Problems(3)
2 turtles are crawling from point A to B (2019 Novosibirsk Oral Geo Oly 7.6)
Source:
4/26/2021
Two turtles, the leader and the slave, are crawling along the plane from point to point . They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to . Prove that the slave turtle crawled no more than the leading one.
geometrygeometric inequality
circles tangent to a fixed circle (2019 Novosibirsk Oral Geo Oly 8.6)
Source:
4/28/2021
Point is located in this circle of radius . An arbitrary chord is drawn through it, and then a circle of radius is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.
geometryfixedtangent circles
polyline of length 200 in a square of side 1 (2019 Novosibirsk Oral Geo Oly 9.6)
Source:
4/28/2021
A square with side contains a non-self-intersecting polyline of length at least . Prove that there is a straight line parallel to the side of the square that has at least points in common with this polyline.
geometrypolyline