i) Let $a,b$ be real numbers such that $b\leq 0$ and $1+ax+bx^{2} \geq 0$ for every $x\in [0,1]$. Prove that $$\lim_{n\to \infty} n \int_{0}^{1}(1+ax+bx^{2})^{n}dx= \begin{cases} -\frac{1}{a} &\text{if}\; a<0,\\ \infty & \text{if}\; a \geq 0. \end{cases}$$ ii) Let $f:[0,1]\rightarrow[0,\infty)$ be a function with a continuous second derivative and let $f''(x)\leq0$ for every $x\in [0,1]$. Suppose that $L=\lim_{n\to \infty} n \int_{0}^{1}(f(x))^{n}dx$ exists and $0<L<\infty$. Prove that $f'$ has a constant sign and $\min_{x\in [0,1]}|f'(x)|=L^{-1}$.

Let be the sequence $ \left( x_n \right)_{n\ge 1} $ defined as $$ x_n= \frac{4009}{4018020} x_{n-1} -\frac{1}{4018020} x_{n-2} + \left( 1+\frac{1}{n} \right)^n. $$ Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent and determine its limit.

Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$ [i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]

Let $f\colon 2^S\rightarrow \mathbb R$ be a function defined on the subsets of a finite set $S$. Prove that if $f(A)=F(S\backslash A)$ and $\max \{ f(A), f(B)\}\geq f(A\cup B)$ for all subsets $A, B$ of $S$, then $f$ assumes at most $|S|$ distinct values.

Show that for all $n \in \mathbb{N}$, \[n = \sum^{}_{d \vert n}\phi(d).\]

Given $ 0< c< 1$, we define $f(x) = \begin{cases} \frac{x}{c} & x \in [0,c] \\ \frac{1-x}{1-c} & x \in [c, 1] \end{cases} $ Let $f^{n}(x)=f(f(...f(x)))$ . Show that for each positive integer $n$, $f^{n}$ has a non-zero finite nunber of fixed points which aren't fixed points of $f^k$ for $k< n$.

Study the convergence of the sequence $$ \left( \sum_{k=2}^{n+1} \sqrt[k]{n+1} -\sum_{k=2}^{n} \sqrt[k]{n} \right)_{n\ge 2} , $$ and calculate its limit. [i]Dan Negulescu[/i]

Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation \[ \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}. \]

Let $a$ and $b$ real numbers. Let $f:[a,b] \rightarrow \mathbb{R}$ a continuous function. We say that f is "smp" if $[a,b]=[c_0,c_1]\cup[c_1,c_2]...\cup[c_{n-1},c_n]$ satisfying $c_0<c_1...<c_n$ and for each $i\in\{0,1,2...n-1\}$: $c_i<x<c_{i+1} \Rightarrow f(c_i)<f(x)<f(c_{i+1})$ or $c_i>x>c_{i+1} \Rightarrow f(c_i)>f(x)>f(c_{i+1})$ Prove that if $f:[a,b] \rightarrow \mathbb{R}$ is continuous such that for each $v\in\mathbb{R}$ there are only finitely many $x$ satisfying $f(x)=v$, then $f$ is "smp".

Prove that for a positive number $ r>1, $ there is a nondecreasing sequence of positive numbers $ \left( a_v\right)_{v\ge 1} $ such that $$ r=\lim_{n\to\infty }\sum_{i=1}^n \frac{a_i}{a_{i+1}} . $$

Let $p_1$ and $p_2$ be positive real numbers. Prove that there exist functions $f_i\colon \mathbb R \rightarrow \mathbb R$ such that the smallest positive period of $f_i$ is $p_i\, (i=1, 2)$, and $f_1-f_2$ is also periodic. [J. Riman]

[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $ [i]Florin Stănescu[/i]

Let $a,b \in \mathbb{R},$ $a<b$ and $f,g:[a,b] \to \mathbb{R}$ two differentiable functions with increasing derivatives and $f'(a)>0,$ $g'(a)>0.$ Prove that there exists $c \in [a,b]$ such that $$\frac{f(b)-f(a)}{b-a} \cdot \frac{g(b)-g(a)}{b-a}=f'(c)g'(c).$$

Suppose that $X$ and $Y$ are two metric spaces and $f:X \longrightarrow Y$ is a continious function. Also for every compact set $K \subseteq Y$, it's pre-image $f^{pre}(K)$ is a compact set in $X$. Prove that $f$ is a closed function, i.e for every close set $C\subseteq X$, it's image $f(C)$ is a closed subset of $Y$.

Let \[T=\{(tq,1-t) \in\mathbb R^{2}| t \in [0,1],q\in\mathbb Q\}\]Prove that each continuous function $f: T\longrightarrow T$ has a fixed point.

Joaquim, José and João participate of the worship of triangle $ABC$. It is well known that $ABC$ is a random triangle, nothing special. According to the dogmas of the worship, when they form a triangle which is similar to $ABC$, they will get immortal. Nevertheless, there is a condition: each person must represent a vertice of the triangle. In this case, Joaquim will represent vertice $A$, José vertice $B$ and João will represent vertice $C$. Thus, they must form a triangle which is similar to $ABC$, in this order. Suppose all three points are in the Euclidean Plane. Once they are very excited to become immortal, they act in the following way: in each instant $t$, Joaquim, for example, will move with constant velocity $v$ to the point in the same semi-plan determined by the line which connects the other two points, and which would create a triangle similar to $ABC$ in the desired order. The other participants act in the same way. If the velocity of all of them is same, and if they initially have a finite, but sufficiently large life, determine if they can get immortal. [i]Observation: Initially, Joaquim, José and João do not represent three collinear points in the plane[/i]

Is is true that if the perfect set $F\subseteq [0,1]$ is of zero Lebesgue measure then those functions in $C^1[0,1]$ which are one-to-one on $F$ form a dense subset of $C^1[0,1]$? (We use the metric $$d(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)| + \sup_{x\in[0,1]} |f'(x)-g'(x)|$$ to define the topology in the space $C^1[0,1]$ of continuously differentiable real functions on $[0,1]$.)

Let be a sequence $ \left( u_n \right)_{n\ge 1} $ given by the recurrence relation $ u_{n+1} =u_n+\sqrt{u_n^2-u_1^2} , $ and the constraints $ u_2\ge u_1>0. $ Calculate $ \lim_{n\to\infty }\frac{2^n}{u_n} . $ [i]Dan Negulescu[/i]

Let $\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f$ continuous $\}$ and $n$ an integer, $n\geq 2$. Find the smallest real constant $c$ such that for any $f\in \mathcal F$ the following inequality takes place \[ \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx. \]

Consider a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that for any third degree polynomial function $P:[0,1]\to [0,1]$, we have \[\int_0^1f(P(x))dx=0\] Prove that $f(x)=0,\ (\forall)x\in [0,1]$. [i]Mihai Piticari[/i]

A function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has property $ \mathcal{F} , $ if for any real number $ a, $ there exists a $ b<a $ such that $ f(x)\le f(a), $ for all $ x\in (b,a) . $ [b]a)[/b] Give an example of a function with property $ \mathcal{F} $ that is not monotone on $ \mathbb{R} . $ [b]b)[/b] Prove that a continuous function that has property $ \mathcal{F} $ is nondecreasing.

Let $f:\mathbb{R}\rightarrow[0,\infty )$ be a function with the property that $|f(x)-f(y)|\le |x-y|$ for every $x,y\in\mathbb{R}$. Show that: a) If $\lim_{n\rightarrow \infty} f(x+n)=\infty$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\infty$. b) If $\lim_{n\rightarrow \infty} f(x+n)=\alpha ,\alpha\in[0,\infty )$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\alpha$.

Let $f:\mathbb{R}\rightarrow [0,\infty)$. Prove that $f(x+y)\ge (y+1)f(x),\ (\forall)x\in \mathbb{R}$ if and only if the function $g:\mathbb{R}\rightarrow [0,\infty),\ g(x)=e^{-x}f(x),\ (\forall)x\in \mathbb{R}$ is increasing.

Let $a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1$. a) Show that $\sum\limits_{n=1}^{+\infty}a_nx^n$ converges for all $x \in (-4, 4)$ and that the function $f(x) = \sum\limits_{n=1}^{+\infty}a_nx^n$ satisfies the differential equation $x(x - 4)f'(x) + (x + 2)f(x) = -x$. b) Prove that $\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}$.

Let $p$ be the [i]potentioral[/i] function, from positive integers to positive integers, defined by $p(1) = 1$ and $p(n + 1) = p(n)$, if $n + 1$ is not a perfect power and $p(n + 1) = (n + 1) \cdot p(n)$, otherwise. Is there a positive integer $N$ such that, for all $n > N,$ $p(n) > 2^n$?