### Archive

Posts Tagged ‘Analysis’

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

## Vojtech Jarnik Competition 2015 – Category 1 Problems

Problem 1. Let ${f:\Bbb{R} \rightarrow \Bbb{R}}$ be differentiable on ${\Bbb{R}}$. Prove that there exists ${x \in [0,1]}$ such that

$\displaystyle \frac{4}{\pi}(f(1)-f(0))=(1+x^2)f'(x).$

Problem 2. Consider the infinite chessboard whose rows and columns are indexed by positive integers. Is it possible to put a single positive rational number into each cell of the chessboard so that each positive rational number appears exactly once and the sum of every row and of every column is finite?

Problem 3. Let ${P(x)=x^{2015}-2x^{2014}+1}$ and ${Q(x) = x^{2015}-2x^{2014}-1}$. Determine for each of the polynomials ${P}$ and ${Q}$ whether it is a divisor of some nonzero polynomial ${c_0+c_1x+...+c_nx^n}$ whose coefficients ${c_i}$ are all in the set ${\{1,-1\}}$.

Problem 4. Let ${m}$ be a positive integer and let ${p}$ be a prime divisor of ${m}$. Suppose that the complex polynomial ${a_0+a_1x+...+a_nx^n}$ with ${n< \displaystyle \frac{p}{p-1}\varphi(m)}$ and ${a_n \neq 0}$ is divisible by the cyclotomic polynomial ${\Phi_m(x)}$. Prove that there at least ${p}$ non-zero coefficients ${a_i}$.

The cyclotomic polynomial ${\Phi_m(x)}$ is the monic polynomial whose roots are the ${m}$-th primitive complex roots of unity. Euler’s totient function ${\varphi(m)}$ denotes the number of positive integers less than or eual to ${m}$ which are coprime to ${m}$.