MathDB

4

Part of 2010 Laurențiu Panaitopol, Tulcea

Problems(4)

Sum of cosines of differences

Source:

10/28/2019
Let be a natural number n, n, and n n real numbers a1,a2,…,an. a_1,a_2,\ldots ,a_n . Then, ∑1≤i<j≤ncos⁡(ai−aj)≥−n/2. \sum_{1\le i<j\le n} \cos\left( a_i-a_j \right)\ge -n/2.
Trigonometric inequalityinequalitiestrigonometry
Vectorial geometry; paralellogram; concurrence

Source:

10/28/2019
On the sides (excluding its endpoints) AB,BC,CD,DA AB,BC,CD,DA of a parallelogram consider the points M,N,P,Q, M,N,P,Q, respectively, such that AP→+AN→+CQ→+CM→=0. \overrightarrow{AP} +\overrightarrow{AN} +\overrightarrow{CQ} +\overrightarrow{CM} = 0. Show that QN,PM,AC QN, PM,AC are concurrent.
Adrian Ivan
geometryparallelogramvectorial geometry
Sum of a matrix with its adjugate

Source:

10/28/2019
Let be an odd integer n≥3 n\ge 3 and an n×n n\times n real matrix A A whose determinant is positive and such that A+adjA=2A−1. A+\text{adj} A=2A^{-1} . Prove that A2010+adj2010A=2A−2010. A^{2010} +\text{adj}^{2010} A =2A^{-2010} .
Lucian Petrescu
linear algebramatrixadjugategeneral linear matrix
Problem in the style of M. Andronache

Source: 2010 contest

10/28/2019
Let be a ring R R which has the property that there exist two distinct natural numbers s,t s,t such that for any element x x of R, R, the equation xs=xt x^s=x^t is true. Show that there exists a polynom in R[X] R[X] of degree ∣s−t∣(1+∣s−t∣) |s-t|\left( 1+|s-t| \right) such that all the elements of R R are roots of it.
abstract algebraRing Theorypolynomialpolynomial ringbeautiful