MathDB

3

Part of 1991 Turkey Team Selection Test

Problems(2)

Turkey TST 1991 - P3

Source:

3/13/2011
Let ff be a function on defined on ∣x∣<1|x|<1 such that f(110)f\left (\tfrac1{10}\right ) is rational and f(x)=∑i=1∞aixif(x)= \sum_{i=1}^{\infty} a_i x^i where ai∈{0,1,2,3,4,5,6,7,8,9}a_i\in{\{0,1,2,3,4,5,6,7,8,9\}}. Prove that ff can be written as f(x)=p(x)q(x)f(x)= \frac{p(x)}{q(x)} where p(x)p(x) and q(x)q(x) are polynomials with integer coefficients.
functionalgebrapolynomialalgebra proposed
Turkey TST 1991 - P6

Source:

3/13/2011
Let UU be the sum of lengths of sides of a tetrahedron (triangular pyramid) with vertices O,A,B,CO,A,B,C. Let VV be the volume of the convex shape whose vertices are the midpoints of the sides of the tetrahedron. Show that V≤(U−∣OA∣−∣BC∣)(U−∣OB∣−∣AC∣)(U−∣OC∣−∣AB∣)(27⋅3)V\leq \frac{(U-|OA|-|BC| )(U-|OB|-|AC| )(U-|OC|-|AB| )}{(2^{7} \cdot 3)}.
geometry3D geometrytetrahedronpyramidgeometry proposed