Olympiad problems

1985 Traian Lălescu, 1.3

Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ and two natural numbers $ m,n. $ Prove that: $$ \det\left( (AB)^m-(BA)^m\right)\cdot\det\left( (AB)^n-(BA)^n\right)\ge 0. $$

ICMC 5, 3

Tags: linear algebra , college contests , Determinants , ICMC

Let $\mathcal M$ be the set of $n\times n$ matrices with integer entries. Find all $A\in\mathcal M$ such that $\det(A+B)+\det(B)$ is even for all $B\in\mathcal M$. [i]Proposed by Ethan Tan[/i]

2017 Korea USCM, 1

Tags: linear algebra , matrix , Determinants , Summation , college contests

$n(\geq 2)$ is a given integer and $T$ is set of all $n\times n$ matrices whose entries are elements of the set $S=\{1,\cdots,2017\}$. Evaluate the following value. \[\sum_{A\in T} \text{det}(A)\]

2019 District Olympiad, 2

Tags: characteristic polynomial , Inequality , Determinants , linear algebra

Let $n \in \mathbb{N},n \ge 2,$ and $A,B \in \mathcal{M}_n(\mathbb{R}).$ Prove that there exists a complex number $z,$ such that $|z|=1$ and $$\Re \left( {\det(A+zB)} \right) \ge \det(A)+\det(B),$$ where $\Re(w)$ is the real part of the complex number $w.$

1984 Spain Mathematical Olympiad, 8

Tags: remainder , polynomial , Determinants , algebra

Find the remainder upon division by $x^2-1$ of the determinant $$\begin{vmatrix} x^3+3x & 2 & 1 & 0 \\ x^2+5x & 3 & 0 & 2 \\x^4 +x^2+1 & 2 & 1 & 3 \\x^5 +1 & 1 & 2 & 3 \\ \end{vmatrix}$$

1996 Romania National Olympiad, 3

Tags: Matrices , Determinants , linear algebra

Let $A, B \in M_2(\mathbb{R})$ such as $det(AB+BA)\leq 0$. Prove that $$det(A^2+B^2)\geq 0$$