Prove that for any integers $n,p$, $0<n\leq p$, all the roots of the polynomial below are real: \[P_{n,p}(x)=\sum_{j=0}^n {p\choose j}{p\choose {n-j}}x^j\]

[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]

Let be a real number $ r $ and two functions $ f:[r,\infty )\longrightarrow\mathbb{R} , F_1:(r,\infty )\longrightarrow\mathbb{R} $ satisfying the following two properties. $ \text{(i)} f $ has Darboux's intermediate value property. $ \text{(ii)} F_1$ is differentiable and $ F'_1=f\bigg|_{(r,\infty )} $ [b]1)[/b] Provide an example of what $ f,F_1 $ could be if $ f $ hasn't a lateral limit at $ r, $ and $ F_1 $ has lateral limit at $ r. $ Moreover, if $ f $ has lateral limit at $ r, $ show that [b]2)[/b] $ F_1 $ has a finite lateral limit at $ r. $ [b]3)[/b] the function $ F:[r,\infty )\longrightarrow\mathbb{R} $ defined as $$ F(x)=\left\{ \begin{matrix} F_1(x) ,& \quad x\in (r,\infty ) \\ \lim_{\stackrel{x\to r}{x>r}} F_1(x), & \quad x=r \end{matrix} \right. $$ is a primitive of $ f. $

Let $f$ be twice continuously differentiable on $(0,\infty)$ such that $\lim_{x \to 0^{+}}f'(x)=-\infty$ and $\lim_{x \to 0^{+}}f''(x)=\infty$. Show that $$\lim_{x\to 0^{+}}\frac{f(x)}{f'(x)}=0.$$

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 $ E$ be a bounded subset of the real line, and let $ \Omega$ be a system of (non degenerate) closed intervals such that for each $ x \in E$ there exists an $ I \in \Omega$ with left endpoint $ x$. Show that for every $ \varepsilon > 0$ there exists a finite number of pairwise non overlapping intervals belonging to $ \Omega$ that cover $ E$ with the exception of a subset of outer measure less than $ \varepsilon$. [J. Czipszer]

Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds: $$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$

Let A = {1,4,6, ...} be a set of natural numbers n for which n is the product of an even number of primes and n+1 is the product of an odd number of primes (taking into account the multiplicity of prime powers). Prove that the series of the reciprocals of the elements of A is divergent. In other words, $A=\{n|\lambda(n)=1$ and $\lambda(n+1)=-1\}$ , where $\lambda$ is the liouville lambda function.

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that: (a) $\lim_{x \to \infty}f(x)$ exists; (b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.