1

Part of 1991 Polish MO Finals

Problems(2)

Source: Problem 1, Polish NO 1991

Prove or disprove that there exist two tetrahedra

T_1

T_2

such that: (i) the volume of

T_1

is greater than that of

T_2

; (ii) the area of any face of

T_1

does not exceed the area of any face of

T_2

geometrygeometry unsolved

Functions operating on vectors

Source: Problem 4, Polish NO 1991

On the Cartesian plane consider the set

V

of all vectors with integer coordinates. Determine all functions

f : V \rightarrow \mathbb{R}

satisfying the conditions: (i)

f(v) = 1

for each of the four vectors

v \in V

of unit length. (ii)

f(v+w) = f(v)+f(w)

for every two perpendicular vectors

v, w \in V

(Zero vector is considered to be perpendicular to every vector).

functionvectorinductionanalytic geometryalgebra unsolvedalgebra