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2

Part of 1994 All-Russian Olympiad

Problems(2)

collinear points starting with two externally tangent circles

Source: All Russian MO 1994 ARO IX P2

7/29/2018
Two circles S1S_1 and S2S_2 touch externally at FF. their external common tangent touches S1S_1 at AA and S2S_2 at BB. A line, parallel to ABAB and tangent to S2S_2 at CC, intersects S1S_1 at DD and EE. Prove that points A,F,CA,F,C are collinear.
(A. Kalinin)
geometrytangent circlescollinear
100-gon with k points, find a 2k-gon

Source: All Russian Math Olympiad 1994

4/3/2015
Inside a convex 100100-gon are selected kk points, 2≤k≤502 \leq k \leq 50. Show that one can choose 2k2k vertices of the 100100-gon so that the convex 2k2k-gon determined by these vertices contains all the selected points.
combinatoricscombinatorics unsolved