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n points create always obtuse triangles, prove n+1 do it also

Source: IMAC Arhimede 2010 p5

May 5, 2019

combinatoricscombinatorial geometrypointsObtuse triangle

Problem Statement

Different points

A_1, A_2,..., A_n

in the plane (

n> 3

) are such that the triangle

A_iA_jA_k

is obtuse for all the different

i,j,k \in\{1,2,...,n\}

. Prove that there is a point

A_{n + 1}

in the plane, such that the triangle

A_iA_jA_{n + 1}

is obtuse for all different

i,j \in\{1,2,...,n\}