This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 3

1982 Tournament Of Towns, (030) 4
Tags: geometry , 3d geometry , vector , regular polygon , regular tetrahedron , tetrahedron
(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon. (b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron. (VV Prasolov, Moscow)

1949-56 Chisinau City MO, 60
Tags: regular tetrahedron , tetrahedron , 3d geometry , geometry , distance
Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.

1986 Tournament Of Towns, (115) 3
Tags: geometry , tetrahedron , regular tetrahedron , 3d geometry , vector
Vectors coincide with the edges of an arbitrary tetrahedron (possibly non-regular). Is it possible for the sum of these six vectors to equal the zero vector? (Problem from Leningrad)