MathDB

4

Part of 2013 Danube Mathematical Competition

Problems(2)

perpendicularity starting with a rectangle, danube junior 2013

Source: Danube 2013 Junior P4

9/7/2018
Let ABCDABCD be a rectangle with AB≠BCAB \ne BC and the center the point OO. Perpendicular from OO on BDBD intersects lines ABAB and BCBC in points EE and FF respectively. Points MM and NN are midpoints of segments [CD][CD] and [AD][AD] respectively. Prove that FM⊥ENFM \perp EN .
geometryrectangleperpendicular
set {nx + S : n \in N}$ is finite, where nx + S ={nx + s : s \in S}

Source: Danube 2013 p4

7/22/2019
Show that there exists a proper non-empty subset SS of the set of real numbers such that, for every real number xx, the set {nx+S:n∈N}\{nx + S : n \in N\} is finite, where nx+S={nx+s:s∈S}nx + S =\{nx + s : s \in S\}
Setsset theorycombinatorics