Let $f : \mathbb R \to \mathbb R$ be a continuous function. Suppose that the restriction of $f$ to the set of irrational numbers is injective. What can we say about $f$? Answer the analogous question if $f$ is restricted to rationals.

A point $ P$ is performing a random walk on the $ X$-axis. At the instant $ t\equal{}0$, $ P$ is at a point $ x_0$ ($ |x_0|\le N$, where $ x_0$ and $ N$ denote integers, $ N>0$). If at an instant $ t$ ($ t$ being a nonnegative integer), $ P$ is at a point of $ x$ integer abscissa and $ |x|<N$, then by the instant $ t\plus{}1$ it reaches either the point $ x\plus{}1$ or the point $ x\minus{}1$, each with probability $ \frac12$. If at the instant $ t$, $ P$ is at the point $ x\equal{}N$ [$ x\equal{}\minus{}N$], then by the instant $ t\plus{}1$ it is certain to reach the point $ N\minus{}1$ [$ \minus{}N\plus{}1$]. Denote by $ P_k(t)$ the probability of $ P$ being at $ x\equal{}k$ at instant $ t$ ($ k$ is an integer). Find $ \lim_{t\to \infty}P_{k}(2t)$ and $ \lim_{t\to \infty}P_k(2t\plus{}1)$ for every fixed $ k$.

Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for all $n\ge 2$ . Define : $P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)$ Compute $\lim_{n\to \infty} P_n$

Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.

$f: \mathbb R\longrightarrow\mathbb R^{+}$ is a non-decreasing function. Prove that there is a point $a\in\mathbb R$ that \[f(a+\frac1{f(a)})<2f(a)\]

Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that $\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.

Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.

[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ [b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.

Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]

Let be a sequence $ \left( x_n \right)_{n\ge 0} $ with $ x_0\in (0,1) $ and defined as $$ 2x_n=x_{n-1}+\sqrt{3-3x_{n-1}^2} . $$ Prove that this sequence is bounded and periodic. Moreover, find $ x_0 $ for which this sequence is convergent. [i]Ovidiu Țâțan[/i]

Let $a_1\in (0,1)$ and $(a_n)_{n\ge 1}$ a sequence of real numbers defined by $a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1$. Evaluate $\lim_{n\to \infty} a_n\sqrt{n}$.

For $a>0$, let $f(a)=\lim_{t\rightarrow +0} \int_{t}^{1} |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$, find the minimum value of $f(a)$.

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\ \cos4 & \cos5 & \cos 6 \\ \cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.) Evaluate $ \lim_{n\to\infty}d_n.$

Let there be $(a_n)_{n\ge1},(b_n)_{n\ge1},a_n,b_n\in\mathbb R^*_+=(0,\infty)$ such that $\lim_{n\to\infty}a_n=a\in\mathbb R^*_+$ and $(b_n)_{n\ge1}$ is a bounded sequence. If $(x_n)_{n\ge1}$, $x_n=\prod_{k=1}^n(ka_h+b_h)$ find: $$\lim_{n\to\infty}\left(\sqrt[n+1]{x_{n+1}}-\sqrt[n]{x_n}\right)$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]

Let $ n\geq 2$ be natural number. Answer the following questions. (1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$ (2) Prove the following inequality. $ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$ (3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$

Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$. Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. [b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \] [b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]

Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

Let $x_0=a, x_1= b, x_2 = c$ be given real numbers and let $x_{n+2} = \frac{x_n + x_{n-1}}{2}$ for all $n\geq 1$. Show that the sequence $(x_n)_{n\geq 0}$ converges and find its limit.

How many $2$ digit number $n=ab$ ($a$ and $b$ are digits) have the property that $$n=a+b+a\times b$$ (A)$20$ (B)$15$ (C)$9$ (D)$8$

For every natural number $ r$, the set of $ r$-tuples of natural numbers is partitioned into finitely many classes. Show that if $ f(r)$ is a function such that $ f(r)\geq 1$ and $ \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty$, then there exists an infinite set of natural numbers that, for all $ r$, contains $ r$-triples from at most $ f(r)$ classes. Show that if $ f(r) \not \rightarrow \plus{}\infty$, then there is a family of partitions such that no such infinite set exists. [i]P. Erdos, A. Hajnal[/i]

Let $ C$ be a nonempty closed bounded subset of the real line and $ f: C\to C$ be a nondecreasing continuous function. Show that there exists a point $ p\in C$ such that $ f(p) \equal{} p$. (A set is closed if its complement is a union of open intervals. A function $ g$ is nondecreasing if $ g(x)\le g(y)$ for all $ x\le y$.)

Let be a sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1>0 $ and satisfying the equality $$ a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , $$ for all natural numbers $ n. $ [b]a)[/b] Find a recurrence relation between two consecutive elements of $ \left( a_n \right)_{n\ge 1} . $ [b]b)[/b] Prove that $ \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2. $