Home > Algebra, Analysis, Linear Algebra, Olympiad, Uncategorized > SEEMOUS 2016 – Problems

## SEEMOUS 2016 – Problems

Problem 1. Let ${f}$ be a continuous and decreasing real valued function defined on ${[0,\pi/2]}$. Prove that

$\displaystyle \int_{\pi/2-1}^{\pi/2} f(x)dx \leq \int_0^{\pi/2} f(x)\cos x dx \leq \int_0^1 f(x) dx.$

When do we have equality?

Problem 2. a) Prove that for every matrix ${X \in \mathcal{M}_2(\Bbb{C})}$ there exists a matrix ${Y \in \mathcal{M}_2(\Bbb{C})}$ such that ${Y^3 = X^2}$.

b) Prove that there exists a matrix ${A \in \mathcal{M}_3(\Bbb{C})}$ such that ${Z^3 \neq A^2}$ for all ${Z \in \mathcal{M}_3(\Bbb{C})}$.

Problem 3. Let ${A_1,A_2,...,A_k}$ be idempotent matrices (${A_i^2 = A_i}$) in ${\mathcal{M}_n(\Bbb{R})}$. Prove that

$\displaystyle \sum_{i=1}^k N(A_i) \geq \text{rank} \left(I-\prod_{i=1}^k A_i\right),$

where ${N(A_i) = n-\text{rank}(A_i)}$ and ${\mathcal{M}_n(\Bbb{R})}$ is the set of ${n \times n}$ matrices with real entries.

Problem 4. Let ${n \geq 1}$ be an integer and set

$\displaystyle I_n = \int_0^\infty \frac{\arctan x}{(1+x^2)^n}dx.$

Prove that

a) ${\displaystyle \sum_{i=1}^\infty \frac{I_n}{n} =\frac{\pi^2}{6}.}$

b) ${\displaystyle \int_0^\infty \arctan x \cdot \ln \left( 1+\frac{1}{x^2}\right) dx = \frac{\pi^2}{6}}$.

Some hints follow.

Hints. 1. Observe that adding a constant to ${f}$ does not affect the inequality. To solve the first inequality, translate ${f}$ so that it is positive and apply Chebyshev’s integral inequality for a couple of two decreasing functions. A change of variables and the fact that ${f}$ is decreasing allow to conclude. For the second inequality, translate ${f}$ into the negatives and change the sign. Apply the same Chebyshev’s inequality for a positive-decreasing couple of functions. A change of variables allows us again to conclude. Equality occurs if ${f}$ is constant.

2. a) If ${X}$ has simple eigenvalues then it is diagonal and we can use its diagonal decomposition to find ${Y}$. If ${X}$ has double eigenvalues then we use Cayley Hamilton to find ${Y}$.

b) Just take ${A}$ to be a Jordan block.

3. Use Sylvester inequality in the form ${n- \text{rank}(A)+\text{rank}(AB) \geq \text{rank}(B)}$ and the fact that the rank of the product is less than the rank of each matrix. Note that starting with ${B = I-A_1...A_k}$ and ${A = A_1}$ allows to reduce the problem to ${k-1}$ matrices. Inductively we only need to show the case ${k=1}$ which is obvious.

4. a) change the variable ${x = \arctan y}$.

b) Use a) and note that the series of functions converges uniformly to the the desired result.

A complete solution for Problem 4 can be found here.