MathDB

2

Part of 2000 Romania Team Selection Test

Problems(3)

ROMANIA 2000 inequality

Source: Romanian TST 2000

1/6/2004
Let n≥1n\ge 1 be a positive integer and x1,x2…,xnx_1,x_2\ldots ,x_n be real numbers such that ∣xk+1−xk∣≤1|x_{k+1}-x_k|\le 1 for k=1,2,…,n−1k=1,2,\ldots ,n-1. Prove that ∑k=1n∣xk∣−∣∑k=1nxk∣≤n2−14\sum_{k=1}^n|x_k|-\left|\sum_{k=1}^nx_k\right|\le\frac{n^2-1}{4}
Gh. Eckstein
inequalitiesinequalities solved
N satisfies ∠ NBA =∠ BAM and ∠ NCA = ∠ CAM

Source: Romanian TST 2000

2/18/2011
Let ABCABC be an acute-angled triangle and MM be the midpoint of the side BCBC. Let NN be a point in the interior of the triangle ABCABC such that ∠NBA=∠BAM\angle NBA=\angle BAM and ∠NCA=∠CAM\angle NCA=\angle CAM. Prove that ∠NAB=∠MAC\angle NAB=\angle MAC.
Gabriel Nagy
geometrycircumcirclegeometry proposed
monic polynomials

Source: Romanian TST 2000

7/17/2005
Let P,QP,Q be two monic polynomials with complex coefficients such that P(P(x))=Q(Q(x))P(P(x))=Q(Q(x)) for all xx. Prove that P=QP=Q.
Marius Cavachi
algebrapolynomialalgebra proposed