Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance of $X$ and $Y$ be maximal?

For each $x>e^e$ define a sequence $S_x=u_0,u_1,\ldots$ recursively as follows: $u_0=e$, and for $n\ge0$, $u_{n+1}=\log_{u_n}x$. Prove that $S_x$ converges to a number $g(x)$ and that the function $g$ defined in this way is continuous for $x>e^e$.

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 $x \in \mathbb R$.

For every $n$ in the set $\mathrm{N} = \{1,2,\dots \}$ of positive integers, let $r_n$ be the minimum value of $|c-d\sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in \mathbb{N}$.

Show that if $ \left( s_n \right)_{n\ge 0} $ is a sequence that tends to $ 6, $ then, the sequence $$ \left( \sqrt[3]{s_n+\sqrt[3]{s_{n-1}+\sqrt[3]{s_{n-2}+\sqrt[3]{\cdots +\sqrt[3]{s_0}}}}} \right)_{n\ge 0} $$ tends to $ 2. $ [i]Mihai Bălună[/i]

(a) Let $n$ is a positive integer. Calculate $\displaystyle \int_0^1 x^{n-1}\ln x\,dx$.\\ (b) Calculate $\displaystyle \sum_{n=0}^{\infty}(-1)^n\left(\frac{1}{(n+1)^2}-\frac{1}{(n+2)^2}+\frac{1}{(n+3)^2}-\dots \right).$

Prove the inequality $$\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2$$ for any sequence of real numbers $x_0, x_1, ..., x_n$ for which $x_0 = x_n = 0.$

Let $ f(t)$ be a continuous function on the interval $ 0 \leq t \leq 1$, and define the two sets of points \[ A_t\equal{}\{(t,0): t\in[0,1]\} , B_t\equal{}\{(f(t),1): t\in [0,1]\}.\] Show that the union of all segments $ \overline{A_tB_t}$ is Lebesgue-measurable, and find the minimum of its measure with respect to all functions $ f$. [A. Csaszar]

Prove that there exist natural numbers $n_k, m_k, k=0,1,2,\ldots$, such that the numbers $n_k+m_k, k=1,2,\ldots$ are pairwise distinct primes and the set of linear combination of the polynomials $x^{n_k}y^{m_k}$ is dense in $C([0,1] \times [0,1])$ under the supremum norm. (translated by Miklós Maróti)

Let $ K_n(n=1,2,\ldots)$ be periodical continuous functions of period $ 2 \pi$, and write \[ k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt .\] Prove that the following statements are equivalent: (i) $ \int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty)$ for all $ f \in \mathcal{L}_1[0,2 \pi]$. (ii) $ k_n(f;0) \rightarrow f(0)$ for all continuous, $ 2 \pi$-periodic functions $ f$. [i]V. Totik[/i]

Determine the positive numbers $a{}$ for which the following statement true: for any function $f:[0,1]\to\mathbb{R}$ which is continuous at each point of this interval and for which $f(0)=f(1)=0$, the equation $f(x+a)-f(x)=0$ has at least one solution. [i]Proposed by I. Yaglom[/i]

Let $f:[0,1]\to\mathbb R$ be a continuous function such that $f(0)=f(1)=0$. Prove that the set $$A:=\{h\in[0,1]:f(x+h)=f(x)\text{ for some }x\in[0,1]\}$$is Lebesgue measureable and has Lebesgue measure at least $\frac12$.

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 $\{x_n\}$ be a sequence of real numbers in the interval $[0,1)$. Prove that there exists a sequence $1 < n_1 < n_2 < n_3 < \cdots$ of positive integers such that the following limit exists $$\lim_{i,j \to \infty} x_{n_i+n_j}. $$ That is, there exists a real number $L$ such that for every $\epsilon > 0,$ there exists a positive integer $N$ such that if $i,j > N$, then $|x_{n_i+n_j}-L| < \epsilon.$

Let $f : [a, b] \rightarrow [a, b]$ be a continuous function and let $p \in [a, b]$. Define $p_0 = p$ and $p_{n+1} = f(p_n)$ for $n = 0, 1, 2,...$. Suppose that the set $T_p = \{p_n : n = 0, 1, 2,...\}$ is closed, i.e., if $x \not\in T_p$ then $\exists \delta > 0$ such that for all $x' \in T_p$ we have $|x'-x|\ge\delta$. Show that $T_p$ has finitely many elements.

Compute $\lim\limits_{n \to \infty} \frac{1}{\log \log n} \sum\limits_{k=1}^n (-1)^k \binom{n}{k} \log k.$

Let $f:\mathbb R\to\mathbb R$ be a continuous function. Do there exist continuous functions $g:\mathbb R\to\mathbb R$ and $h:\mathbb R\to\mathbb R$ such that $f(x)=g(x)\sin x+h(x)\cos x$ holds for every $x\in\mathbb R$?

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

By a partition $\pi$ of an integer $n\ge 1$, we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$, then the partitions $\pi$ are $1+1+1+1$, $1+1+2$, $1+3, 2+2$, and $4$). For any partition $\pi$, define $A(\pi)$ to be the number of $1$'s which appear in $\pi$, and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$. (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$, then $A(\pi)=2$ and $B(\pi) = 3$). Prove that, for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$.

A cat is trying to catch a mouse in the non-negative quadrant \[N=\{(x_1,x_2)\in \mathbb{R}^2: x_1,x_2\geq 0\}.\] At time $t=0$ the cat is at $(1,1)$ and the mouse is at $(0,0)$. The cat moves with speed $\sqrt{2}$ such that the position $c(t)=(c_1(t),c_2(t))$ is continuous, and differentiable except at finitely many points; while the mouse moves with speed $1$ such that its position $m(t)=(m_1(t),m_2(t))$ is also continuous, and differentiable except at finitely many points. Thus $c(0)=(1,1)$ and $m(0)=(0,0)$; $c(t)$ and $m(t)$ are continuous functions of $t$ such that $c(t),m(t)\in N$ for all $t\geq 0$; the derivatives $c'(t)=(c'_1(t),c'_2(t))$ and $m'(t)=(m'_1(t),m'_2(t))$ each exist for all but finitely many $t$ and \[(c'_1(t)^2+(c'_2(t))^2=2 \qquad (m'_1(t)^2+(m'_2(t))^2=1,\] whenever the respective derivative exists. At each time $t$ the cat knows both the mouse's position $m(t)$ and velocity $m'(t)$. Show that, no matter how the mouse moves, the cat can catch it by time $t=1$; that is, show that the cat can move such that $c(\tau)=m(\tau)$ for some $\tau\in[0,1]$.

Let be a sequence $ \left( x_n \right)_{n\ge 1} $ having the property that $$ \lim_{n\to\infty } \left( 14(n+2)x_{n+2} -15(n+1)x_{n+1} +nx_n \right) =13. $$ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent and calculate its limit. [i]Cosmin Nițu[/i]

Let be two nonzero real numbers $ a,b $ such that $ |a|\neq |b| $ and let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function satisfying the functional relation $$ af(x)+bf(-x)=(x^3+x)^5+\sin^5 x . $$ Calculate $ \int_{-2019}^{2019}f(x)dx . $ [i]Constantin Rusu[/i]

Study the convergence of a sequence $ \left( x_n\right)_{n\ge 0} $ for which $ x_0\in\mathbb{R}\setminus\mathbb{Q} , $ and $ x_{n+1}\in \left\{ \frac{x_n+1}{x_n} , \frac{x_n+2}{2x_n-1}\right\} , $ for all $ n\ge 1. $

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 $n \geq 2$ be an integer. Calculate$$\int \limits_{0}^{\frac{\pi}{2}}\frac{\sin x}{\sin^{2n-1}x+\cos^{2n-1}x}dx$$