Found problems: 884
1999 IMC, 3
Let $x_i\ge -1$ and $\sum^n_{i=1}x_i^3=0$. Prove $\sum^n_{i=1}x_i \le \frac{n}{3}$.
1978 Miklós Schweitzer, 4
Let $ \mathbb{Q}$ and $ \mathbb{R}$ be the set of rational numbers and the set of real numbers, respectively, and let $ f : \mathbb{Q} \rightarrow \mathbb{R}$ be a function with the following property. For every $ h \in \mathbb{Q} , \;x_0 \in \mathbb{R}$, \[ f(x\plus{}h)\minus{}f(x) \rightarrow 0\] as $ x \in \mathbb{Q}$ tends to $ x_0$. Does it follow that $ f$ is bounded on some interval?
[i]M. Laczkovich[/i]
2007 Nicolae Păun, 3
Construct a function $ f:[0,1]\longrightarrow\mathbb{R} $ that is primitivable, bounded, and doesn't touch its bounds.
[i]Dorian Popa[/i]
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 $\lim_{n\to \infty} a_n\sqrt{n}$.
2012 USAMO, 6
For integer $n\geq2$, let $x_1, x_2, \ldots, x_n$ be real numbers satisfying \[x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.\]For each subset $A\subseteq\{1, 2, \ldots, n\}$, define\[S_A=\sum_{i\in A}x_i.\](If $A$ is the empty set, then $S_A=0$.)
Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$. For which choices of $x_1, x_2, \ldots, x_n, \lambda$ does equality hold?
2008 Moldova National Olympiad, 12.6
Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.
2004 Unirea, 4
Let be a real number $ a\in (0,1) $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that:
$$ \lim_{x\to 0} f(x) =0= \lim_{x\to 0} \frac{f(x)-f(ax)}{x} $$
Prove that $ \lim_{x\to\infty } \frac{f(x)}{x} =0. $
1954 Miklós Schweitzer, 1
[b]1.[/b] Given a positive integer $r>1$, prove that there exists an infinite number of infinite geometrical series, with positive terms, having the sum 1 and satisfying the following condition: for any positive real numbers $S_{1},S_{2},\dots,S_{r}$ such that $S_{1}+S_{2}+\dots+S_{r}=1$, any of these infinite geometrical series can be divided into $r$ infinite series(not necessarily geometrical) having the sums $S_{1},S_{2},\dots,S_{r}$, respectively. [b](S. 6)[/b]
2015 District Olympiad, 1
Let $ f:[0,1]\longrightarrow [0,1] $ a function with the property that, for all $ y\in [0,1] $ and $ \varepsilon >0, $ there exists a $ x\in [0,1] $ such that $ |f(x)-y|<\varepsilon . $
[b]a)[/b] Prove that if $ \left. f\right|_{[0,1]} $ is continuos, then $ f $ is surjective.
[b]b)[/b] Give an example of a function with the given property, but which isn´t surjective.
1998 Miklós Schweitzer, 4
For any measurable set $H \subset R$ , we define the sequence $a_n(H)$ by the formula:
$$a_n(H) = \lambda \bigg([0,1] \setminus \bigcup_{k = n}^{2n} (H + \log_2 k) \bigg)$$
where $\lambda$ denotes the Lebesgue measure and $\log_2$ denotes the binary logarithm. Prove that there is a measurable, 1-periodic, positive measure set $H \subset R$ , such that the sequence $a_n( H )$ does not belong to any space $l_p$ ($1 \leq p < \infty$).
[hide=not sure about this part]For what numbers $1 \leq p <\infty$ is it true that whenever H is 1-periodic, positive measure, the sequence $a_n( H )$ belongs to the space $l_p$?[/hide]
2005 Grigore Moisil Urziceni, 3
Let be a sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1>0 $ and satisfying the equality
$$ a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , $$
for all natural numbers $ n. $
[b]a)[/b] Find a recurrence relation between two consecutive elements of $ \left( a_n \right)_{n\ge 1} . $
[b]b)[/b] Prove that $ \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2. $
2021 Miklós Schweitzer, 5
Let $f(x)=\frac{1+\cos(2 \pi x)}{2}$, for $x \in \mathbb{R}$, and $f^n=\underbrace{ f \circ \cdots \circ f}_{n}$. Is it true that for Lebesgue almost every $x$, $\lim_{n \to \infty} f^n(x)=1$?
1994 Miklós Schweitzer, 6
Show that if n is an arbitrary natural number and $\sqrt n \leq K \leq \frac{n}{2}$, then there exist n distinct integers, $k_j$ ( j = 1, ..., n ) such that $\bigg | \sum_ {j = 1} ^ ne ^ {ik_jt} \bigg | \geq K$ is satisfied on a subset of the interval $(- \pi, \pi)$ with Lebesgue measure at least $\frac{cn}{K^2}$ , where c is a suitable absolute constant.
2011 District Olympiad, 3
Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous and nondecreasing function.
[b]a)[/b] Show that the sequence $ \left( \frac{1}{2^n}\sum_{i=1}^{2^n} f\left(\frac{i}{2^n}\right) \right)_{n\ge 1} $ is nonincreasing.
[b]b)[/b] Prove that, if there exists some natural index at which the sequence above is equal to $ \int_0^1 f(x)dx, $ then $ f $ is constant.
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]
1964 Miklós Schweitzer, 9
Let $ E$ be the set of all real functions on $ I\equal{}[0,1]$. Prove that one cannot define a topology on $ E$ in which $ f_n\rightarrow f$ holds if and only if $ f_n$ converges to $ f$ almost everywhere.
2018 IMC, 4
Find all differentiable functions $f:(0,\infty) \to \mathbb{R}$ such that
$$f(b)-f(a)=(b-a)f’(\sqrt{ab}) \qquad \text{for all}\qquad a,b>0.$$
[i]Proposed by Orif Ibrogimov, National University of Uzbekistan[/i]
2008 Gheorghe Vranceanu, 1
Prove that for a positive number $ r>1, $ there is a nondecreasing sequence of positive numbers $ \left( a_v\right)_{v\ge 1} $ such that $$ r=\lim_{n\to\infty }\sum_{i=1}^n \frac{a_i}{a_{i+1}} . $$
2004 Gheorghe Vranceanu, 2
Let be two real numbers $ a<b, $ a nonempty and non-maximal subset $ K $ of the interval $ (a,b) $ and three functions
$$ f:(a,b)\longrightarrow\mathbb{R}, g,h:\mathbb{R}\longrightarrow\mathbb{R} $$
satisfying the following relations.
$ \text{(i)} g $ and $ h $ are primitivable.
$ \text{(ii)} g-h $ hasn't any root in $ (a,b). $
$ \text{(iii)} $ The restrictions of $ f $ at $ K $ and $ (a,b)\setminus K $ are equal to $ g,h, $ respectively.
Prove that $ f $ is not primitivable.
1959 Miklós Schweitzer, 7
[b]7.[/b] Let $(z_n)_{n=1}^{\infty}$ be a sequence of complex numbers tending to zero. Prove that there exists a sequence $(\epsilon_n)_{n=1}^{\infty}$ (where $\epsilon_n = +1$ or $-1$) such that the series
$\sum_{n=1}^{\infty} \epsilon_n z_n$
is convergente. [b](F. 9)[/b]
2014 Romania National Olympiad, 2
Let $ I,J $ be two intervals, $ \varphi :J\longrightarrow\mathbb{R} $ be a continuous function whose image doesn't contain $ 0, $ and $ f,g:I\longrightarrow J $ be two differentiable functions such that $ f'=\varphi\circ f,g'=\varphi\circ g $ and such that the image of $ f-g $ contains $ 0. $
Show that $ f $ and $ g $ are the same function.
2021 Brazil Undergrad MO, Problem 4
For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$.
Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$
1954 Miklós Schweitzer, 9
[b]9.[/b] Lep $p$ be a connected non-closed broken line without self-intersection in the plane $\varphi $. Prove that if $v$ is a non-zero vector in $\varphi $ and $p$ has a commom point with the broken line $p+v$, then $p$ has a common point with the broken line $p+\alpha v$ too, where $\alpha =\frac{1}{n}$ and $n$ is a positive integer. Does a similar statemente hold for other positive values of $\alpha$? ($p+v$ denotes the broken line obtained from $p$ through displacemente by the vector $v$.) [b](G. 1)[/b]
2009 District Olympiad, 4
a) Prove that the function $F:\mathbb{R}\rightarrow \mathbb{R},\ F(x)=2\lfloor x\rfloor-\cos(3\pi\{x\})$ is continuous over $\mathbb{R}$ and for any $y\in \mathbb{R}$, the equation $F(x)=y$ has exactly three solutions.
b) Let $k$ a positive even integer. Prove that there is no function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is continuous over $\mathbb{R}$ and that for any $y\in \text{Im}\ f$, the equation $f(x)=y$ has exactly $k$ solutions $(\text{Im}\ f=f(\mathbb{R}))$.
2002 IMC, 9
For each $n\geq 1$ let
$$a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \;\; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.$$
Show that $a_{n}\cdot b_{n}$ is an integer.