For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$. [list=1] [*] Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$? [*] Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?[/list] [i]Brian Hamrick.[/i]

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

[list=1] [*] Prove that $$\lim \limits_{n \to \infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)=0$$ [*] Calculate $$\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)$$ [/list]

Evaluate $\sum_{k=0}^{37}(-1)^k\binom{75}{2k}$.

Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$. Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$.

Prove that $$\sum_{k=0}^\infty x^k\frac{1+x^{2k+2}}{(1-x^{2k+2})^2}=\sum_{k=0}^\infty(-1)^k\frac{x^k}{(1-x^{k+1})^2}$$for all $x\in(-1,1)$.

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

a) Is it true that for every bijection $f:\mathbb N\to\mathbb N$ the series $$\sum_{n=1}^\infty\frac1{nf(n)}$$is convergent? b) Prove that there exists a bijection $f:\mathbb N\to\mathbb N$ such that the series $$\sum_{n=1}^\infty\frac1{n+f(n)}$$is convergent. ($\mathbb N$ is the set of all positive integers.)

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Find $\sum_{k=1}^\infty\frac{k^2-2}{(k+2)!}$.

Let $f$ be a function satisfying $f(x+y+z)=f(x)+f(y)+f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1)=1$, $f(2)=2$. Then $\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals [list=1] [*] 4 [*] 6 [*] 12 [*] 24 [/list]

Given that $a_1, a_2, \dots, a_{10}$ are positive real numbers, determine the smallest possible value of \[\sum \limits_{i = 1}^{10} \left\lfloor \frac{7a_i}{a_i+a_{i+1}}\right\rfloor\] where we define $a_{11} = a_1$. [i]Proposed by Sutanay Bhattacharya[/i]

A given natural number $N$ is being decomposed in a sum of some consecutive integers. [b]a.)[/b] Find all such decompositions for $N=500.$ [b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only? [b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$ [b]Note by Darij:[/b] The $0$ is not considered as a natural number.

Let $n \in \mathbb{N}$, $n \geq 2$, $a_1, a_2, \cdots , a_n \in \mathbb{R}$ and $a_n = max \{a_1, a_2,\cdots , a_n\}$ [list=1] [*]If $t_k$, $t'_k \in \mathbb{R}$, $k \in \{1, 2,\cdots , n\}$ , $t_k \leq t'_k$, for any $k \in \{1, 2, \cdots, n - 1\}$ and $$\sum \limits_{k=1}^nt_k=\sum \limits_{k=1}^nt'_k$$Prove that $$\sum \limits_{k=1}^nt_ka_k\geq \sum \limits_{k=1}^nt'_ka_k$$ [*] If $b_k$, $c_k \in \mathbb{R}^*_+$, $k \in \{1, 2,\cdots , n\}$ , $b_k \leq c_k$ for any $k \in \{1, 2,\cdots, k - 1\}$ and $$b_1b_2\cdots b_n=c_1c_2\cdots c_n$$Prove that $$\prod \limits_{k=1}^n b_k^{a_k}\geq \prod \limits_{k=1}^nc_k^{a_k}$$ [/list]

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Let $x_1, x_2,\geq , x_n$ be a positive numbers, $k \geq 1$. Then the following inequality is true: $$\left(x_1^k+x_2^k+\cdots +x_n^k\right)^{k+1}\geq \left(x_1^{k+1}+x_2^{k+1}\cdots +x_n^{k+1}\right)^k+2\left(\sum \limits_{1\leq i<j\leq n}x_i^kx_j\right)^k$$

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

Let $n$ and $k$ be positive integers. Evaluate the following sum $$\sum_{j=0}^k\binom kj^2\binom{n+2k-j}{2k}$$where $\binom nk=\frac{n!}{k!(n-k)!}$.

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

Suppose that $a_1 = 2$ and the sequence $(a_n)$ satisfies the recurrence relation \[\frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}\] for all $n \ge 2.$ What is the greatest integer less than or equal to \[\sum^{100}_{n=1} a_n^2?\] $\textbf{(A) } 338{,}550 \qquad \textbf{(B) } 338{,}551 \qquad \textbf{(C) } 338{,}552 \qquad \textbf{(D) } 338{,}553 \qquad \textbf{(E) } 338{,}554$

The number $n$ is called a composite number if it can be written in the form $n = a\times b$, where $a, b$ are positive integers greater than $1$. Write number $2013$ in a sum of $m$ composite numbers. What is the largest value of $m$? (A): $500$, (B): $501$, (C): $502$, (D): $503$, (E): None of the above.

Let $a_n>0$, $n\ge1$. Consider the right triangles $\triangle A_0A_1A_2$, $\triangle A_0A_2A_3,\ldots$, $\triangle A_0A_{n-1}A_n,\ldots,$ as in the figure. (More precisely, for every $n\ge2$ the hypotenuse $A_0A_n$ of $\triangle A_0A_{n-1}A_n$ is a leg of $\triangle A_0A_nA_{n+1}$ 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 $n\ge1$.) [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8yL2M1ZjAxM2I1ZWU0N2E4MzQyYWIzNmQ5OGM3NjJlZjljODdmMTliLnBuZw==&rn=U0VFTU9VUyAyMDEyLnBuZw==[/img] Is it possible for the set of points $\{A_n\mid n\ge0\}$ to be unbounded but the series $\sum_{n=2}^\infty m\angle A_{n-1}A_0A_n$ to be convergent? [i]Note.[/i] A subset $B$ of the plane is bounded if and only if there is a disk $D$ such that $B\subseteq D$.

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

Each of the numbers $x_1, x_2, \ldots, x_{101}$ is $\pm 1$. What is the smallest positive value of $\sum_{1\leq i < j \leq 101} x_i x_j$ ?