11.6

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(2)

homothetic polyhedrons

Source: VI Soros Olympiad 1999-00 Round 1 11.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

It is known that a

n

-vertex contains within itself a polyhedron

M

with a center of symmetry at some point

Q

and is itself contained in a polyhedron homothetic to

M

with a homothety center at a point

Q

and coefficient

k

. Find the smallest value of

k

n = 4

n = 5

geometry3D geometryhomothety

integer polynomial with root \sqrt2+\sqrt3 has also \sqrt2 - \sqrt3

Source: VI Soros Olympiad 1990-00 R3 11.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

P(x)

be a polynomial with integer coefficients. It is known that the number

\sqrt2+\sqrt3

is its root. Prove that the number

\sqrt2-\sqrt3

is also its root.

algebrapolynomialInteger Polynomial