Olympiad problems

2001 SNSB Admission, 3

Let be an $ n\times n $ positive-definite symmetric real matrix $ A. $ Prove the following equality. $$ \tiny\int_{\mathbb{R}^n} \exp\left( -\begin{pmatrix} x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}^\intercal A\begin{pmatrix} x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}\right) dx_1dx_2\cdots dx_n=\normalsize\frac{\pi^{n/2}}{\sqrt{\det A} } $$

2005 Miklós Schweitzer, 11

Tags: real analysis , homogeneous function , positive definite , linear algebra

Let $E: R^n \backslash \{0\} \to R^+$ be a infinitely differentiable, quadratic positive homogeneous (that is, for any λ>0 and $p \in R^n \backslash \{0\}$ , $E (\lambda p) = \lambda^2 E (p)$). Prove that if the second derivative of $E''(p): R^n \times R^n \to R$ is a non-degenerate bilinear form at any point $p \in R^n \backslash \{0\}$, then $E''(p)$ ($p \in R^n \backslash \{0\}$) is positive definite.

2018 Korea USCM, 7

Tags: matrix , linear algebra , positive definite , vector , eigenvalue

Suppose a $3\times 3$ matrix $A$ satisfies $\mathbf{v}^t A \mathbf{v} > 0$ for any vector $\mathbf{v} \in\mathbb{R}^3 -\{0\}$. (Note that $A$ may not be a symmetric matrix.) (1) Prove that $\det(A)>0$. (2) Consider diagonal matrix $D=\text{diag}(-1,1,1)$. Prove that there's exactly one negative real among eigenvalues of $AD$.