Olympiad problems

1969 Miklós Schweitzer, 8

Let $ f$ and $ g$ be continuous positive functions defined on the interval $ [0, +\infty)$, and let $ E \subset[0,+\infty)$ be a set of positive measure. Prove that the range of the function defined on $ E \times E$ by the relation \[ F(x,y)= %Error. "dispalymath" is a bad command. \int_0^xf(t)dt+ %Error. "dispalymath" is a bad command. \int_0^y g(t)dt\] has a nonvoid interior. [i]L. Losonczi[/i]

2005 Brazil Undergrad MO, 4

Tags: logarithms , limit , ratio , real analysis , real analysis unsolved

Let $a_{n+1} = a_n + \frac{1}{{a_n}^{2005}}$ and $a_1=1$. Show that $\sum^{\infty}_{n=1}{\frac{1}{n a_n}}$ converge.

2004 Romania National Olympiad, 4

Tags: function , real analysis , real analysis unsolved

(a) Build a function $f : \mathbb R \to \mathbb R_+$ with the property $\left( \mathcal P \right)$, i.e. all $x \in \mathbb Q$ are local, strict minimum points. (b) Build a function $f : \mathbb Q \to \mathbb R_+$ such that every point is a local, strict minimum point and such that $f$ is unbounded on $I \cap \mathbb Q$, where $I$ is a non-degenerate interval. (c) Let $f: \mathbb R \to \mathbb R_+$ be a function unbounded on every $I \cap \mathbb Q$, where $I$ is a non-degenerate interval. Prove that $f$ doesn't have the property $\left( \mathcal P \right)$.

2005 District Olympiad, 4

Tags: function , algebra , polynomial , irrational number , real analysis , real analysis unsolved

Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$ a monotonic bijective function. a)Prove that there exist a unique continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ such that $F(x)=f(x),\ (\forall)x\in \mathbb{Q}$. b)Give an example of a non-injective polynomial function $G:\mathbb{R}\rightarrow \mathbb{R}$ such that $G(\mathbb{Q})\subset \mathbb{Q}$ and it's restriction defined on $\mathbb{Q}$ is injective.

2014 District Olympiad, 4

Tags: induction , real analysis , real analysis unsolved

Let $f\colon\mathbb{N}\rightarrow\mathbb{N}^{\ast}$ be a strictly increasing function. Prove that: [list=a] [*]There exists a decreasing sequence of positive real numbers, $(y_{n})_{n\in\mathbb{N}}$, converging to $0$, such that $y_{n}\leq2y_{f(n)}$, for all $n\in\mathbb{N}$. [*]If $(x_{n})_{n\in\mathbb{N}}$ is a decreasing sequence of real numbers, converging to $0$, then there exists a decreasing sequence of real numbers $(y_{n})_{n\in\mathbb{N}}$, converging to $0$, such that $x_{n}\leq y_{n} \leq2y_{f(n)}$, for all $n\in\mathbb{N}$.[/list]

1974 Miklós Schweitzer, 5

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

Let $ \{f_n \}_{n=0}^{\infty}$ be a uniformly bounded sequence of real-valued measurable functions defined on $ [0,1]$ satisfying \[ \int_0^1 f_n^2=1.\] Further, let $ \{ c_n \}$ be a sequence of real numbers with \[ \sum_{n=0}^{\infty} c_n^2= +\infty.\] Prove that some re-arrangement of the series $ \sum_{n=0}^{\infty} c_nf_n$ is divergent on a set of positive measure. [i]J. Komlos[/i]

2014 IMS, 8

Tags: trigonometry , real analysis , real analysis unsolved

Is $\sum_{n=1}^{+\infty}\frac{\cos n}{n}(1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}})$ convergent? why?

2004 District Olympiad, 3

Tags: function , real analysis , real analysis unsolved

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a function such that $f\left(\frac{a+b}{2}\right)\in \{f(a),f(b)\},\ (\forall)a,b\in \mathbb{R}$. a) Give an example of a non-constant function that satisfy the hypothesis. b)If $f$ is continuous, prove that $f$ is constant.

2001 District Olympiad, 3

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

Consider a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that for any third degree polynomial function $P:[0,1]\to [0,1]$, we have \[\int_0^1f(P(x))dx=0\] Prove that $f(x)=0,\ (\forall)x\in [0,1]$. [i]Mihai Piticari[/i]

2012 District Olympiad, 1

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

Let $a,b,c$ three positive distinct real numbers. Evaluate: \[\lim_{t\to \infty} \int_0^t \frac{1}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}dx\]

2008 Moldova National Olympiad, 12.8

Tags: integration , trigonometry , logarithms , real analysis , real analysis unsolved

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

2007 Moldova National Olympiad, 12.8

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

Find all continuous functions $f\colon [0;1] \to R$ such that \[\int_{0}^{1}f(x)dx = 2\int_{0}^{1}(f(x^{4}))^{2}dx+\frac{2}{7}\]

1969 Miklós Schweitzer, 10

Tags: real analysis , analytic geometry , real analysis unsolved

In $ n$-dimensional Euclidean space, the square of the two-dimensional Lebesgue measure of a bounded, closed, (two-dimensional) planar set is equal to the sum of the squares of the measures of the orthogonal projections of the given set on the $ n$-coordinate hyperplanes. [i]L. Tamassy[/i]

2012 Pre-Preparation Course Examination, 1

Tags: function , real analysis , real analysis unsolved

Suppose that $X$ and $Y$ are two metric spaces and $f:X \longrightarrow Y$ is a continious function. Also for every compact set $K \subseteq Y$, it's pre-image $f^{pre}(K)$ is a compact set in $X$. Prove that $f$ is a closed function, i.e for every close set $C\subseteq X$, it's image $f(C)$ is a closed subset of $Y$.

1997 Romania National Olympiad, 4

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

Suppose that $(f_n)_{n\in N}$ be the sequence from all functions $f_n:[0,1]\rightarrow \mathbb{R^+}$ s.t. $f_0$ be the continuous function and $\forall x\in [0,1] , \forall n\in \mathbb {N} , f_{n+1}(x)=\int_0^x \frac {1}{1+f_n (t)}dt$. Prove that for every $x\in [0,1]$ the sequence of $(f_n(x))_{n\in N}$ be the convergent sequence and calculate the limitation.

2012 District Olympiad, 4

Tags: function , integration , geometry , rhombus , real analysis , real analysis unsolved

Let $f:[0,1]\rightarrow \mathbb{R}$ a differentiable function such that $f(0)=f(1)=0$ and $|f'(x)|\le 1,\ \forall x\in [0,1]$. Prove that: \[\left|\int_0 ^1f(t)dt\right|<\frac{1}{4}\]

2006 Moldova National Olympiad, 11.2

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 \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

2012 Pre-Preparation Course Examination, 3

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

Consider the set $\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}$. Prove that there is no function in this function space that gives us the minimum of $S=\int_{-1}^1x^2f'(x)^2dx$. What is the infimum of $S$ for the functions of this space?

1979 Miklós Schweitzer, 10

Tags: function , real analysis , real analysis unsolved

Prove that if $ a_i(i=1,2,3,4)$ are positive constants, $ a_2-a_4 > 2$, and $ a_1a_3-a_2 > 2$, then the solution $ (x(t),y(t))$ of the system of differential equations \[ \.{x}=a_1-a_2x+a_3xy,\] \[ \.{y}=a_4x-y-a_3xy \;\;\;(x,y \in \mathbb{R}) \] with the initial conditions $ x(0)=0, y(0) \geq a_1$ is such that the function $ x(t)$ has exactly one strict local maximum on the interval $ [0, \infty)$. [i]L. Pinter, L. Hatvani[/i]

2001 Romania National Olympiad, 3

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

Let $f:\mathbb{R}\rightarrow[0,\infty )$ be a function with the property that $|f(x)-f(y)|\le |x-y|$ for every $x,y\in\mathbb{R}$. Show that: a) If $\lim_{n\rightarrow \infty} f(x+n)=\infty$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\infty$. b) If $\lim_{n\rightarrow \infty} f(x+n)=\alpha ,\alpha\in[0,\infty )$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\alpha$.

1949 Miklós Schweitzer, 4

Tags: real analysis , real analysis unsolved , college contests , Miklos Schweitzer

Let $ A$ and $ B$ be two disjoint sets in the interval $ (0,1)$ . Denoting by $ \mu$ the Lebesgue measure on the real line, let $ \mu(A)>0$ and $ \mu(B)>0$ . Let further $ n$ be a positive integer and $ \lambda \equal{}\frac1n$ . Show that there exists a subinterval $ (c,d)$ of $ (0,1)$ for which $ \mu(A\cap (c,d))\equal{}\lambda \mu(A)$ and $ \mu(B\cap (c,d))\equal{}\lambda \mu(B)$ . Show further that this is not true if $ \lambda$ is not of the form $ \frac1n$.

1950 Miklós Schweitzer, 9

Tags: conics , real analysis , real analysis unsolved

Find the necessary and sufficient conditions for two conics that every tangent to one of them contains a real point of the other.

1978 Miklós Schweitzer, 4

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

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]

1962 Miklós Schweitzer, 6

Tags: real analysis , real analysis unsolved

Let $ E$ be a bounded subset of the real line, and let $ \Omega$ be a system of (non degenerate) closed intervals such that for each $ x \in E$ there exists an $ I \in \Omega$ with left endpoint $ x$. Show that for every $ \varepsilon > 0$ there exists a finite number of pairwise non overlapping intervals belonging to $ \Omega$ that cover $ E$ with the exception of a subset of outer measure less than $ \varepsilon$. [J. Czipszer]

1983 Miklós Schweitzer, 5

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

Let $ g : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $ x+g(x)$ is strictly monotone (increasing or decreasing), and let $ u : [0,\infty) \rightarrow \mathbb{R}$ be a bounded and continuous function such that \[ u(t)+ \int_{t-1}^tg(u(s))ds\] is constant on $ [1,\infty)$. Prove that the limit $ \lim_{t\rightarrow \infty} u(t)$ exists. [i]T. Krisztin[/i]