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SEEMOUS 2018 – Problems

March 1, 2018 Leave a comment

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).

Source: official site of SEEMOUS 2018 

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.

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SEEMOUS 2016 Problem 4 – Solution

March 6, 2016 3 comments

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}}.

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SEEMOUS 2016 – Problems

March 5, 2016 3 comments

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.

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SEEMOUS 2014

March 18, 2014 Leave a comment

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

March 23, 2013 9 comments

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.

Categories: Algebra, Analysis, Undergraduate Tags:

SEEMOUS 2012 Problem 3

March 8, 2012 Leave a comment

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?

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SEEMOUS 2012 Problem 2

March 8, 2012 Leave a comment

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.

(A subset of the plane is bounded if, for example is contained in a disk.)

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