Found problems: 265
1952 Miklós Schweitzer, 9
Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$.
(The latter means that
$ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$
holds for every integrable function $ h(x)$.)
1949 Miklós Schweitzer, 2
Compute
$ \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx$ .
1997 Traian Lălescu, 4
Compute the limit: \[ \lim_{n\to\infty} \frac{1}{n^2}\sum\limits_{1\leq i <j\leq n}\sin \frac{i+j}{n}\].
2008 IMS, 9
Let $ \gamma: [0,1]\rightarrow [0,1]\times [0,1]$ be a mapping such that for each $ s,t\in [0,1]$
\[ |\gamma(s) \minus{} \gamma(t)|\leq M|s \minus{} t|^\alpha
\]
in which $ \alpha,M$ are fixed numbers. Prove that if $ \gamma$ is surjective, then $ \alpha\leq\frac12$
1949 Miklós Schweitzer, 1
Let an infinite sequence of measurable sets be given on the interval $ (0,1)$ the measures of which are $ \geq \alpha>0$. Show that there exists a point of $ (0,1)$ which belongs to infinitely many terms of the sequence.
1968 Miklós Schweitzer, 2
Let $ a_1,a_2,...,a_n$ be nonnegative real numbers. Prove that \[ ( \sum_{i=1}^na_i)( \sum_{i=1}^na_i^{n-1}) \leq n \prod_{i=1}^na_i+ (n-1) ( \sum_{i=1}^na_i^n).\]
[i]J. Suranyi[/i]
2010 IMC, 3
Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute
$\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.
1997 IMC, 1
Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]
1983 Miklós Schweitzer, 7
Prove that if the function $ f : \mathbb{R}^2 \rightarrow [0,1]$ is continuous and its average on every circle of radius $ 1$ equals the function value at the center of the circle, then $ f$ is constant.
[i]V. Totik[/i]
1971 Miklós Schweitzer, 2
Prove that there exists an ordered set in which every uncountable subset contains an uncountable, well-ordered subset and that cannot be represented as a union of a countable family of well-ordered subsets.
[i]A. Hajnal[/i]
2006 Romania National Olympiad, 3
We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$, which have different centers of mass. Prove that there is a point $P$ such that \[ PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n . \]
1997 Romania National Olympiad, 2
Prove that:
$\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2$
Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D
2012 Romania National Olympiad, 3
[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by
\[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\]
Determine the following two sets:
[list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$
[b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]
1968 Miklós Schweitzer, 5
Let $ k$ be a positive integer, $ z$ a complex number, and $ \varepsilon <\frac12$ a positive number. Prove that the following inequality holds for infinitely many positive integers $ n$: \[ \mid \sum_{0\leq l \leq \frac{n}{k+1}} \binom{n-kl}{l}z^l \mid \geq (\frac 12-\varepsilon)^n.\]
[i]P. Turan[/i]
2010 IberoAmerican Olympiad For University Students, 2
Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.
2010 District Olympiad, 3
Let $ f: \mathbb{R}\rightarrow \mathbb{R}$ a strictly increasing function such that $ f\circ f$ is continuos. Prove that $ f$ is continuos.
1969 Miklós Schweitzer, 4
Show that the following inequality hold for all $ k \geq 1$, real numbers $ a_1,a_2,...,a_k$, and positive numbers $ x_1,x_2,...,x_k.$
\[ \ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} .
\]
[i]L. Losonczi[/i]
1975 Miklós Schweitzer, 6
Let $ f$ be a differentiable real function and let $ M$ be a positive real number. Prove that if \[ |f(x\plus{}t)\minus{}2f(x)\plus{}f(x\minus{}t)| \leq Mt^2 \; \textrm{for all}\ \;x\ \; \textrm{and}\ \;t\ , \] then \[ |f'(x\plus{}t)\minus{}f'(x)| \leq M|t|.\]
[i]J. Szabados[/i]
1951 Miklós Schweitzer, 3
Consider the iterated sequence
(1) $ x_0,x_1 \equal{} f(x_0),\dots,x_{n \plus{} 1} \equal{} f(x_n),\dots$,
where $ f(x) \equal{} 4x \minus{} x^2$. Determine the points $ x_0$ of $ [0,1]$ for which (1) converges and find the limit of (1).
1969 Miklós Schweitzer, 3
Let $ f(x)$ be a nonzero, bounded, real function on an Abelian group $ G$, $ g_1,...,g_k$ are given elements of $ G$ and $ \lambda_1,...,\lambda_k$ are real numbers. Prove that if \[ \sum_{i=1}^k \lambda_i f(g_ix) \geq 0\] holds for all $ x \in G$, then \[ \sum_{i=1}^k \lambda_i \geq 0.\]
[i]A. Mate[/i]
2006 District Olympiad, 4
We say that a function $f: \mathbb R \to \mathbb R$ has the property $(P)$ if, for any real numbers $x$, \[ \sup_{t\leq x} f(x) = x. \] a) Give an example of a function with property $(P)$ which has a discontinuity in every real point.
b) Prove that if $f$ is continuous and satisfies $(P)$ then $f(x) = x$, for all $x\in \mathbb R$.
2011 Romania National Olympiad, 2
[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]
1962 Miklós Schweitzer, 5
Let $ f$ be a finite real function of one variable. Let $ \overline{D}f$ and $ \underline{D}f$ be its upper and lower derivatives, respectively, that is, \[ \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}\] ,
\[ \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}.\] Show that $ \overline{D}f$ and $ \underline{D}f$ are Borel-measurable functions. [A. Csaszar]
2010 Olympic Revenge, 2
Joaquim, José and João participate of the worship of triangle $ABC$. It is well known that $ABC$ is a random triangle, nothing special. According to the dogmas of the worship, when they form a triangle which is similar to $ABC$, they will get immortal. Nevertheless, there is a condition: each person must represent a vertice of the triangle. In this case, Joaquim will represent vertice $A$, José vertice $B$ and João will represent vertice $C$. Thus, they must form a triangle which is similar to $ABC$, in this order.
Suppose all three points are in the Euclidean Plane. Once they are very excited to become immortal, they act in the following way: in each instant $t$, Joaquim, for example, will move with constant velocity $v$ to the point in the same semi-plan determined by the line which connects the other two points, and which would create a triangle similar to $ABC$ in the desired order. The other participants act in the same way.
If the velocity of all of them is same, and if they initially have a finite, but sufficiently large life, determine if they can get immortal.
[i]Observation: Initially, Joaquim, José and João do not represent three collinear points in the plane[/i]
1999 IMC, 4
Prove that there's no function $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ such that $f(x)^2\ge f(x+y)\left(f(x)+y\right)$ for all $x,y>0$.