Found problems: 119
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 $p$ 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 $S(x,n) = 0.$
Prove that for all positive integers $m$ and $n$,
$$\frac1m\cdot\binom{2n}0-\frac1{m+1}\cdot\binom{2n}1+\frac1{m+2}\cdot\binom{2n}2-\ldots+\frac1{m+2n}\cdot\binom{2n}{n2}>0$$
Given a positive integer \(n\), let \(\phi(n)\) denote the number of positive integers less than or equal to \(n\) that are relatively prime to \(n\). Find all possible positive integers \(k\) for which there exist positive integers \(1 \leq a_1 < a_2 < \dots < a_k\) such that:
\[
\left\lfloor \frac{\phi(a_1)}{a_1} + \frac{\phi(a_2)}{a_2} + \dots + \frac{\phi(a_k)}{a_k} \right\rfloor = 2024
\]
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 $S(x,n) = 0.$
Which of the following series are convergent?
$\textbf{(A)}~\sum_{n=1}^\infty\sqrt{\frac{2n^2+3}{5n^3+1}}$
$\textbf{(B)}~\sum_{n=1}^\infty\frac{(n+1)^n}{n^{n+3/2}}$
$\textbf{(C)}~\sum_{n=1}^\infty n^2x\left(1-x^2\right)^n$
$\textbf{(D)}~\text{None of the above}$
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?
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by
\[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right),
\] where $\lfloor x\rfloor$ denotes the integer part of $x$.
[b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often.
[b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often.
[i]Proposed by Johan Meyer, South Africa[/i]
Express $\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}$ as a rational number.
Find the value of
\[\sum_{a = 1}^{\infty} \sum_{b = 1}^{\infty} \sum_{c = 1}^{\infty} \frac{ab(3a + c)}{4^{a+b+c} (a+b)(b+c)(c+a)}.\]
What is the value of the series $\sum_{1 \leq l <m<n} \frac{1}{5^l3^m2^n}$
Compute $$\lim \limits_{n \to \infty}\frac{1}{n}\sum \limits_{k=1}^n\frac{\sqrt[n+k+1]{n+1}-\sqrt[n+k]{n}}{\sqrt[n+k]{n+1}-\sqrt[n+k]{n}}$$
Consider the sequence of real numbers $(a_n)_{n\ge1}$ such that
$$\lim_{n\to\infty}\frac1{n^r}\sum_{k=1}^n\frac{a_k}k=l\in\mathbb R,r\in\mathbb N^*$$
Show that:
$$\lim_{n\to\infty}\left(\dfrac{\displaystyle\sum_{p=n+1}^{2n}\sum_{k=1}^p\sum_{i=1}^k\frac{a_i}{p\cdot i}}{n^{r+1}}\right)=l\left(\frac{2^{r+1}}{r(r+1)}-\frac{2^r}{(r+1)^2}\right)$$
[i]Proposed by Florin Stănescu and Şerban Cioculescu[/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.
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]
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]
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]
[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]
If $\sum_{i=1}^n\cos^{-1}(\alpha_i)=0$, then find $\sum_{i=1}^n\alpha_i$.
$\textbf{(A)}~\frac n2$
$\textbf{(B)}~n$
$\textbf{(C)}~n\pi$
$\textbf{(D)}~\frac{n\pi}2$
Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u_k > a_k$ and $v_k > b_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1 \leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$, then $$\sum \limits_{k=1}^n(lu_kv_k-a_kb_k)\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^2\right)\right)^\frac{1}{2}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^2\right)\right)^\frac{1}{2}$$where$$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$
Let $p$ and $n$ be positive integers. Suppose that the numbers $c_{hk}$ ($h=1,2,\ldots,n$ ; $k=1,2,\ldots,ph$) satisfy $0 \leq c_{hk} \leq 1.$ Prove that
$$ \left( \sum \frac{ c_{hk} }{h} \right)^2 \leq 2p \sum c_{hk} ,$$
where each summation is over all admissible ordered pairs $(h,k).$
For each integer $ n\geq 2 $, let $ S_n $ be the sum of all products $ jk $, where $ j $ and $ k $ are integers and $ 1\leq j<k\leq n $. What is the sum of the 10 least values of $ n $ such that $ S_n $ is divisible by $ 3 $?
$\textbf{(A) }196\qquad\textbf{(B) }197\qquad\textbf{(C) }198\qquad\textbf{(D) }199\qquad\textbf{(E) }200$
Find three consecutive natural numbers, each of which is divisible by the square of the sum of its digits. Prove that there are no five such numbers in a row.
For each $n\geq 1$ let
$$a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \;\; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.$$
Show that $a_{n}\cdot b_{n}$ is an integer.
Prove that $ \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}$