Found problems: 265
1970 Miklós Schweitzer, 9
Construct a continuous function $ f(x)$, periodic with period $ 2 \pi$, such that the Fourier series of $ f(x)$ is divergent at $ x\equal{}0$, but the Fourier series of $ f^2(x)$ is uniformly convergent on $ [0,2 \pi].$
[i]P. Turan[/i]
2006 District Olympiad, 4
Let $\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f$ continuous $\}$ and $n$ an integer, $n\geq 2$. Find the smallest real constant $c$ such that for any $f\in \mathcal F$ the following inequality takes place \[ \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx. \]
2010 Contests, 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]
2012 IFYM, Sozopol, 5
Let $c_0,c_1>0$. And suppose the sequence $\{c_n\}_{n\ge 0}$ satisfies
\[ c_{n+1}=\sqrt{c_n}+\sqrt{c_{n-1}}\quad \text{for} \;n\ge 1 \]
Prove that $\lim_{n\to \infty}c_n$ exists and find its value.
[i]Proposed by Sadovnichy-Grigorian-Konyagin[/i]
1996 Romania National Olympiad, 2
Suppose that $ f: [a,b]\rightarrow \mathbb{R} $ be a monotonic function and for every $ x_1,x_2\in [a,b] $ that $ x_1<x_2 $ ,there exist $ c\in (a,b) $ such that $ \int _{x_1}^{x_2}f(x)dx=f(c)(x_1-x_2) $
a) Show that $ f $ be the continuous function on interval $ (a,b) $
b) Suppose that $ f $ is integrable function on interval $ [a,b] $ but $ f $ isn't a monotonic function then ,is it the result of part a) right?
2010 Contests, 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
$\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.
2006 IberoAmerican Olympiad For University Students, 4
Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval
\[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\]
such that
\[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\]
for all polynomials $f$ with real coefficients and degree less than $n$.
1997 IMC, 2
Let $a_n$ be a sequence of reals. Suppose $\sum a_n$ converges. Do these sums converge aswell?
(a) $a_1+a_2+(a_4+a_3)+(a_8+...+a_5)+(a_{16}+...+a_9)+...$
(b) ${a_1+a_2+(a_3)+(a_4)+(a_5+a_7)+(a_6+a_8)+(a_9+a_{11}+a_{13}+a_{15})+(a_{10}+a_{12}+a_{14}+a_{16})+(a_{17}+a_{19}+...}$
1998 Romania National Olympiad, 1
Suppose that $a,b\in\mathbb{R}^+$ which $a+b<1$ and $f:[0,+\infty) \rightarrow [0,+\infty) $ be the increasing function s.t. $\forall x\geq 0 ,\int _0^x f(t)dt=\int _0^{ax} f(t)dt+\int _0^{bx} f(t)dt$. Prove that $\forall x\geq 0 , f(x)=0$
2005 IberoAmerican Olympiad For University Students, 6
A smooth function $f:I\to \mathbb{R}$ is said to be [i]totally convex[/i] if $(-1)^k f^{(k)}(t) > 0$ for all $t\in I$ and every integer $k>0$ (here $I$ is an open interval).
Prove that every totally convex function $f:(0,+\infty)\to \mathbb{R}$ is real analytic.
[b]Note[/b]: A function $f:I\to \mathbb{R}$ is said to be [i]smooth[/i] if for every positive integer $k$ the derivative of order $k$ of $f$ is well defined and continuous over $\mathbb{R}$. A smooth function $f:I\to \mathbb{R}$ is said to be [i]real analytic[/i] if for every $t\in I$ there exists $\epsilon> 0$ such that for all real numbers $h$ with $|h|<\epsilon$ the Taylor series
\[\sum_{k\geq 0}\frac{f^{(k)}(t)}{k!}h^k\]
converges and is equal to $f(t+h)$.
1977 Miklós Schweitzer, 6
Let $ f$ be a real function defined on the positive half-axis for which $ f(xy)\equal{}xf(y)\plus{}yf(x)$ and $ f(x\plus{}1) \leq f(x)$ hold for every positive $ x$ and $ y$. Show that if $ f(1/2)\equal{}1/2$, then \[ f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x)\] for every $ x\in (0,1)$.
[i]Z. Daroczy, Gy. Maksa[/i]
1964 Miklós Schweitzer, 6
Let $ y_1(x)$ be an arbitrary, continuous, positive function on $ [0,A]$, where $ A$ is an arbitrary positive number. Let \[ y_{n+1}=2 \int_0^x \sqrt{y_n(t)}dt \;(n=1,2,...)\ .\] Prove that the functions $ y_n(x)$ converge to the function $ y=x^2$ uniformly on $ [0,A]$.
2000 IMC, 5
Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.
2006 Romania National Olympiad, 4
Let $f: [0,1]\to\mathbb{R}$ be a continuous function such that \[ \int_{0}^{1}f(x)dx=0. \] Prove that there is $c\in (0,1)$ such that \[ \int_{0}^{c}xf(x)dx=0. \]
[i]Cezar Lupu, Tudorel Lupu[/i]
2014 District Olympiad, 1
For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.$
2014 IMS, 4
Let $(X,d)$ be a metric space and $f:X \to X$ be a function such that $\forall x,y\in X : d(f(x),f(y))=d(x,y)$.
$\text{a})$ Prove that for all $x \in X$, $\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n}$ exists, where $f^n(x)$ is $\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots ))$.
$\text{b})$ Prove that the amount of the limit does [b][u]not[/u][/b] depend on choosing $x$.
2012 Pre-Preparation Course Examination, 2
Suppose that $\lim_{n\to \infty} a_n=a$ and $\lim_{n\to \infty} b_n=b$. Prove that
$\lim_{n\to \infty}\frac{1}{n}(a_1b_n+a_2b_{n-1}+...+a_nb_1)=ab$.
2008 Moldova MO 11-12, 2
Find the exact value of $ E\equal{}\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.
1974 Miklós Schweitzer, 6
Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$.
[i]G. Halasz[/i]
2013 ISI Entrance Examination, 3
Let $f:\mathbb R\to\mathbb R$ satisfy
\[|f(x+y)-f(x-y)-y|\leq y^2\]
For all $(x,y)\in\mathbb R^2.$ Show that $f(x)=\frac x2+c$ where $c$ is a constant.
1968 Miklós Schweitzer, 3
Let $ K$ be a compact topological group, and let $ F$ be a set of continuous functions defined on $ K$ that has cardinality greater that continuum. Prove that there exist $ x_0 \in K$ and $ f \not\equal{}g \in F$ such that
\[ f(x_0)\equal{}g(x_0)\equal{}\max_{x\in K}f(x)\equal{}\max_{x \in K}g(x).\]
[i]I. Juhasz[/i]
2009 District Olympiad, 3
Let $(x_n)_{n\ge 1}$ a sequence defined by $x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*$. Prove that $\lim_{n\to \infty} x_n=1$ and evaluate $\lim_{n\to \infty} x_n^n$.
2013 Romania National Olympiad, 3
A function \[\text{f:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] is called contract if, for every numbers $x,y\in \text{(0,}\infty \text{)}$ we have, $\underset{n\to \infty }{\mathop{\lim }}\,\left( {{f}^{n}}\left( x \right)-{{f}^{n}}\left( y \right) \right)=0$ where ${{f}^{n}}=\underbrace{f\circ f\circ ...\circ f}_{n\ f\text{'s}}$
a) Consider \[f:\text{(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] a function contract, continue with the property that has a fixed point, that existing ${{x}_{0}}\in \text{(0,}\infty \text{) }$ there so that $f\left( {{x}_{0}} \right)={{x}_{0}}.$ Show that $f\left( x \right)>x,$ for every $x\in \text{(0,}{{x}_{0}}\text{)}\,$ and $f\left( x \right)<x$, for every $x\in \text{(}{{x}_{0}}\text{,}\infty \text{)}\,$.
b) Show that the given function \[f\text{:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] given by $f\left( x \right)=x+\frac{1}{x}$ is contracted but has no fix number.
2002 District Olympiad, 4
Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that:
1. $f$ has one-side limits in any $a\in \mathbb{R}$ and $f(a-0)\le f(a)\le f(a+0)$.
2. for any $a,b\in \mathbb{R},\ a<b$, we have $f(a-0)<f(b-0)$.
Prove that $f$ is strictly increasing.
[i]Mihai Piticari & Sorin Radulescu[/i]
2002 IMC, 2
Does there exist a continuously differentiable function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for every $x \in \mathbb{R}$ we have $f(x) > 0$ and $f'(x) = f(f(x))$?