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.

  1. aqws
    March 6, 2016 at 5:21 am

    Can you please explain exactly how to solve #4?

  2. aqws
    March 6, 2016 at 5:22 am

    How does one solve #4 exactly? In particular what is the full solution?

  3. March 6, 2016 at 2:58 pm

    I’ve posted a solution to Problem 4 here

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