[b]a)[/b] Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of real numbers having the property that $ \left| x_{n+1} -x_n \right|\leqslant 1/2^n, $ for any $ n\geqslant 1. $ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent. [b]b)[/b] Create a sequence $ \left( y_n \right)_{n\ge 1} $ of real numbers that has the following properties: $ \text{(i) } \lim_{n\to\infty } \left( y_{n+1} -y_n \right) = 0 $ $ \text{(ii) } $ is bounded $ \text{(iii) } $ is divergent [i]Eugen Popa[/i]

Determine the pairs of sets $X,Y\subset\mathbb{R}$ for which the following is true: if $f(x, y)$ is a function on $X\times Y{}$ such that for every $x\in X$ it is equal to a polynomial in $y$ on $Y$ and for every $y\in Y$ it is equal to a polynomial in $x$ on $X$ then $f$ is a bivariate polynomial on $X\times Y.$

Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds: $\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$

Let $p(n)\geq 0$ for all positive integers $n$. Furthermore, $x(0)=0, v(0)=1$, and \[x(n)=x(n-1)+v(n-1), \qquad v(n)=v(n-1)-p(n)x(n) \qquad (n=1,2,\dots).\] Assume that $v(n)\to 0$ in a decreasing manner as $n \to \infty$. Prove that the sequence $x(n)$ is bounded if and only if $\sum_{n=1}^{\infty}n\cdot p(n)<\infty$.

Let $M, \alpha, \beta \in \mathbb{R} $ with $M > 0$ and $\alpha, \beta \in (0,1)$. If $R>1$ is a real number, we say that a sequence of positive real numbers $\{ C_n \}_{n\geq 0}$ is $R$-[i]inoceronte[/i] if $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$ for all $n \geq 1$. Determine the smallest real $R>1$ for which exists a $R$-[i]inoceronte[/i] sequence $ \{ C_n \}_{n\geq 0}$ such that $\sum_{n=1}^{\infty} \beta ^n C_n^{\alpha}$ diverges.

A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given. Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$ \[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\] Then for every integer $n\geq s,$ the condition \[a_{n+1}=\max_{0\leq k<n}(f_n(k))\] is satisfied. Prove that this sequence must be eventually constant.

Let $p(x) = (x- x_1)(x- x_2)(x- x_3)$, where $x_1, x_2$ and $x_3$ are real. Show that $p(x) p''(x) \le p'(x)^2$ for all $x$.

Show that the sum of the (local) maximum and minimum values of the function $\frac{tan(3x)}{tan^3x}$ on the interval $\big(0, \frac{\pi }{2}\big)$ is rational.

How many roots does equation $\sin x = \frac{x}{100}$ have?

Evaluate \[\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots\] [i]Proposed by Ethan Tan[/i]

The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous. (A.I. Plotkin, Leningrad)

The shorthand of a clock has the length 3, the longhand has the length 4. Determine the distance between the endpoints of the hands at the time, where their distance increases the most.

Let $\varepsilon$ be positive constant and $u$ satisfies that \[ \begin{cases} (\partial_t-\varepsilon\partial_x^2-\partial_y^2)u=0, & (t,x,y) \in \mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}_+,\\ \partial_y u\vert_{y=0}=\partial_x h, &\\u\vert_{t=0}=0. & \end{cases}\] Here $h(t,x)$ is a smooth Schwartz function. Define the operator $e^{a\langle D\rangle}$ \[\mathcal{F}_x(e^{a\langle D\rangle} f)(k)=e^{a\langle k\rangle} \mathcal{F}_x(f)(k), \quad \langle k\rangle=1+\vert k\vert,\] where $\mathcal{F}_x$ stands for the Fourier transform in $x$. Show that \[\int_0^T \|e^{(1-s)\langle D\rangle} u\|_{L_{x,y}^2}^2 ds \le C \int_0^T \|e^{(1-s)\langle D\rangle} h\|_{H_x^{\frac{1}{4}}}^2 ds\] with constant $C$ independent of $\varepsilon, T$ and $h$.

For any natural number $ m, \quad\lim_{n\to\infty } n^{1+m} \int_{0}^1 e^{-nx}\ln \left( 1+x^m \right) dx =m! . $ [i]Gheorghe Iurea[/i]

Let $a$ and $b$ satisfy $a \ge b >0, a + b = 1$. i) Prove that if $m$ and $n$ are positive integers with $m < n$, then $a^m - a^n \ge b^m- b^n > 0$. ii) For each positive integer $n$, consider a quadratic function $f_n(x) = x^2 - b^nx- a^n$. Show that $f(x)$ has two roots that are in between $-1$ and $1$.