[b]Problem 2[/b] Determine all pairs $(n, m)$ of positive integers satisfying the equation $$5^n = 6m^2 + 1\ . $$

Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$. [i]G. Halasz[/i]

For $p \in \mathbb{R}$, let $(a_n)_{n \ge 1}$ be the sequence defined by \[ a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx. \] Determine all possible values of $p$ for which the series $\sum_{n=1}^\infty a_n$ converges.

Find all continous functions $f:[0,1]\to[0,2]$ with the property that $\left(\int_{\frac1{n+1}}^{\frac1n}xf(x)dx\right)^2=\int_{\frac1{n+1}}^{\frac1n}x^2f(x)dx, \forall n\in\mathbb N^*$. [i]Gabriel Marsanu, Andrei Nedelcu[/i]

Define a sequence by $a_1=1,a_2=\frac12$, and $a_{n+2}=a_{n+1}-\frac{a_na_{n+1}}2$ for $n$ a positive integer. Find $\lim_{n\to\infty}na_n$.

Show that a continuous function $f : \mathbb{R} \to \mathbb{R}$ is increasing if and only if \[(c - b)\int_a^b f(x)\, \text{d}x \le (b - a) \int_b^c f(x) \, \text{d}x,\] for any real numbers $a < b < c$.

Determine all continuous, strictly monotone functions $\phi : \mathbb{R}^+\to\mathbb{R}$ such that $$F(x,y)=\phi^{-1} \left(\frac{x\phi(x)+y\phi(y)}{x+y}\right) + \phi^{-1} \left(\frac{y\phi(x)+x\phi(y)}{x+y}\right) $$ is homogeneous of degree 1, ie $F(tx,ty)=tF(x,y) , \forall x,y,t\in\mathbb{R}^+$ [hide=Note]F(x,y)=F(y,x) and F(x,x)=2x[/hide]

Let be a twice-differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the properties that: $ \text{(i) supp} f''=f\left(\mathbb{R}\right) $ $ \text{(ii)}\exists g:\mathbb{R}\longrightarrow\mathbb{R}\quad\forall x\in\mathbb{R}\quad f(x+1)=f(x)+f'\left( g(x)\right)\text{ and } f'(x+1)=f'(x)+f''\left( g(x)\right) $ Prove that: [b]a)[/b] any such $ g $ is injective. [b]b)[/b] $ f $ is of class $ C^{\infty } , $ and for any natural number $ n, $ any real number $ x $ and any such $ g, $ $$f^{(n)}(x+1)=f^{(n)}(x)+f^{(n+1)}\left( g(x)\right) . $$ [i]Laurențiu Panaitopol[/i]

Let $p>1$. Show that there exists a constant $K_{p} >0$ such that for every $x,y\in \mathbb{R}$ with $|x|^{p}+|y|^{p}=2$, we have $$(x-y)^{2} \leq K_{p}(4-(x+y)^{2}).$$

Find $\sum_{k=1}^\infty\frac{k^2-2}{(k+2)!}$.

For a set $A$, we define $A + A = \{a + b: a, b \in A \}$. Determine whether there exists a set $A$ of positive integers such that $$\sum_{a \in A} \frac{1}{a} = +\infty \quad \text{and} \quad \lim_{n \rightarrow +\infty} \frac{|(A+A) \cap \{1,2,\cdots,n \}|}{n}=0.$$ [hide=Note]Google translated from [url=http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales]http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales[/url][/hide]

[b]a)[/b] Prove that $ \lim_{x\to\infty } \sqrt{x}\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x}=1. $ [b]b)[/b] Show that $ \lim_{x\to\infty } \left( -\left\lfloor\sqrt{x}\right\rfloor +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) =\frac{-1}{2} $ [b]c)[/b] What about $ \lim_{x\to\infty } \left( -\sqrt{x} +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) ? $

[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality $$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$ holds. [b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?

Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ be two points in the image of $f$ (that is, there exists $x,y$ such that $f(x)=a$ and $f(y)=b$). Show that there is an interval $I$ such that $f(I)=[a,b]$.

Let $ \pi_n(x)$ be a polynomial of degree not exceeding $ n$ with real coefficients such that \[ |\pi_n(x)| \leq \sqrt{1\minus{}x^2} \;\textrm{for}\ \;\minus{}1\leq x \leq 1 \ .\] Then \[ |\pi'_n(x)| \leq 2(n\minus{}1).\] [i]P. Turan[/i]

A function \[\text{f:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] is called contract if, for every numbers $x,y\in \text{(0,}\infty \text{)}$ we have, $\underset{n\to \infty }{\mathop{\lim }}\,\left( {{f}^{n}}\left( x \right)-{{f}^{n}}\left( y \right) \right)=0$ where ${{f}^{n}}=\underbrace{f\circ f\circ ...\circ f}_{n\ f\text{'s}}$ a) Consider \[f:\text{(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] a function contract, continue with the property that has a fixed point, that existing ${{x}_{0}}\in \text{(0,}\infty \text{) }$ there so that $f\left( {{x}_{0}} \right)={{x}_{0}}.$ Show that $f\left( x \right)>x,$ for every $x\in \text{(0,}{{x}_{0}}\text{)}\,$ and $f\left( x \right)<x$, for every $x\in \text{(}{{x}_{0}}\text{,}\infty \text{)}\,$. b) Show that the given function \[f\text{:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] given by $f\left( x \right)=x+\frac{1}{x}$ is contracted but has no fix number.

Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if \[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\] for every prime number $p$ and every real number $y.$

Let be a real number $ a, $ and a function $ f:\mathbb{R}_{>0 }\longrightarrow\mathbb{R}_{>0 } . $ Show that the following relations are equivalent. $ \text{(i)}\quad\varepsilon\in\mathbb{R}_{>0 } \implies\left( \lim_{x\to\infty } \frac{f(x)}{x^{a+\varepsilon }} =0\wedge \lim_{x\to\infty } \frac{f(x)}{x^{a-\varepsilon }} =\infty \right) $ $ \text{(ii)}\quad\lim_{x\to\infty } \frac{\ln f(x)}{\ln x } =a $

Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

Find all $a$ real number such that $x_n=n\{an! \}$ is convergeant Gabriel Dospinescu

Let \( n \) be a positive integer. Determine the largest possible value of \( k \) with the following property: there exists a bijective function \( \phi: [0, 1] \to [0, 1]^k \) and a constant \( C > 0 \) such that, for all \( x, y \in [0, 1] \), \[ \| \phi(x) - \phi(y) \| \leq C \| x - y \|^k. \] Note: \( \| \cdot \| \) denotes the Euclidean norm, that is, \( \| (a_1, \ldots, a_n) \| = \sqrt{a_1^2 + \cdots + a_n^2} \).

Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.

Assume that $C$ is a convex subset of $\mathbb R^{d}$. Suppose that $C_{1},C_{2},\dots,C_{n}$ are translations of $C$ that $C_{i}\cap C\neq\emptyset$ but $C_{i}\cap C_{j}=\emptyset$. Prove that \[n\leq 3^{d}-1\] Prove that $3^{d}-1$ is the best bound. P.S. In the exam problem was given for $n=3$.

Given the polynomial $$P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}.$$ Express the numerical value of its derivative of order $325$ for $x = 0$.