### Archive

Posts Tagged ‘SEEMOUS’

## SEEMOUS 2018 – Problems

Problem 1. Let ${f:[0,1] \rightarrow (0,1)}$ be a Riemann integrable function. Show that

$\displaystyle \frac{\displaystyle 2\int_0^1 xf^2(x) dx }{\displaystyle \int_0^1 (f^2(x)+1)dx }< \frac{\displaystyle \int_0^1 f^2(x) dx}{\displaystyle \int_0^1 f(x)dx}.$

Problem 2. Let ${m,n,p,q \geq 1}$ and let the matrices ${A \in \mathcal M_{m,n}(\Bbb{R})}$, ${B \in \mathcal M_{n,p}(\Bbb{R})}$, ${C \in \mathcal M_{p,q}(\Bbb{R})}$, ${D \in \mathcal M_{q,m}(\Bbb{R})}$ be such that

$\displaystyle A^t = BCD,\ B^t = CDA,\ C^t = DAB,\ D^t = ABC.$

Prove that ${(ABCD)^2 = ABCD}$.

Problem 3. Let ${A,B \in \mathcal M_{2018}(\Bbb{R})}$ such that ${AB = BA}$ and ${A^{2018} = B^{2018} = I}$, where ${I}$ is the identity matrix. Prove that if ${\text{tr}(AB) = 2018}$ then ${\text{tr}(A) = \text{tr}(B)}$.

Problem 4. (a) Let ${f: \Bbb{R} \rightarrow \Bbb{R}}$ be a polynomial function. Prove that

$\displaystyle \int_0^\infty e^{-x} f(x) dx = f(0)+f'(0)+f''(0)+...$

(b) Let ${f}$ be a function which has a Taylor series expansion at ${0}$ with radius of convergence ${R=\infty}$. Prove that if ${\displaystyle \sum_{n=0}^\infty f^{(n)}(0)}$ converges absolutely then ${\displaystyle \int_0^{\infty} e^{-x} f(x)dx}$ converges and

$\displaystyle \sum_{n=0}^\infty f^{(n)}(0) = \int_0^\infty e^{-x} f(x).$

Hints: 1. Just use $2f(x) \leq f^2(x)+1$ and $xf^2(x) < f^2(x)$. The strict inequality comes from the fact that the Riemann integral of strictly positive function cannot be equal to zero. This problem was too simple…

2. Use the fact that $ABCD = AA^t$, therefore $ABCD$ is symmetric and positive definite. Next, notice that $(ABCD)^3 = ABCDABCDABCD = D^tC^tB^tA^t = (ABCD)^t = ABCD$. Notice that $ABCD$ is diagonalizable and has eigenvalues among $-1,0,1$. Since it is also positive definite, $-1$ cannot be an eigenvalue. This allows to conclude.

3. First note that the commutativity allows us to diagonalize $A,B$ using the same basis. Next, note that $A,B$ both have eigenvalues of modulus one. Then the trace of $AB$ is simply the sum $\sum \lambda_i\mu_i$ where $\lambda_i,\mu_i$ are eigenvalues of $A$ and $B$, respectively. The fact that the trace equals $2018$ and the triangle inequality shows that eigenvalues of $A$ are a multiple of eigenvalues of $B$. Finish by observing that they have the same eigenvalues.

4. (a) Integrate by parts and use a recurrence. (b) Use (a) and the approximation of a continuous function by polynomials on compacts to conclude.

I’m not sure about what others think, but the problems of this year seemed a bit too straightforward.

## SEEMOUS 2016 Problem 4 – Solution

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

## 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.

## SEEMOUS 2014

Problem 1. Let ${n}$ be a nonzero natural number and ${f:\Bbb{R} \rightarrow \Bbb{R}\setminus \{0\}}$ be a function such that ${f(2014)=1-f(2013)}$. Let ${x_1,..,x_n}$ be distinct real numbers. If

$\displaystyle \left| \begin{matrix} 1+f(x_1)& f(x_2)&f(x_3) & \cdots & f(x_n) \\ f(x_1) & 1+f(x_2) & f(x_3) & \cdots & f(x_n)\\ f(x_1) & f(x_2) &1+f(x_3) & \cdots & f(x_n) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ f(x_1)& f(x_2) & f(x_3) & \cdots & 1+f(x_n) \end{matrix} \right|=0$

prove that ${f}$ is not continuous.

Problem 2. Consider the sequence ${(x_n)}$ given by

$\displaystyle x_1=2,\ \ x_{n+1}= \frac{x_n+1+\sqrt{x_n^2+2x_n+5}}{2},\ n \geq 2.$

Prove that the sequence ${y_n = \displaystyle \sum_{k=1}^n \frac{1}{x_k^2-1} ,\ n \geq 1}$ is convergent and find its limit.

Problem 3. Let ${A \in \mathcal{M}_n (\Bbb{C})}$ and ${a \in \Bbb{C},\ a \neq 0}$ such that ${A-A^* =2aI_n}$, where ${A^* = (\overline A)^t}$ and ${\overline A}$ is the conjugate matrix of ${A}$.

(a) Show that ${|\det(A)| \geq |a|^n}$.

(b) Show that if ${|\det(A)|=|a|^n}$ then ${A=aI_n}$.

Problem 4. a) Prove that ${\displaystyle \lim_{n \rightarrow \infty} n \int_0^n \frac{\arctan(x/n)}{x(x^2+1)}dx=\frac{\pi}{2}}$.

b) Find the limit ${\displaystyle \lim_{n \rightarrow \infty} n\left(n \int_0^n \frac{\arctan(x/n)}{x(x^2+1)}dx-\frac{\pi}{2} \right)}$

## SEEMOUS 2013 + Solutions

Here are some of the problems of SEEMOUS 2013. Update: the 4th problem has arrived; it is number 3 below.

1. Let ${f:[1,8] \rightarrow \Bbb{R}}$ be a continuous mapping, such that

$\displaystyle \int_1^2 f^2(t^3)dt+2\int_1^2f(t^3)dt=\frac{2}{3}\int_1^8 f(t)dt-\int_1^2 (t^2-1)^2 dt.$

Find the form of the map ${f}$.

Solution: Change the variable from ${t}$ to ${t^3}$ in the RHS integral and DO NOT calculate the last integral in the RHS. Get all the terms in the left and find that it is in fact the integral of a square equal to zero.

2. Let ${M,N \in \mathcal{M}_2(\Bbb{C})}$ be nonzero matrices such that ${M^2=N^2=0}$ and ${MN+NM=I_2}$. Prove that there is an invertible matrix ${A \in \mathcal{M}_2(\Bbb{C})}$ such that ${M=A\begin{pmatrix} 0&1\\ 0&0\end{pmatrix}A^{-1}}$ and ${N=A\begin{pmatrix} 0&0 \\ 1&0\end{pmatrix}A^{-1}}$.

Solution: One solution can be given using the fact that ${M,N}$ can be written in that form, but for different matrices ${A}$.

Another way to do it is to consider applications ${f,g: \Bbb{C}^2 \rightarrow \Bbb{C}^2,\ f(x)=Mx,\ g(x)=Nx}$. We get at once ${f^2=0,g^2=0,fg+gf=Id}$ and from these we deduce that ${(fg)^2=fg}$ and ${(gf)^2=gf}$. First note that ${fg}$ is not the zero application. Then there exists ${u \in Im(fg) \setminus\{0\}}$, i.e. there exists ${w (\neq 0)}$ such that ${f(g(w))=v}$. We have ${fg(u)=(fg)^2(w)=fg(w)=u}$. Consider ${v=g(u)}$.

Then ${u,v}$ are not collinear, ${f(u)=0,f(v)=u, g(u)=v,g(v)=0}$. Consider now the basis formed by ${u,v}$ and take ${A}$ to be the change of base matrix from the canonical base to ${\{u,v\}}$.

3. Find the maximum possible value of

$\displaystyle \int_0^1 |f'(x)|^2|f(x)|\frac{1}{\sqrt{x}}dx$

over all continuously differentiable functions ${f:[0,1] \rightarrow \Bbb{R}}$ with ${f(0)=0}$ and ${\int_0^1|f'(x)|^2 dx\leq 1}$.

4. Let ${A \in \mathcal{M}_2(\Bbb{Q})}$ such that there is ${n \in \Bbb{N},\ n\neq 0}$, with ${A^n=-I_2}$. Prove that either ${A^2=-I_2}$ or ${A^3=-I_2}$.

Solution: Consider ${p \in \Bbb{Q}[X]}$ the minimal polynomial of ${A}$, which has degree at most ${2}$. The eigenvalues of ${A}$ satisfy ${\lambda_1^n=\lambda_2^n=-1}$. We have two cases: either the eigenvalues are real and therefore they are both equal to ${-1}$ either they are complex and conjugate of modulus one. In both cases the determinant of ${A}$ is equal to ${1}$. Therefore, by Cayley Hamiltoh theorem ${A}$ satisfies an equation of the type ${A^2-qA+I_2=0}$.

By hypothesis the minimal polynomial ${p}$ divides ${X^n+1}$. If ${p}$ has degree one then ${A=\lambda I_2}$ and ${\lambda \in \Bbb{Q},\lambda^n=-1}$ so ${A=-I_2}$.

If not, then the minimal polynomial is ${X^2-qX+1}$ and we must have

$\displaystyle X^2-qX+1 | X^n+1.$

It can be proved that ${q}$ is in fact an integer. (using the fact that the product of two primitive polynomials is primitive)

Since ${q=2\cos \theta}$ it follows that ${q \in \{0,\pm 1,\pm 2\}}$. The rest is just casework.

## SEEMOUS 2012 Problem 3

a) Prove that if $k$ is an even positive integer and $A$ is a real symmetric $n \times n$ matrix such that $(\text{Tr}\, (A^k))^{k+1}=(\text{Tr}\, (A^{k+1}))^k$ then $A^n=\text{Tr}\, (A)A^{n-1}$.

b) Does this assertion from a) also hold for odd positive integers $k$?

## SEEMOUS 2012 Problem 2

Let $a_n>0, \ n \geq 1$. Consider the right triangles $\Delta A_0A_1A_2,\Delta A_0A_2A_3,...,\Delta A_0A_{n-1}A_n,...$ as in the figure. (More precisely, for every $n \geq 2$ the hypotenuse $A_0A_n$ of $\Delta A_0A_{n-1}A_n$ is a leg of $\Delta A_0A_nA_{n+1}$ with right angle $\angle A_0A_nA_{n+1}$, and the vertices $A_{n-1}$ and $A_{n+1}$ lie on the opposite sides of the straight line $A_0A_n$; also $|A_{n-1}A_n|=a_n$ for every $n \geq 1$.)
It is possible for the set of points $\{A_n | n \geq 0\}$ to be unbounded, but the series $\sum_{n=2}^\infty m(\angle A_{n-1}A_0A_n)$ to be convergent? Here $m(\angle ABC)$ denotes the measure of $\angle ABC$.