Find $\lim_{n\to\infty} \frac{1}{n}\sqrt[n]{\frac{(4n)!}{(3n)!}}.$

Evaluate $ \lim_{n\to\infty} \int_0^1 \int_0^1 \cdots \int_0^1 \cos ^ 2 \left\{\frac{\pi}{2n}(x_1\plus{}x_2\plus{}\cdots \plus{}x_n)\right\} dx_1dx_2\cdots dx_n.$

Given a real number $ c > 0$, a sequence $ (x_n)$ of real numbers is defined by $ x_{n \plus{} 1} \equal{} \sqrt {c \minus{} \sqrt {c \plus{} x_n}}$ for $ n \ge 0$. Find all values of $ c$ such that for each initial value $ x_0$ in $ (0, c)$, the sequence $ (x_n)$ is defined for all $ n$ and has a finite limit $ \lim x_n$ when $ n\to \plus{} \infty$.

If $(a_n)$ is a sequence of real numbers such that for $n \geq 1$ $$(2-a_n )a_{n+1} =1,$$ prove that $\lim_{n\to \infty} a_n =1.$

$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions: [b]I.[/b] $f(1) = 2$ [b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$ Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.

Evaluate the following limit. \[\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx\]

The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by \[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\] Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$

Let $ A$ and $ B$ be two Abelian groups, and define the sum of two homomorphisms $ \eta$ and $ \chi$ from $ A$ to $ B$ by \[ a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ .\] With this addition, the set of homomorphisms from $ A$ to $ B$ forms an Abelian group $ H$. Suppose now that $ A$ is a $ p$-group ( $ p$ a prime number). Prove that in this case $ H$ becomes a topological group under the topology defined by taking the subgroups $ p^kH \;(k\equal{}1,2,...)$ as a neighborhood base of $ 0$. Prove that $ H$ is complete in this topology and that every connected component of $ H$ consists of a single element. When is $ H$ compact in this topology? [L. Fuchs]

A function $f:[0,\infty)\to\mathbb R\setminus\{0\}$ is called [i]slowly changing[/i] if for any $t>1$ the limit $\lim_{x\to\infty}\frac{f(tx)}{f(x)}$ exists and is equal to $1$. Is it true that every slowly changing function has for sufficiently large $x$ a constant sign (i.e., is it true that for every slowly changing $f$ there exists an $N$ such that for every $x,y>N$ we have $f(x)f(y)>0$?)

Calculate $\displaystyle \sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right)$.

For positive integers $n, p$ with $n \geq p$, define real number $K_{n, p}$ as follows: $K_{n, 0} = \frac{1}{n+1}$ and $K_{n, p} = K_{n-1, p-1} -K_{n, p-1}$ for $1 \leq p \leq n.$ (i) Define $S_n = \sum_{p=0}^n K_{n,p} , \ n = 0, 1, 2, \ldots$ . Find $\lim_{n \to \infty} S_n.$ (ii) Find $T_n = \sum_{p=0}^n (-1)^p K_{n,p} , \ n = 0, 1, 2, \ldots$.

Find, with proof, all possible values of $t$ such that \[\lim_{n \to \infty} \left( \frac{1 + 2^{1/3} + 3^{1/3} + \dots + n^{1/3}}{n^t} \right ) = c\] for some real $c>0$. Also find the corresponding values of $c$.

[b]a)[/b] Provide an example of a sequence $ \left( a_n \right)_{n\ge 1} $ of positive real numbers whose series converges, and has the property that each member (sequence) of the family of sequences $ \left(\left( n^{\alpha } a_n \right)_{n\ge 1}\right)_{\alpha >0} $ is unbounded. [b]b)[/b] Let $ \left( b_n \right)_{n\ge 1} $ be a sequence of positive real numbers, having the property that $$ nb_{n+1}\leqslant b_1+b_2+\cdots +b_n, $$ for any natural number $ n. $ Prove that the following relations are equivalent: $\text{(i)} $ there exists a convergent member (series) of the family of series $ \left( \sum_{i=1}^{\infty } b_i^{\beta } \right)_{\beta >0} $ $ \text{(ii)} $ there exists a member (sequence) of the family of sequences $ \left(\left( n^{\beta } b_n \right)_{n\ge 1}\right)_{\beta >0} $ that is convergent to $ 0. $ [i]Eugen Păltănea[/i]

Find the following limit. \[\lim_{n\to\infty} \frac{1}{n} \log \left(\sum_{k=2}^{2^n} k^{1/n^2} \right)\]

Let $a_{n+1} = \frac{4}{7}a_n + \frac{3}{7}a_{n-1}$ and $a_0 = 1$, $a_1 = 2$. Find $\lim_{n \to \infty} a_n$.

On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at any speed, let $ M$ be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when $ M$ is divided by 10.

Find the limit $\lim \limits_{n \to \infty} \sin{n!}$ [list=1] [*] 1 [*] 0 [*] $\frac{\pi}{4}$ [*] None of the above [/list]

Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that (i) $m<2a$; (ii) $2n|(2am-m^2+n^2)$; (iii) $n^2-m^2+2mn\leq2a(n-m)$. For $(m, n)\in A$, let \[f(m,n)=\frac{2am-m^2-mn}{n}.\] Determine the maximum and minimum values of $f$.

Calculate $ \lim_{n\to\infty } \frac{f(1)+(f(2))^2+\cdots +(f(n))^n}{(f(n))^n} , $ where $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ is an unbounded and nondecreasing function. [i]Dan Popescu[/i]

Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$. Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$.

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function. a) Prove that $f$ is continous; b) Prove that there exists an unique function $g:[0,\infty)\to\mathbb{R}$ such that for all $x\geq 0$ we have \[ f(x+g(x)) = f(g(x)) - g(x) . \]

Let $f:\mathbb{N}\to\mathbb{N}$ be a bijection of the positive integers. Prove that at least one of the following limits is true: \[\lim_{N\to\infty}\sum_{n=1}^{N}\frac{1}{n+f(n)}=\infty;\qquad\lim_{N\to\infty}\sum_{n=1}^N\left(\frac{1}{n}-\frac{1}{f(n)}\right)=\infty.\][i]Proposed by Dylan Toh[/i]

Consider a right-angled triangle with $AB=1,\ AC=\sqrt{3},\ \angle{BAC}=\frac{\pi}{2}.$ Let $P_1,\ P_2,\ \cdots\cdots,\ P_{n-1}\ (n\geq 2)$ be the points which are closest from $A$, in this order and obtained by dividing $n$ equally parts of the line segment $AB$. Denote by $A=P_0,\ B=P_n$, answer the questions as below. (1) Find the inradius of $\triangle{P_kCP_{k+1}}\ (0\leq k\leq n-1)$. (2) Denote by $S_n$ the total sum of the area of the incircle for $\triangle{P_kCP_{k+1}}\ (0\leq k\leq n-1)$. Let $I_n=\frac{1}{n}\sum_{k=0}^{n-1} \frac{1}{3+\left(\frac{k}{n}\right)^2}$, show that $nS_n\leq \frac {3\pi}4I_n$, then find the limit $\lim_{n\to\infty} I_n$. (3) Find the limit $\lim_{n\to\infty} nS_n$.

Let $n\ge 2$ be a positive integer. Find the positive integers $x$ \[\sqrt{x+\sqrt{x+\ldots +\sqrt{x}}}<n \] for any number of radicals.

Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says $0$. In each turn except the first the possible moves are determined from the previous number $n$ in the following way: write \[n =\sum_{m\in O_n}2^m;\] the valid numbers are the elements $m$ of $O_n$. That way, for example, after Arnaldo says $42= 2^5 + 2^3 + 2^1$, Bernaldo must respond with $5$, $3$ or $1$. We define the sets $A,B\subset \mathbb{N}$ in the following way. We have $n\in A$ iff Arnaldo, saying $n$ in his first turn, has a winning strategy; analogously, we have $n\in B$ iff Bernaldo has a winning strategy if Arnaldo says $n$ during his first turn. This way, \[A =\{0, 2, 8, 10,\cdots\}, B = \{1, 3, 4, 5, 6, 7, 9,\cdots\}\] Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=|A\cap \{0,1,\cdots,n-1\}|$. For example, $f(8) = 2$ and $f(11)=4$. Find \[\lim_{n\to\infty}\frac{f(n)\log(n)^{2005}}{n}\]