Olympiad problems

1975 Miklós Schweitzer, 8

Prove that if \[ \sum_{n=1}^m a_n \leq Na_m \;(m=1,2,...)\] holds for a sequence $ \{a_n \}$ of nonnegative real numbers with some positive integer $ N$, then $ \alpha_{i+p} \geq p \alpha_i$ for $ i,p=1,2,...,$ where \[ \alpha_i= \sum_{n=(i-1)N+1}^{iN} a_n \;(i=1,2,...)\ .\] [i]L. Leindler[/i]

2006 Moldova MO 11-12, 2

Tags: function , limit , real analysis , real analysis unsolved

Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

1966 Miklós Schweitzer, 9

Tags: limit , real analysis , real analysis unsolved

If $ \sum_{m=-\infty}^{+\infty} |a_m| < \infty$, then what can be said about the following expression? \[ \lim_{n \rightarrow \infty} \frac{1}{2n+1} \sum_{m=-\infty}^{+\infty} |a_{m-n}+a_{m-n+1}+...+a_{m+n}|.\] [i]P. Turan[/i]

2004 District Olympiad, 1

Tags: limit , Putnam , real analysis , real analysis unsolved

Let $(x_n)_{n\ge 0}$ a sequence of real numbers defined by $x_0>0$ and $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$. Compute $\lim_{n\to \infty}x_n$ and $\lim_{n\to \infty} \frac{x_n^3}{n^2}$.

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

Tags: calculus , integration , limit , logarithms , real analysis , real analysis unsolved

For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function. Prove that : (i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$ (ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$

1971 Miklós Schweitzer, 10

Tags: function , real analysis , real analysis unsolved

Let $ \{\phi_n(x) \}$ be a sequence of functions belonging to $ L^2(0,1)$ and having norm less that $ 1$ such that for any subsequence $ \{\phi_{n_k}(x) \}$ the measure of the set \[ \{x \in (0,1) : \;|\frac{1}{\sqrt{N}} \sum _{k=1}^N \phi_{n_k}(x)| \geq y\ \}\] tends to $ 0$ as $ y$ and $ N$ tend to infinity. Prove that $ \phi_n$ tends to $ 0$ weakly in the function space $ L^2(0,1).$ [i]F. Moricz[/i]

1967 Miklós Schweitzer, 4

Tags: logarithms , real analysis , real analysis unsolved , Miklos Schweitzer , inequalities

Let $ a_1,a_2,...,a_N$ be positive real numbers whose sum equals $ 1$. For a natural number $ i$, let $ n_i$ denote the number of $ a_k$ for which $ 2^{1-i} \geq a_k \geq 2^{-i}$ holds. Prove that \[ \sum_{i=1}^{\infty} \sqrt{n_i2^{-i}} \leq 4+\sqrt{\log_2 N}.\] [i]L. Leinder[/i]

1965 Miklós Schweitzer, 7

Tags: induction , geometry , 3D geometry , sphere , real analysis , real analysis unsolved

Prove that any uncountable subset of the Euclidean $ n$-space contains an countable subset with the property that the distances between different pairs of points are different (that is, for any points $ P_1 \not\equal{} P_2$ and $ Q_1\not\equal{} Q_2$ of this subset, $ \overline{P_1P_2}\equal{}\overline{Q_1Q_2}$ implies either $ P_1\equal{}Q_1$ and $ P_2\equal{}Q_2$, or $ P_1\equal{}Q_2$ and $ P_2\equal{}Q_1$). Show that a similar statement is not valid if the Euclidean $ n$-space is replaced with a (separable) Hilbert space.

2009 Romania National Olympiad, 4

Tags: function , calculus , derivative , real analysis , real analysis unsolved

Let $f,g,h:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is differentiable, $g$ and $h$ are monotonic, and $f'=f+g+h$. Prove that the set of the points of discontinuity of $g$ coincides with the respective set of $h$.

2007 Romania National Olympiad, 2

Tags: integration , calculus , real analysis , real analysis unsolved

Let $f: [0,1]\rightarrow(0,+\infty)$ be a continuous function. a) Show that for any integer $n\geq 1$, there is a unique division $0=a_{0}<a_{1}<\ldots<a_{n}=1$ such that $\int_{a_{k}}^{a_{k+1}}f(x)\, dx=\frac{1}{n}\int_{0}^{1}f(x)\, dx$ holds for all $k=0,1,\ldots,n-1$. b) For each $n$, consider the $a_{i}$ above (that depend on $n$) and define $b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$. Show that the sequence $(b_{n})$ is convergent and compute it's limit.

2005 District Olympiad, 2

Tags: function , real analysis , real analysis unsolved

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $a,b\in \mathbb{R}$, with $a<b$ such that $f(a)=f(b)$, there exist some $c\in (a,b)$ such that $f(a)=f(b)=f(c)$. Prove that $f$ is monotonic over $\mathbb{R}$.

1982 Miklós Schweitzer, 5

Tags: function , real analysis , real analysis unsolved

Find a perfect set $ H \subset [0,1]$ of positive measure and a continuous function $ f$ defined on $ [0,1]$ such that for any twice differentiable function $ g$ defined on $ [0,1]$, the set $ \{ x \in H : \;f(x)\equal{}g(x)\ \}$ is finite. [i]M. Laczkovich[/i]

2014 Contests, 4

Tags: real analysis , real analysis unsolved

For a positive integer $n$, define $f(n)$ to be the number of sequences $(a_1,a_2,\dots,a_k)$ such that $a_1a_2\cdots a_k=n$ where $a_i\geq 2$ and $k\ge 0$ is arbitrary. Also we define $f(1)=1$. Now let $\alpha>1$ be the unique real number satisfying $\zeta(\alpha)=2$, i.e $ \sum_{n=1}^{\infty}\frac{1}{n^\alpha}=2 $ Prove that [list] (a) \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\alpha) \] (b) There is no real number $\beta<\alpha$ such that \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\beta) \] [/list]

1969 Miklós Schweitzer, 5

Tags: function , integration , calculus , real analysis , real analysis unsolved

Find all continuous real functions $ f,g$ and $ h$ defined on the set of positive real numbers and satisfying the relation \[ f(x\plus{}y)\plus{}g(xy)\equal{}h(x)\plus{}h(y)\] for all $ x>0$ and $ y>0$. [i]Z. Daroczy[/i]

2013 Romania National Olympiad, 1

Tags: function , integration , calculus , real analysis , real analysis unsolved

Determine continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $\left( {{a}^{2}}+ab+{{b}^{2}} \right)\int\limits_{a}^{b}{f\left( x \right)dx=3\int\limits_{a}^{b}{{{x}^{2}}f\left( x \right)dx,}}$ for every $a,b\in \mathbb{R}$ .

2009 IberoAmerican Olympiad For University Students, 4

Tags: function , real analysis , real analysis unsolved

Given two positive integers $m,n$, we say that a function $f : [0,m] \to \mathbb{R}$ is $(m,n)$-[i]slippery[/i] if it has the following properties: i) $f$ is continuous; ii) $f(0) = 0$, $f(m) = n$; iii) If $t_1, t_2\in [0,m]$ with $t_1 < t_2$ are such that $t_2-t_1\in \mathbb{Z}$ and $f(t_2)-f(t_1)\in\mathbb{Z}$, then $t_2-t_1 \in \{0,m\}$. Find all the possible values for $m, n$ such that there is a function $f$ that is $(m,n)$-slippery.

2008 Moldova MO 11-12, 4

Tags: limit , real analysis , real analysis unsolved

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)$.

1966 Miklós Schweitzer, 5

Tags: geometry , rectangle , real analysis , real analysis unsolved

A "letter $ T$" erected at point $ A$ of the $ x$-axis in the $ xy$-plane is the union of a segment $ AB$ in the upper half-plane perpendicular to the $ x$-axis and a segment $ CD$ containing $ B$ in its interior and parallel to the $ x$-axis. Show that it is impossible to erect a letter $ T$ at every point of the $ x$-axis so that the union of those erected at rational points is disjoint from the union of those erected at irrational points. [i]A.Csaszar[/i]

2014 Miklós Schweitzer, 7

Tags: function , real analysis , topology , real analysis unsolved

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and let $g : \mathbb{R} \to \mathbb{R}$ be arbitrary. Suppose that the Minkowski sum of the graph of $f$ and the graph of $g$ (i.e., the set $\{( x+y; f(x)+g(y) ) \mid x, y \in \mathbb{R}\}$) has Lebesgue measure zero. Does it follow then that the function $f$ is of the form $f(x) = ax + b$ with suitable constants $a, b \in \mathbb{R}$ ?

1993 Vietnam National Olympiad, 3

Tags: invariant , trigonometry , real analysis , real analysis unsolved

Define the sequences $a_{0}, a_{1}, a_{2}, ...$ and $b_{0}, b_{1}, b_{2}, ...$ by $a_{0}= 2, b_{0}= 1, a_{n+1}= 2a_{n}b_{n}/(a_{n}+b_{n}), b_{n+1}= \sqrt{a_{n+1}b_{n}}$. Show that the two sequences converge to the same limit, and find the limit.

2003 District Olympiad, 4

Tags: function , real analysis , real analysis unsolved

Let $\alpha>1$ and $f:\left[\frac{1}{\alpha},\alpha\right]\rightarrow \left[\frac{1}{\alpha},\alpha\right]$, a bijective function. If $f^{-1}(x)=\frac{1}{f(x)},\ \forall x\in \left[\frac{1}{\alpha},\alpha\right]$, prove that: a)$f$ has at least one point of discontinuity; b)if $f$ is continuous in $1$, then $f$ has an infinity points of discontinuity; c)there is a function $f$ which satisfies the conditions from the hypothesis and has a finite number of points of dicontinuity. [i]Radu Mortici [/i]

1976 Miklós Schweitzer, 6

Tags: real analysis , real analysis unsolved

Let $ 0 \leq c \leq 1$, and let $ \eta$ denote the order type of the set of rational numbers. Assume that with every rational number $ r$ we associate a Lebesgue-measurable subset $ H_r$ of measure $ c$ of the interval $ [0,1]$. Prove the existence of a Lebesgue-measurable set $ H \subset [0,1]$ of measure $ c$ such that for every $ x \in H$ the set \[ \{r : \;x \in H_r\ \}\] contains a subset of type $ \eta$. [i]M. Laczkovich[/i]

2001 Romania National Olympiad, 4

Tags: function , limit , real analysis , real analysis unsolved

The continuous function $f:[0,1]\rightarrow\mathbb{R}$ has the property: \[\lim_{x\rightarrow\infty}\ n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)=0 \] for every $x\in [0,1)$. Show that: a) For every $\epsilon >0$ and $\lambda\in (0,1)$, we have: \[ \sup\ \{x\in[0,\lambda )\mid |f(x)-f(0)|\le \epsilon x \}=\lambda \] b) $f$ is a constant function.

1998 IberoAmerican Olympiad For University Students, 1

Tags: calculus , integration , function , inequalities , real analysis , real analysis unsolved

The definite integrals between $0$ and $1$ of the squares of the continuous real functions $f(x)$ and $g(x)$ are both equal to $1$. Prove that there is a real number $c$ such that \[f(c)+g(c)\leq 2\]

2005 Brazil Undergrad MO, 2

Tags: function , integration , inequalities , induction , real analysis , real analysis unsolved

Let $f$ and $g$ be two continuous, distinct functions from $[0,1] \rightarrow (0,+\infty)$ such that $\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$ Let $y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq 0$, natural. Prove that $(y_n)$ is an increasing and divergent sequence.