### Archive

Posts Tagged ‘Analysis’

## Putnam 2017 A3 – Solution

Problem A3. Denote ${\phi = f-g}$. Then ${\phi}$ is continuous and ${\int_a^b \phi = 0}$. We can see that

$\displaystyle I_{n+1}-I_n = \int_a^b (f/g)^n \phi = \int_{\phi\geq 0} (f/g)^n \phi+ \int_{\phi<0} (f/g)^n \phi$

Now note that on ${\{ \phi>=0\}}$ we have ${f/g \geq 1}$ so ${(f/g)^n \phi \geq \phi}$. Furthermore, on ${\{\phi<0\}}$ we have ${(f/g)^n <1}$ so multiplying with ${\phi<0}$ we get ${(f/g)^n \phi \geq \phi}$. Therefore

$\displaystyle I_{n+1}-I_n \geq \int_{\phi \geq 0} \phi + \int_{\phi<0} \phi = \int \phi = 0.$

To prove that ${I_n}$ goes to ${+\infty}$ we can still work with ${I_{n+1}-I_n}$. Note that the negative part cannot get too big:

$\displaystyle \left|\int_{ \phi <0 } (f/g)^n \phi \right| \leq \int_{\phi<0} |\phi| \leq \int_a^b |f-g|.$

As for the positive part, taking ${0<\varepsilon< \max_{[a,b]} \phi}$ we have

$\displaystyle \int_{\phi\geq 0} (f/g)^n \phi \geq \int_{\phi>\varepsilon}(f/g)^n \varepsilon.$

Next, note that on ${\{ \phi \geq \varepsilon\}}$

$\displaystyle \frac{f}{g} = \frac{g+\phi}{g} \geq \frac{g+ \varepsilon}{g}.$

We would need that the last term be larger than ${1+\delta}$. This is equivalent to ${g\delta <\varepsilon}$. Since ${g}$ is continuous on ${[a,b]}$, it is bounded above, so some delta small enough exists in order for this to work.

Grouping all of the above we get that

$\displaystyle I_{n+1}-I_n \geq \int_{\phi \geq 0} (f/g)^n \phi \geq \int_{\phi>\varepsilon} \varepsilon (1+\delta)^n.$

Since ${|\phi>\varepsilon|>0}$ we get that ${I_{n+1}-I_n}$ goes to ${+\infty}$.

## Some of the easy Putnam 2016 Problems

Here are a few of the problems of the Putnam 2016 contest. I choose to only list problems which I managed to solve. Most of them are pretty straightforward, so maybe the solutions posted here may be very similar to the official ones. You can find a complete list of the problems on other sites, for example here.

A1. Find the smallest integer ${j}$ such that for every polynomial ${p}$ with integer coefficients and every integer ${k}$, the number

$\displaystyle p^{(j)}(k),$

that is the ${j}$-th derivative of ${p}$ evaluated at ${k}$, is divisible by ${2016}$.

Hints. Successive derivatives give rise to terms containing products of consecutive numbers. The product of ${j}$ consecutive numbers is divisible by ${j!}$. Find the smallest number such that ${2016 | j!}$. Prove that ${j-1}$ does not work by choosing ${p = x^{j-1}}$. Prove that ${j}$ works by working only on monomials…

## IMC 2016 – Day 2 – Problem 6

July 28, 2016 1 comment

Problem 6. Let ${(x_1,x_2,...)}$ be a sequence of positive real numbers satisfying ${\displaystyle \sum_{n=1}^\infty \frac{x_n}{2n-1}=1}$. Prove that

$\displaystyle \sum_{k=1}^\infty \sum_{n=1}^k \frac{x_n}{k^2} \leq 2.$

## IMC 2016 – Day 1 – Problem 1

Problem 1. Let ${f:[a,b] \rightarrow \Bbb{R}}$ be continuous on ${[a,b]}$ and differentiable on ${(a,b)}$. Suppose that ${f}$ has infinitely many zeros, but there is no ${x \in (a,b)}$ with ${f(x)=f'(x) = 0}$.

• (a) Prove that ${f(a)f(b)=0}$.
• (b) Give an example of such a function.

## 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.

## Vojtech Jarnik Competition 2015 – Problems Category 2

Problem 1. Let ${A}$ and ${B}$ be two ${3 \times 3}$ matrices with real entries. Prove that

$\displaystyle A - (A^{-1}+(B^{-1}-A)^{-1})^{-1} = ABA,$

provided all the inverses appearing on the left-hand side of the equality exist.

Problem 2. Determine all pairs ${(n,m)}$ of positive integers satisfying the equation

$\displaystyle 5^n = 6m^2+1.$

Problem 3. Determine the set of real values ${x}$ for which the following series converges, and find its sum:

$\displaystyle \sum_{n=1}^\infty \left( \sum_{k_i \geq 0, k_1+2k_2+...+nk_n = n} \frac{(k_1+...+k_n)!}{k_1!...k_n!} x^{k_1+...+k_n}\right).$

Problem 4. Find all continuously differentiable functions ${f : \Bbb{R} \rightarrow \Bbb{R}}$, such that for every ${a \geq 0}$ the following relation holds:

$\displaystyle \int_{D(a)} xf\left( \frac{ay}{\sqrt{x^2+y^2}}\right) dxdydz = \frac{\pi a^3}{8}(f(a)+\sin a -1),$

where ${D(a) = \left\{ (x,y,z) : x^2+y^2+z^2 \leq a^2,\ |y| \leq \frac{x}{\sqrt{3}}\right\}}$