Olympiad problems

2007 IMS, 8

Let \[T=\{(tq,1-t) \in\mathbb R^{2}| t \in [0,1],q\in\mathbb Q\}\]Prove that each continuous function $f: T\longrightarrow T$ has a fixed point.

1950 Miklós Schweitzer, 7

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

Examine the behavior of the expression $ \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n$ as $ n\rightarrow \infty$

1975 Miklós Schweitzer, 5

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

Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$ [i]J. Szucs[/i]

1969 Miklós Schweitzer, 9

Tags: real analysis , real analysis unsolved

In $ n$-dimensional Euclidean space, the union of any set of closed balls (of positive radii) is measurable in the sense of Lebesgue. [i]A. Csaszar[/i]

2001 IMC, 3

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

Find $\lim_{t\rightarrow 1^-} (1-t) \sum_{n=1}^{\infty}\frac{t^n}{1+t^n}$.

2006 Pre-Preparation Course Examination, 3

Tags: function , real analysis , real analysis unsolved

Show that if $f: [0,1]\rightarrow [0,1]$ is a continous function and it has topological transitivity then periodic points of $f$ are dense in $[0,1]$. Topological transitivity means there for every open sets $U$ and $V$ there is $n>0$ such that $f^n(U)\cap V\neq \emptyset$.

2010 Contests, 1

Tags: integration , real analysis , real analysis unsolved

Let $0 < a < b$. Prove that $\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.

2010 Romania National Olympiad, 1

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

Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic function and $F:\mathbb{R}\to\mathbb{R}$ given by \[F(x)=\int_0^xf(t)\ \text{d}t.\] Prove that if $F$ has a finite derivative, then $f$ is continuous. [i]Dorin Andrica & Mihai Piticari[/i]

1971 Miklós Schweitzer, 6

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

Let $ a(x)$ and $ r(x)$ be positive continuous functions defined on the interval $ [0,\infty)$, and let \[ \liminf_{x \rightarrow \infty} (x-r(x)) >0.\] Assume that $ y(x)$ is a continuous function on the whole real line, that it is differentiable on $ [0, \infty)$, and that it satisfies \[ y'(x)=a(x)y(x-r(x))\] on $ [0, \infty)$. Prove that the limit \[ \lim_{x \rightarrow \infty}y(x) \exp \left\{ -%Error. "diaplaymath" is a bad command. \int_0^x a(u)du \right \}\] exists and is finite. [i]I. Gyori[/i]

2013 Miklós Schweitzer, 9

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

Prove that there is a function ${f: (0,\infty) \rightarrow (0,\infty)}$ which is nowhere continuous and for all ${x,y \in (0,\infty)}$ and any rational ${\alpha}$ we have \[ \displaystyle f\left( \left(\frac{x^\alpha+y^\alpha}{2}\right)^{\frac{1}{\alpha}}\right)\leq \left(\frac{f(x)^\alpha +f(y)^\alpha }{2}\right)^{\frac{1}{\alpha}}. \] Is there such a function if instead the above relation holds for every ${x,y \in (0,\infty)}$ and for every irrational ${\alpha}?$ [i]Proposed by Maksa Gyula and Zsolt Páles[/i]

1952 Miklós Schweitzer, 8

Tags: real analysis , real analysis unsolved

For which values of $ z$ does the series $ \sum_{n\equal{}1}^{\infty}c_1c_2\cdots c_n z^n$ converge, provided that $ c_k>0$ and $ \sum_{k\equal{}1}^{\infty} \frac{c_k}{k}<\infty$ ?

1998 IMC, 5

Tags: algebra , polynomial , induction , real analysis , real analysis unsolved

Let $P$ be a polynomial of degree $n$ with only real zeros and real coefficients. Prove that for every real $x$ we have $(n-1)(P'(x))^2\ge nP(x)P''(x)$. When does equality occur?

1963 Miklós Schweitzer, 5

Tags: function , real analysis , real analysis unsolved

Let $ H$ be a set of real numbers that does not consist of $ 0$ alone and is closed under addition. Further, let $ f(x)$ be a real-valued function defined on $ H$ and satisfying the following conditions: \[ \;f(x)\leq f(y)\ \mathrm{if} \;x \leq y\] and \[ f(x\plus{}y)\equal{}f(x)\plus{}f(y) \;(x,y \in H)\ .\] Prove that $ f(x)\equal{}cx$ on $ H$, where $ c$ is a nonnegative number. [M. Hosszu, R. Borges]

2011 District Olympiad, 1

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

a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$. b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by \[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\] Prove that the sequence $(x_n)_{n\ge 1}$ is convergent and find it's limit.

2014 IMS, 2

Tags: topology , real analysis , real analysis unsolved

Let $(X,d)$ be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that $X$ has exactly one element.

2011 IMC, 4

Tags: function , probability , real analysis , real analysis unsolved

Let $A_1,A_2,\dots, A_n$ be finite, nonempty sets. Define the function \[f(t)=\sum_{k=1}^n \sum_{1\leq i_1<i_2<\dots<i_k\leq n} (-1)^{k-1}t^{|A_{i_1}\cup A_{i_2}\cup \dots\cup A_{i_k}|}.\] Prove that $f$ is nondecreasing on $[0,1].$ ($|A|$ denotes the number of elements in $A.$)

1999 IMC, 3

Tags: real analysis , real analysis unsolved

Let $x_i\ge -1$ and $\sum^n_{i=1}x_i^3=0$. Prove $\sum^n_{i=1}x_i \le \frac{n}{3}$.

2002 Romania National Olympiad, 2

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

Let $f:[0,1]\rightarrow\mathbb{R}$ be an integrable function such that: \[0<\left\vert \int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1.\] Show that there exists $x_1\not= x_2, x_1,x_2\in [0,1]$, such that: \[\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}\]

1966 Miklós Schweitzer, 6

Tags: function , real analysis , real analysis unsolved

A sentence of the following type if often heard in Hungarian weather reports: "Last night's minimum temperatures took all values between $ \minus{}3$ degrees and $ \plus{}5$ degrees." Show that it would suffice to say, "Both $ \minus{}3$ degrees and $ \plus{}5$ degrees occurred among last night's minimum temperatures." (Assume that temperature as a two-variable function of place and time is continuous.) [i]A.Csaszar[/i]

2005 Unirea, 3

Tags: real analysis , real analysis unsolved

$a_1=b_1=1$ $a_{n+1}=b_n+\frac{1}{n}$ $b_{n+1}=a_n-\frac{1}{n}$ Prove that $a_n$, $b_n$ is not convergent, but $a_nb_n$ is convergent Laurentin Panaitopol

1964 Miklós Schweitzer, 7

Tags: invariant , function , real analysis , real analysis unsolved

Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution $ f(t)$ and any real number $ c,f(t\plus{}c)$ is also a solution.

2010 Romania National Olympiad, 4

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

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and \[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\] Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$. b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$. [i]Calin Popescu[/i]

2001 District Olympiad, 3

Tags: function , real analysis , real analysis unsolved

Let $f:\mathbb{R}\to \mathbb{R}$ a function which transforms any closed bounded interval in a closed bounded interval and any open bounded interval in an open bounded interval. Prove that $f$ is continuous. [i]Mihai Piticari[/i]

2012 Pre-Preparation Course Examination, 4

Tags: topology , real analysis , real analysis unsolved

Suppose that $X$ and $Y$ are metric spaces and $f:X \longrightarrow Y$ is a continious function. Also $f_1: X\times \mathbb R \longrightarrow Y\times \mathbb R$ with equation $f_1(x,t)=(f(x),t)$ for all $x\in X$ and $t\in \mathbb R$ is a closed function. Prove that for every compact set $K\subseteq Y$, its pre-image $f^{pre}(K)$ is a compact set in $X$.

1997 IMC, 6

Tags: function , trigonometry , real analysis , real analysis unsolved

Let $f: [0,1]\rightarrow \mathbb{R}$ continuous. We say that $f$ crosses the axis at $x$ if $f(x)=0$ but $\exists y,z \in [x-\epsilon,x+\epsilon]: f(y)<0<f(z)$ for any $\epsilon$. (a) Give an example of a function that crosses the axis infinitely often. (b) Can a continuous function cross the axis uncountably often?