MathDB

7

Part of 1978 Romania Team Selection Test

Problems(2)

polynomials L of deg(L)=3,4

Source: Romanian TST 1978, Day 1, P7

9/28/2018
Let P,Q,R P,Q,R be polynomials of degree 3 3 with real coefficients such that P(x)≤Q(x)≤R(x), P(x)\le Q(x)\le R(x) , for every real x. x. Suppose P−R P-R admits a root. Show that Q=kP+(1−k)R, Q=kP+(1-k)R, for some real number k∈[0,1]. k\in [0,1] . What happens if P,Q,R P,Q,R are of degree 4, 4, under the same circumstances?
algebrapolynomial
Barycenter at TST

Source: Romanian TST 1978, Day 3, P7

9/30/2018
a) Prove that for any natural number n≥1, n\ge 1, there is a set M \mathcal{M} of n n points from the Cartesian plane such that the barycenter of every subset of M \mathcal{M} has integral coordinates (both coordinates are integer numbers).
b) Show that if a set N \mathcal{N} formed by an infinite number of points from the Cartesian plane is given such that no three of them are collinear, then there exists a finite subset of N, \mathcal{N} , the barycenter of which has non-integral coordinates.
analytic geometrylinear algebrageometrybarycenter