A double Putnam 2016 series

Evaluate the series:

$$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}$$

(Putnam 2016)

Solution

A special integral

For $x \geq 1$ we define $f(x)$ as the unique number $c$ such that $c^c = x $. Evaluate the integral:

$$\mathcal{J}=\int_0^e f(x) \, {\rm d}x$$

(Euler Competition 2013)

Solution

All distances are integer

Prove that for every $n \geq 3$ there exist $n$ points on the plane , not all colinear, such that the distance between any of them is actually an integer number.

Solution

An identity with integer part of $\log_2$

Let $n \in \mathbb{N}$. Prove that:

$$\sum_{k=1}^{2^n} \left \lfloor \log_2 k \right \rfloor =\left ( n-2 \right )2^n +n +2$$

Solution

Inequality

Let $x, y,z>0$ such that $x+y+z+2=xyz$ . Prove that:

$$x+y+z+6 \geq 2\left( \sqrt{xy}+ \sqrt{yz}+\sqrt{zx}\right)$$

Solution

Inequality

For all $x_1, x_2, \dots, x_n \geq 0$ , let $x_{n+1}=x_1$. Prove that:

$$\sum_{k=1}^{n}\sqrt{\frac{1}{\left ( x_k+1 \right )^2} + \frac{x_{k+1}^2}{\left ( x_{k+1}+1 \right )^2}} \geq \frac{n}{\sqrt{2}}$$

Solution

Double series

Evaluate the double series:

$$\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \frac{1}{ m^2n + mn^2 + 2mn}$$

Solution

Series with Euler's totient function

Let $\varphi$ denote Euler's totient function. Evaluate the series:

$$\mathcal{S}=\sum_{n=1}^{\infty} \frac{\varphi(n)}{2^n-1}$$

Solution

$1-1$ function

Let $f:\mathbb{N} \rightarrow [0, 1)$ be a function defined as $f(n)=\{ 2^{n+1/2} \}$ where $\{ \cdot\}$ denotes the fractional part. Prove that $f$ is $1-1$.

Solution

Sum with radicals

Evaluate the sum

$$\mathcal{S}= \sqrt{1+\frac{1}{1^2}+ \frac{1}{2^2}}+ \sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\cdots + \sqrt{1+\frac{1}{2011^2}+\frac{1}{2012^2}}$$

Solution

The series diverges

Let $a_n$ be a decreasing sequence such that $a_n>0$ and $\lim a_n =0$. Prove that the series:

$$\sum_{n=1}^{\infty} \frac{a_n - a_{n+1}}{a_n}$$

diverges.

Solution

Putnam 2015/A6

Let $n$ be a positive integer. Suppose that $A,B,$ and $M$ are $n\times n$ matrices with real entries such that $AM=MB,$ and such that $A$ and $B$ have the same characteristic polynomial. Prove that $\det(A-MX)=\det(B-XM)$ for every $n\times n$ matrix $X$ with real entries.

Solution

Putnam 2015/A3

Compute:

\[\log_2\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{2\pi iab/2015}\right)\right)\]

Solution

Rational (!) number

Prove that 

$$\mathscr{A}= \sqrt{1+ 1999\sqrt{1+2000\sqrt{4+2000\sqrt{1+2003\cdot 2005}}}}$$

is rational.

Solution

Square of rational

Let $x, y, z \in \mathbb{R}^*$ such that it holds $x+y+z=0$. Prove that the number

$$\mathcal{A}= \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$$

is a square of rational number.

Solution

Rationals or irrationals?

Determine whether the following numbers are rationals or irrationals.

a) $\displaystyle \mathcal{A}= \sqrt{33\cdot 5.\overline{15}+11\cdot 20.\overline{90}}$

b) $\displaystyle \mathcal{B}= \sqrt{\frac{\left | 33\cdot 5.\overline{15}-11\cdot 20.\overline{90} \right |}{10}}$

Solution

Floor function from India

For every $x \in \mathbb{R}$ let $[x]$ denote the floor function (that is the greater integer that is less or equal to $x$.) We define the real function $f:[-10, 10 ]\rightarrow \mathbb{R}$ as:

$$f(x)= \left\{\begin{matrix}
x-[x] & , & [x]=2k+1\\
 1+[x]-x&,  & [x]=2k
\end{matrix}\right., \quad k \in \mathbb{Z}$$

Evaluate the integral: $\displaystyle \mathcal{J}= \frac{\pi^2}{10}\int_{-10}^{10}f(x)\cos \pi x \, {\rm d}x$.

(IIT JEE 2010, India)

Solution

Solving equation using Geometry

Solve the equation:

$$\sqrt{x^2+y^2}+ \sqrt{x^2 + \left ( y-4 \right )^2}+ \sqrt{\left ( x-2 \right )^2+ y^2}+ \sqrt{\left ( x-2 \right )^2+\left ( y-4 \right )^2}=4 \sqrt{5}$$

Solution

Strategy to win

Two players take turns choosing one number at a time (without replacement) from the set  $\mathcal{A}=\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$.  The first player to obtain three numbers (out of three, four, or five) which sum to $0$  wins.

Does either player have a forced win?

Solution

A special sum

Evaluate the sum:

$$\mathbf{ \sum_{d \mid 10!} \frac{1}{d+\sqrt{10!}}}$$

Solution