Found problems: 265
2007 District Olympiad, 1
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 $\lim_{n\to \infty} a_n\sqrt{n}$.
2013 Romania National Olympiad, 3
Given $a\in (0,1)$ and $C$ the set of increasing functions
$f:[0,1]\to [0,\infty )$ such that $\int\limits_{0}^{1}{f(x)}dx=1$ . Determine:
$(a)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{f(x)dx}$
$(b)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{{{f}^{2}}(x)dx}$
2010 Paenza, 3
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 $\lim_{n \to \infty} x_n$.
1983 Miklós Schweitzer, 9
Prove that if $ E \subset \mathbb{R}$ is a bounded set of positive Lebesgue measure, then for every $ u < 1/2$, a point $ x\equal{}x(u)$ can be found so that \[ |(x\minus{}h,x\plus{}h) \cap E| \geq uh\] and \[ |(x\minus{}h,x\plus{}h) \cap (\mathbb{R} \setminus E)| \geq uh\] for all sufficiently small positive values of $ h$.
[i]K. I. Koljada[/i]
1979 Miklós Schweitzer, 8
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]
2010 Contests, 2
Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.
1950 Miklós Schweitzer, 1
Let $ a>0$, $ d>0$ and put
$ f(x)\equal{}\frac{1}{a}\plus{}\frac{x}{a(a\plus{}d)}\plus{}\cdots\plus{}\frac{x^n}{a(a\plus{}d)\cdots(a\plus{}nd)}\plus{}\cdots$
Give a closed form for $ f(x)$.
2008 District Olympiad, 3
Let $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ a sequence of positive real numbers, such that:
\[x_{n+1}\ge \frac{x_n+y_n}{2},\ y_{n+1}\ge \sqrt{\frac{x_n^2+y_n^2}{2}},\ (\forall)n\in \mathbb{N}^*\]
a) Prove that the sequences $(x_n+y_n)_{n\ge 1}$ and $(x_ny_n)_{n\ge 1}$ have limit.
b) Prove that the sequences $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ have limit and that their limits are equal.
2010 Romania National Olympiad, 3
Let $f:\mathbb{R}\rightarrow [0,\infty)$. Prove that $f(x+y)\ge (y+1)f(x),\ (\forall)x\in \mathbb{R}$ if and only if the function $g:\mathbb{R}\rightarrow [0,\infty),\ g(x)=e^{-x}f(x),\ (\forall)x\in \mathbb{R}$ is increasing.
1976 Miklós Schweitzer, 5
Let $ S_{\nu}\equal{}\sum_{j\equal{}1}^n b_jz_j^{\nu} \;(\nu\equal{}0,\pm 1, \pm 2 ,...) $, where the $ b_j$ are arbitrary and the $ z_j$ are nonzero complex numbers . Prove that \[ |S_0| \leq n \max_{0<|\nu| \leq n} |S_{\nu}|.\]
[i]G. Halasz[/i]
1981 Miklós Schweitzer, 6
Let $ f$ be a strictly increasing, continuous function mapping $ I=[0,1]$ onto itself. Prove that the following inequality holds for all pairs $ x,y \in I$: \[ 1-\cos (xy) \leq \int_0^xf(t) \sin (tf(t))dt + \int_0^y f^{-1}(t) \sin (tf^{-1}(t)) dt .\]
[i]Zs. Pales[/i]
2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 3
Let $a$ be a positive real number.
Evaluate $I=\int_0^{+\infty} \frac{\sin x\cos x}{x(x^2+a^2)}dx.$
1997 IMC, 5
For postive integer $n$ consider the hyperplane \[ R_0^n = {x=(x_1x_2...x_n)\in\mathbb{R}^n : \sum\limits^n_{i=1}x_i=0} \] and the lattice \[ Z_0^n = \{y \in R^n_0 : \ (\forall i: y_i \in \mathbb{N})\} \]
Define the quasi-norm in $\mathbb{R}^n$ by $\|x\|_p= \sqrt[p]{\sum\limits^{n}_{i=1}|x_i|^p}$ if $0<p<\infty$ and $\|x\|_{\infty} = \max\limits_i |x_i|$.
(a) If $x\in R^n_0$ so that $\max x_i - \min x_i \le 1$ then prove that $\forall p \in [1,\infty], \forall y \in Z^n_0$ we have $\|x\|_p\le\|x+y\|_p$
(b) Prove that for every $p\in ]0,1[$, there exist $n \in \mathbb{N}, x\in R^n_0, y\in Z^n_0$ with $\max x_i - \min x_i \le 1$ and $\|x\|_p>\|x+y\|_p$
1975 Miklós Schweitzer, 9
Let $ l_0,c,\alpha,g$ be positive constants, and let $ x(t)$ be the solution of the differential equation \[ ([l_0\plus{}ct^{\alpha}] ^2x')'\plus{}g[l_0\plus{}ct^{\alpha}] \sin x\equal{}0, \;t \geq 0,\ \;\minus{}\frac{\pi}{2} <x< \frac{\pi}{2},\] satisfying the initial conditions $ x(t_0)\equal{}x_0, \;x'(t_0)\equal{}0$. (This is the equation of the mathematical pendulum whose length changes according to the law $ l\equal{}l_0\plus{}ct^{\alpha}$.) Prove that $ x(t)$ is defined on the interval $ [t_0,\infty)$; furthermore, if $ \alpha >2$ then for every $ x_0 \not\equal{} 0$ there exists a $ t_0$ such that \[ \liminf_{t \rightarrow \infty} |x(t)| >0.\]
[i]L. Hatvani[/i]
2008 Teodor Topan, 4
Let $ (a_n)_{n \in \mathbb{N}^*}$ be a sequence of real positive numbers such that $ a_n>a_0,n\in \mathbb{N}$. Prove that $ \displaystyle\lim_{n\to\infty}\displaystyle\sum_{k\equal{}0}^{n}(\frac{a_k}{a_{n\minus{}k}})^k\equal{}\infty$.
1980 Miklós Schweitzer, 7
Let $ n \geq 2$ be a natural number and $ p(x)$ a real polynomial of degree at most $ n$ for which \[ \max _{ \minus{}1 \leq x \leq 1} |p(x)| \leq 1, \; p(\minus{}1)\equal{}p(1)\equal{}0 \ .\] Prove that then \[ |p'(x)| \leq \frac{n \cos \frac{\pi}{2n}}{\sqrt{1\minus{}x^2 \cos^2 \frac{\pi}{2n}}} \;\;\;\;\; \left( \minus{}\frac{1}{\cos \frac{\pi}{2n}} < x < \frac{1}{\cos \frac{\pi}{2n}} \\\\\ \right).\]
[i]J. Szabados[/i]
2012 Romania National Olympiad, 1
[color=darkred]Let $f\colon [0,\infty)\to\mathbb{R}$ be a continuous function such that $\int_0^nf(x)f(n-x)\ \text{d}x=\int_0^nf^2(x)\ \text{d}x$ , for any natural number $n\ge 1$ . Prove that $f$ is a periodic function.[/color]
2009 Miklós Schweitzer, 9
Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$
\[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]
1965 Miklós Schweitzer, 8
Let the continuous functions $ f_n(x), \; n\equal{}1,2,3,...,$ be defined on the interval $ [a,b]$ such that every point of $ [a,b]$ is a root of $ f_n(x)\equal{}f_m(x)$ for some $ n \not\equal{} m$. Prove that there exists a subinterval of $ [a,b]$ on which two of the functions are equal.
1963 Miklós Schweitzer, 7
Prove that for every convex function $ f(x)$ defined on the interval $ \minus{}1\leq x \leq 1$ and having absolute value at most $ 1$,
there is a linear function $ h(x)$ such that \[ \int_{\minus{}1}^1|f(x)\minus{}h(x)|dx\leq 4\minus{}\sqrt{8}.\] [L. Fejes-Toth]
1951 Miklós Schweitzer, 2
Denote by $ \mathcal{H}$ a set of sequences $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}$ of real numbers having the following properties:
(i) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$, then $ S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H}$;
(ii) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ T\equal{}\{t_n\}_{n\equal{}1}^{\infty}$, then
$ S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$;
(iii) $ \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}$.
A real valued function $ f(S)$ defined on $ \mathcal{H}$ is called a quasi-limit of $ S$ if it has the following properties:
If $ S\equal{}{c,c,\dots,c,\dots}$, then $ f(S)\equal{}c$;
If $ s_i\geq 0$, then $ f(S)\geq 0$;
$ f(S\plus{}T)\equal{}f(S)\plus{}f(T)$;
$ f(ST)\equal{}f(S)f(T)$,
$ f(S')\equal{}f(S)$
Prove that for every $ S$, the quasi-limit $ f(S)$ is an accumulation point of $ S$.
2013 Miklós Schweitzer, 7
Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function $($that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}})$ for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous.
[i]Proposed by Zoltán Boros[/i]
1999 IMC, 3
Suppose that $f: \mathbb{R}\rightarrow\mathbb{R}$ fulfils $\left|\sum^n_{k=1}3^k\left(f(x+ky)-f(x-ky)\right)\right|\le1$ for all $n\in\mathbb{N},x,y\in\mathbb{R}$. Prove that $f$ is a constant function.
2007 IMS, 3
Prove that $\mathbb R^{2}$ has a dense subset such that has no three collinear points.
1967 Miklós Schweitzer, 6
Let $ A$ be a family of proper closed subspaces of the Hilbert space $ H\equal{}l^2$ totally ordered with respect to inclusion (that is
, if $ L_1,L_2 \in A$, then either $ L_1\subset L_2$ or $ L_2\subset L_1$). Prove that there exists a vector $ x \in H$ not contaied in any of the subspaces $ L$ belonging to $ A$.
[i]B. Szokefalvi Nagy[/i]