad
Radical axis of circles is concur

Source: Sharygin 2019 Finals Day 2 Grade 9 P4

7/31/2019
A hexagon A1A2A3A4A5A6A_1A_2A_3A_4A_5A_6 has no four concyclic vertices, and its diagonals A1A4A_1A_4, A2A5A_2A_5 and A3A6A_3A_6 concur. Let lil_i be the radical axis of circles AiAi+1Ai−2A_iA_{i+1}A_{i-2} and AiAi−1Ai+2A_iA_{i-1}A_{i+2} (the points AiA_i and Ai+6A_{i+6} coincide). Prove that li,i=1,⋯ ,6l_i, i=1,\cdots,6, concur.
Sharygin 2019 Finals day2 P4Sharygin Geometry Olympiad
Collinear points

Source: Sharygin 2019 Finals Day 2 Grade 10 P4

8/4/2019
Several points and planes are given in the space. It is known that for any two of given points there exactly two planes containing them, and each given plane contains at least four of given points. Is it true that all given points are collinear?