Found problems: 373
Find the integer part of $ \frac{1}{\sqrt1}+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+...+\frac{1}{\sqrt{1000}} $
Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$
\[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \]
Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.
For each $k$, $\mathcal{C}_k$ is biased so that, when tossed, it has probability $\tfrac{1}{(2k+1)}$ of falling heads. If the $n$ coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function $n$.
Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \]
Let $ a_1\geq \cdots \geq a_n \geq a_{n \plus{} 1} \equal{} 0$ be real numbers. Show that
\[ \sqrt {\sum_{k \equal{} 1}^n a_k} \leq \sum_{k \equal{} 1}^n \sqrt k (\sqrt {a_k} \minus{} \sqrt {a_{k \plus{} 1}}).
\]
[i]Proposed by Romania[/i]
What is the value of the product $$\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?$$
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{50}{99}\qquad\textbf{(C) }\frac{9800}{9801}\qquad\textbf{(D) }\frac{100}{99}\qquad\textbf{(E) } 50$
Find this:
$ (1+\frac{1}{5})(1+\frac{1}{6})...(1+\frac{1}{2023})(1+\frac{1}{2024}) $
Calculate the sum
$\frac{5}{2.7}+\frac{5}{7.12}+...+\frac{5}{2002.2007}$
Suppose that
\[S_n=\frac 59 \times \frac{14}{20} \times \frac{27}{35} \times \cdots \times \frac{2n^2-n-1}{2n^2+n-1}\]
Find $\lim_{n \to \infty} S_n.$
Consider the sequence defined by $a_k=\frac 1{k^2+k}$ for $k\ge 1.$ Given that $a_m+a_{m+1}+\cdots+a_{n-1}=1/29,$ for positive integers $m$ and $n$ with $m<n$, find $m+n.$
Let $ a\equal{}\pi/2008$. Find the smallest positive integer $ n$ such that
\[ 2[\cos(a)\sin(a)\plus{}\cos(4a)\sin(2a)\plus{}\cos(9a)\sin(3a)\plus{}\cdots\plus{}\cos(n^2a)\sin(na)]\] is an integer.
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
Given:
\[ S = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1} {6}} + \cdots + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + \cdots + \frac{1} {1993006}} \]
where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. $k=\frac{n(n+1)}{2}$ for $n = 1$, $2$, $\ldots$,$1996$). Prove that $S>1001$.
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
What is the sum $ \frac{1}{1 \cdot 2 \cdot 3} \plus{} \frac{1}{2 \cdot 3 \cdot 4} \plus{} \cdots \plus{} \frac{1}{1996 \cdot 1997 \cdot 1998}$?
A. $ \frac{2 \cdot 1997}{3 \cdot 1996 \cdot 1998}$
B. $ \frac{1}{3} \minus{} \frac{1}{3 \cdot 1998}$
C. $ \frac{1}{4} \minus{} \frac{1}{1997^2}$
D. $ \frac{1}{3} \minus{} \frac{1}{3 \cdot 1997 \cdot 1998}$
E. $ \frac{1}{4} \minus{} \frac{1}{2 \cdot 1997 \cdot 1998}$
For a positive integer $n>2$, consider the $n-1$ fractions $$\dfrac21, \dfrac32, \cdots, \dfrac{n}{n-1}$$ The product of these fractions equals $n$, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of $n$ for which this is possible and prove that you have found them all.
Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$.
[i]Victor Wang[/i]
Prove that $\sqrt{a_1}+\sqrt{a_2}+\cdots+\sqrt{a_{119}}$ is an integer, where \[a_n=2-\frac{1}{n^2+\sqrt{n^4+1/4}}.\]
Prove that
$$S_n=\frac1{1^2}+\frac1{2^2}+\ldots+\frac1{n^2}<2$$for every $n\in\mathbb N$.
Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has
\[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]
Evaluate $\textstyle\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$.
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
Find the sum \[\frac{1}{1(3)}+\frac{1}{3(5)}+\dots+\frac{1}{(2n-1)(2n+1)}+\dots+\frac{1}{255(257)}.\]
$\textbf{(A) }\frac{127}{255}\qquad\textbf{(B) }\frac{128}{255}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{128}{257}\qquad \textbf{(E) }\frac{129}{257}$
[b] Problem 5. [/b]Denote with $d(a,b)$ the numbers of the divisors of natural $a$, which are greater or equal to $b$. Find all natural $n$, for which
$d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$
[i]Ivan Landgev[/i]