Let $ f:\mathbb{R} \to \mathbb{R} $ be an increasing function satisfying the following conditions: 1. $ f(x+1) = f(x) + 1 $ for each $ x \in \mathbb{R} $, 2. there exists an integer p such that $ f(f(f(O))) = p $. Prove that for every real number $ x $ $$ \lim_{n\to \infty} \frac{x_n}{n} = \frac{p}{3}.$$ where $ x_1 = x $ and $ x_n =f(x_{n-1}) $ for $ n = 2, 3, \ldots $.

Define a sequence $(a_n)$ by $a_0 =0$ and $a_n = 1 +\sin(a_{n-1}-1)$ for $n\geq 1$. Evaluate $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k.$$

Let $n$ be positive integer. Define a sequence $\{a_k\}$ by \[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\] (1) Find $a_2$ and $a_3$. (2) Find the general term $a_k$. (3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$. 50 points

Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

Let $f:\;\mathbb{R}\to\mathbb{R}$ be a continuously differentiable function that satisfies $f'(t)>f(f(t))$ for all $t\in\mathbb{R}$. Prove that $f(f(f(t)))\le0$ for all $t\ge0$. [i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]

Let $C$ be a fixed circle, $u > 0$ be a fixed real and let $v_0 , v_1 , v_2 , \ldots$ be a sequence of positive real numbers. Two ants $A$ and $B$ walk around the perimeter of $C$ in opposite directions, starting from the same starting point. Ant $A$ has a constant speed $u$, while ant $B$ has an initial speed $v_0$. For each positive integer $n$, when the two ants collide for the $n$−th time, they change the directions in which they walk around the perimeter of $C$, with ant $A$ remaining at speed $u$ and ant $B$ stops walking at speed $v_{n-1}$ to walk at speed $v_n$. (a) If the sequence $\{v_n\}$ is strictly increasing, with $\lim_{n\rightarrow \infty} v_n = +\infty$, prove that there is exactly one point in $C$ that ant $A$ will pass "infinitely" many times. (b) Prove that there is a sequence $\{v_n\}$ with $\lim_{n\rightarrow\infty} v_n = +\infty$, such that ant $A$ will pass "infinitely" many times through all points on the circle $C$.

Given a convex, $n$-sided polygon $P$, form a $2n$-sided polygon $\text{clip}(P)$ by cutting off each corner of $P$ at the edges' trisection points. In other words, $\text{clip}(P)$ is the polygon whose vertices are the $2n$ edge trisection points of $P$, connected in order around the boundary of $P$. Let $P_1$ be an isosceles trapezoid with side lengths $13,13,13,$ and $3$, and for each $i\geq 2$, let $P_i=\text{clip}(P_{i-1}).$ This iterative clipping process approaches a limiting shape $P_\infty=\lim_{i\to\infty}P_i$. If the difference of the areas of $P_{10}$ and $P_\infty$ is written as a fraction $\tfrac xy$ in lowest terms, calculate the number of positive integer factors of $x\cdot y$.

Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's launched up. Find $\lim_{r\to0} \left(f(r)\right) $. Express your answer is meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$. [i](B. Dejean, 6 points)[/i]

Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$ Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $ [i]Radu Gologan[/i]

Let $ n$ be positive integer. For $ n \equal{} 1,\ 2,\ 3,\ \cdots n$, let denote $ S_k$ be the area of $ \triangle{AOB_k}$ such that $ \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k$. Find the limit $ \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k$.

Let $f(x)$ be a real-valued function defined for $0<x<1.$ If $$ \lim_{x \to 0} f(x) =0 \;\; \text{and} \;\; f(x) - f \left( \frac{x}{2} \right) =o(x),$$ prove that $f(x) =o(x),$ where we use the O-notation.

Let $(a_n)^\infty_{n=1}$ be an unbounded and strictly increasing sequence of positive reals such that the arithmetic mean of any four consecutive terms $a_n,a_{n+1},a_{n+2},a_{n+3}$ belongs to the same sequence. Prove that the sequence $\frac{a_{n+1}}{a_n}$ converges and find all possible values of its limit.

$(1)$ Start with $a$ white balls and $b$ black balls. $(2)$ Draw one ball at random. $(3)$ If the ball is white, then stop. Otherwise, add two black balls and go to step $2$. Let $S$ be the number of draws before the process terminates. For the cases $a = b = 1$ and $a = b = 2$ only, find $a_n = P(S = n), b_n = P(S \le n), \lim_{n\to\infty} b_n$, and the expectation value of the number of balls drawn: $E(S) =\displaystyle\sum_{n\ge1} na_n.$

In a $4\times 4$ chessboard, in how many ways can you place $3$ rooks and one bishop such that none of these pieces threaten another piece?

[b](a)[/b] Let $n$ be a positive integer, prove that \[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\] [b](b)[/b] Find a positive integer $n$ for which \[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]

The ratio $ \dfrac{10^{2000}\plus{}10^{2002}}{10^{2001}\plus{}10^{2001}}$ is closest to which of the following numbers? $ \text{(A)}\ 0.1\qquad \text{(B)}\ 0.2\qquad \text{(C)}\ 1\qquad \text{(D)}\ 5\qquad \text{(E)}\ 10$

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $\lim \limits_{x\to \infty}f'(x)=1$, then [list=1] [*] $f$ is increasing [*] $f$ is unbounded [*] $f'$ is bounded [*] All of these [/list]

5) f twice cont diff, $|f''(x)+2xf'(x)+(x^{2}+1)f(x)|\leq 1$. prove $\lim_{x\rightarrow +\infty} f(x) = 0$

Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) \equal{} 0, f(1) \equal{} 1,$ and \[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\] Determine $ f \left( \frac{1}{7} \right).$

Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic function and $F:\mathbb{R}\to\mathbb{R}$ given by \[F(x)=\int_0^xf(t)\ \text{d}t.\] Prove that if $F$ has a finite derivative, then $f$ is continuous. [i]Dorin Andrica & Mihai Piticari[/i]

Let $A$ be a nonempty set of positive integers, and let $N(x)$ denote the number of elements of $A$ not exceeding $x$. Let $B$ denote the set of positive integers $b$ that can be written in the form $b=a-a^{\prime}$ with $a\in A$ and $a^{\prime}\in A$. Let $b_1<b_2<\cdots$ be the members of $B$, listed in increasing order. Show that if the sequence $b_{i+1}-b_i$ is unbounded, then $\lim_{x\to \infty}\frac{N(x)}{x}=0$.

Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.

Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.

Let $0<t<\frac{\pi}{2}$. Evaluate \[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]

Given infinite sequence $a_n$. It is known that the limit of $$b_n=a_{n+1}-a_n/2$$ equals zero. Prove that the limit of $a_n$ equals zero.