MathDB

6

Part of 2014 Tuymaada Olympiad

Problems(2)

From 2n squares, n of them have a common point

Source: Tuymaada 2014, Day 2, Problem 3 Juniors, Problem 2 Seniors

7/12/2014
Each of nn black squares and nn white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to nn squares.
(V. Dolnikov)
geometrygeometric transformationanalytic geometryrectanglecombinatorics unsolvedcombinatoricsTuymaada
If two circles are tangent, then all three are

Source: Tuymaada 2014, Day 2, Problem 2, Junior League

7/12/2014
Radius of the circle ωA\omega_A with centre at vertex AA of a triangle △ABC\triangle{ABC} is equal to the radius of the excircle tangent to BCBC. The circles ωB\omega_B and ωC\omega_C are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other.
(L. Emelyanov)
trigonometrygeometry unsolvedgeometryTuymaada