Found problems: 265
2007 Iran MO (3rd Round), 4
a) Let $ n_{1},n_{2},\dots$ be a sequence of natural number such that $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$ be a sequence such that $ \epsilon_{i}\in\{1,2\}$. Prove that the sequence: \[ \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots\plus{}\sqrt[n_{k}]{\epsilon_{k}}}}\]is convergent and its limit is in $ (1,2]$. Define $ \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}$ to be this limit.
b) Prove that for each $ x\in(1,2]$ there exist sequences $ n_{1},n_{2},\dots\in\mathbb N$ and $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$, such that $ n_{i}\geq2$ and $ \epsilon_{i}\in\{1,2\}$, and $ x\equal{}\sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}$
1963 Miklós Schweitzer, 9
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]
2008 Moldova National Olympiad, 12.4
Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$.
Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$.
1999 IMC, 4
Find all strictly monotonic functions $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ for which $f\left(\frac{x^2}{f(x)}\right)=x$ for all $x$.
1976 Miklós Schweitzer, 8
Prove that the set of all linearly combinations (with real coefficients) of the system of polynomials $ \{ x^n\plus{}x^{n^2} \}_{n\equal{}0}^{\infty}$ is dense in $ C[0,1]$.
[i]J. Szabados[/i]
2010 Contests, 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 $\lim_{n \to \infty} x_n$.
1966 Miklós Schweitzer, 7
Does there exist a function $ f(x,y)$ of two real variables that takes natural numbers as its values and for which $ f(x,y)\equal{}f(y,z)$ implies $ x\equal{}y\equal{}z?$
[i]A. Hajnal[/i]
1982 Miklós Schweitzer, 6
For every positive $ \alpha$, natural number $ n$, and at most $ \alpha n$ points $ x_i$, construct a trigonometric polynomial $ P(x)$ of degree at most $ n$ for which \[ P(x_i) \leq 1, \; \int_0^{2 \pi} P(x)dx=0,\ \; \textrm{and}\ \; \max P(x) > cn\ ,\] where the constant $ c$ depends only on $ \alpha$.
[i]G. Halasz[/i]
1998 IMC, 4
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ is twice differentiable and satisfies $f(0)=2,f'(0)=-2,f(1)=1$.
Prove that there is a $\xi \in ]0,1[$ for which we have $f(\xi)\cdot f'(\xi)+f''(\xi)=0$.
1962 Miklós Schweitzer, 9
Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].
1998 Romania National Olympiad, 3
Suppose $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function for which the inequality $f'(x) \leq f'(x+\frac{1}{n})$ holds for every $x\in\mathbb{R}$ and every $n\in\mathbb{N}$.Prove that f is continiously differentiable
2013 Miklós Schweitzer, 8
Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which
\[ \displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right) \]
for all ${x,y \in \Bbb{R}} ({f^{-1}}$ denotes the inverse of ${f})$. Prove that there exist real constants ${a \neq 0}$ and ${b}$ such that ${f(x)=ax+b}$ for all ${x \in \Bbb{R}}.$
[i]Proposed by Zoltán Daróczy[/i]
1998 IMC, 3
Let $f(x)=2x(1-x), x\in\mathbb{R}$ and denote $f_n=f\circ f\circ ... \circ f$, $n$ times.
(a) Find $\lim_{n\rightarrow\infty} \int^1_0 f_n(x)dx$.
(b) Now compute $\int^1_0 f_n(x)dx$.
2014 District Olympiad, 2
[list=a]
[*]Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
$g\colon\mathbb{R}\rightarrow\mathbb{R}$, $g(x)=f(x)+f(2x)$, and
$h\colon\mathbb{R}\rightarrow\mathbb{R}$, $h(x)=f(x)+f(4x)$, are continuous
functions. Prove that $f$ is also continuous.
[*]Give an example of a discontinuous function $f\colon\mathbb{R}
\rightarrow\mathbb{R}$, with the following property: there exists an interval
$I\subset\mathbb{R}$, such that, for any $a$ in $I$, the function $g_{a}
\colon\mathbb{R}\rightarrow\mathbb{R}$, $g_{a}(x)=f(x)+f(ax)$, is continuous.[/list]
1980 Miklós Schweitzer, 8
Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$.
[i]G. Halasz[/i]
2002 Romania National Olympiad, 3
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and bounded function such that
\[x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t,\quad\text{for any}\ x\in\mathbb{R}.\]
Prove that $f$ is a constant function.
2011 IMC, 1
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function. A point $x$ is called a [i]shadow[/i] point if there exists a point $y\in \mathbb{R}$ with $y>x$ such that $f(y)>f(x).$ Let $a<b$ be real numbers and suppose that
$\bullet$ all the points of the open interval $I=(a,b)$ are shadow points;
$\bullet$ $a$ and $b$ are not shadow points.
Prove that
a) $f(x)\leq f(b)$ for all $a<x<b;$
b) $f(a)=f(b).$
[i]Proposed by José Luis Díaz-Barrero, Barcelona[/i]
1970 IMO Longlists, 25
A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.
2009 IMS, 5
Suppose that $ f: \mathbb R^2\rightarrow \mathbb R$ is a non-negative and continuous function that $ \iint_{\mathbb R^2}f(x,y)dxdy\equal{}1$. Prove that there is a closed disc $ D$ with the least radius possible such that $ \iint_D f(x,y)dxdy\equal{}\frac12$.
1963 Miklós Schweitzer, 8
Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be
absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]
2008 Grigore Moisil Intercounty, 2
Let $ n\in \mathbb{N^*}$ and $ f: [0,1]\rightarrow \mathbb{R}$ a continuos function with the prop. $ \int_{0}^{1}(1\minus{}x^n)f(x)dx\equal{}0$.
Prove that $ \int_{0}^{1}f^2(x)dx \geq 2(n\plus{}1)\left(\int_{0}^{1}f(x)dx\right)^2$
1965 Miklós Schweitzer, 9
Let $ f$ be a continuous, nonconstant, real function, and assume the existence of an $ F$ such that $ f(x\plus{}y)\equal{}F[f(x),f(y)]$ for all real $ x$ and $ y$. Prove that $ f$ is strictly monotone.
2007 Romania National Olympiad, 2
Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ be two points in the image of $f$ (that is, there exists $x,y$ such that $f(x)=a$ and $f(y)=b$).
Show that there is an interval $I$ such that $f(I)=[a,b]$.
1950 Miklós Schweitzer, 9
Find the sum of the series
$ x\plus{}\frac{x^3}{1\cdot 3}\plus{}\frac{x^5}{1\cdot 3\cdot 5}\plus{}\cdots\plus{}\frac{x^{2n\plus{}1}}{1\cdot 3\cdot 5\cdot \cdots \cdot (2n\plus{}1)}\plus{}\cdots$
2003 District Olympiad, 2
Let $f:[0,1]\rightarrow [0,1]$ a continuous function in $0$ and in $1$, which has one-side limits in any point and $f(x-0)\le f(x)\le f(x+0),\ (\forall)x\in (0,1)$. Prove that:
a)for the set $A=\{x\in [0,1]\ |\ f(x)\ge x\}$, we have $\sup A\in A$.
b)there is $x_0\in [0,1]$ such that $f(x_0)=x_0$.
[i]Mihai Piticari[/i]