MathDB

2

Part of 2004 Unirea

Problems(4)

sup(x+y+z) , where (cyc) x²+yz <=2

Source:

12/17/2019
Find the maximum value of the expression x+y+z, x+y+z, where x,y,z x,y,z are real numbers satisfying {x2+yz≤2y2+zx≤2z2+xy≤2. \left\{ \begin{matrix} x^2+yz\le 2 \\y^2+zx\le 2\\ z^2+xy\le 2 \end{matrix} \right. .
inequalitiesinequality systemalgebra
Find all arithmetic progressions

Source:

12/17/2019
Find the arithmetic sequences of 5 5 integers n1,n2,n3,n4,n5 n_1,n_2,n_3,n_4,n_5 that verify 5∣n1,2∣n2,11∣n3,7∣n4,17∣n5. 5|n_1,2|n_2,11|n_3,7|n_4,17|n_5.
arithmetic sequenceDivisibilityalgebra
2x2 nilpotent matrix

Source:

12/17/2019
Let be two matrices A,N∈M2(R) A,N\in\mathcal{M}_2(\mathbb{R}) that commute and such that N N is nilpotent. Show that:
a) det⁡(A+N)=det⁡(A) \det (A+N)=\det (A) b) if A A is general linear, then the matrix A+N A+N is invertible and (A+N)−1=(A−N)A−2. (A+N)^{-1}=(A-N)A^{-2} .
linear algebramatrix
Properties of groups of elements of same order

Source:

12/17/2019
Consider a group G G which has the property that any element of it, with the exception of the identity, has order p≥2. p\ge 2. Prove that
a) p p is prime. b) G G is commutative if any subset of G G having p2−1 p^2-1 elements contains at least p p elements that commute between themselves pairwise.
group theory