Found problems: 884
1996 Miklós Schweitzer, 6
Let $\{a_n\}$ be a bounded real sequence.
(a) Prove that if X is a positive-measure subset of $\mathbb R$, then for almost all $x\in X$, there exist a subsequence $\{y_n\}$ of X such that $$\sum_{n=1}^\infty (n(y_n-x)-a_n)=1$$
(b) construct an unbounded sequence $\{a_n\}$ for which the above equation is also true.
2000 Moldova National Olympiad, Problem 2
For $n\in\mathbb N$, define
$$a_n=\frac1{\binom n1}+\frac1{\binom n2}+\ldots+\frac1{\binom nn}.$$
(a) Prove that the sequence $b_n=a_n^n$ is convergent and determine the limit.
(b) Show that $\lim_{n\to\infty}b_n>\left(\frac32\right)^{\sqrt3+\sqrt2}$.
2018 Romania National Olympiad, 3
Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$
$\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$
$\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$
[i]Nicolae Bourbacut[/i]
2014 Cezar Ivănescu, 1
For a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers that are at least $ 1, $ prove that the series $ \sum_{i=1}^{\infty } \frac{1}{x_i} $ converges if and only if the series $ \sum_{i=1}^{\infty } \frac{1}{1+x_i} $ converges if and only if the series $ \sum_{i=1}^{\infty } \frac{1}{\lfloor x_i\rfloor } $ converges.
2003 District Olympiad, 4
Consider the continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , $ where $
f $ has a finite limit at $ \infty . $ Show that:
$$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$
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]
2007 Today's Calculation Of Integral, 184
(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality.
\[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \]
(2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$
2010 Contests, 4
A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.
2010 IMC, 1
Let $0 < a < b$. Prove that
$\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.
2003 VJIMC, Problem 3
Let $\{a_n\}^\infty_{n=0}$ be the sequence of real numbers satisfying $a_0=0$, $a_1=1$ and
$$a_{n+2}=a_{n+1}+\frac{a_n}{2^n}$$for every $n\ge0$. Prove that
$$\lim_{n\to\infty}a_n=1+\sum_{n=1}^\infty\frac1{2^{\frac{n(n-1)}2}\displaystyle\prod_{k=1}^n(2^k-1)}.$$
2012 Grigore Moisil Intercounty, 2
Let $ \left( x_n \right)_{n\ge 0} $ be a sequence of positive real numbers with $ x_0=1 $ and defined recursively:
$$ x_{n+1}=x_n+\frac{x_0}{x_1+x_2+\cdots +x_n} $$
[b]a)[/b] Show that $ \lim_{n\to\infty } x_n=\infty . $
[b]b)[/b] Calculate $ \lim_{n\to\infty }\frac{x_n}{\sqrt{\ln n}} . $
[i]Ovidiu Furdui[/i]
2014 Cezar Ivănescu, 2
Let be a function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0} $ that satisfies the relation
$$ \sqrt{x^2-x+1}\le f(x) e^{f(x)}\le \sqrt{x^2+x+1} , $$
for any positive real number $ x. $ Prove that
[b]a)[/b] $ \lim_{x\to\infty } f(x)=\infty . $
[b]b)[/b] $ \lim_{x\to\infty } (1/x)^{1/f(x)} =1/e. $
2007 District Olympiad, 3
Let $a,b\in \mathbb{R}$. Evaluate:
\[\lim_{n\to \infty}\left(\sqrt{a^2n^2+bn}-an\right)\]
Consider the sequence $(x_n)_{n\ge 1}$, defined by $x_n=\sqrt{n}-\lfloor \sqrt{n}\rfloor$. Denote by $A$ the set of the points $x\in \mathbb{R}$, for which there is a subsequence of $(x_n)_{n\ge 1}$ tending to $x$.
a) Prove that $\mathbb{Q}\cap [0,1]\subset A$.
b) Find $A$.
1968 Miklós Schweitzer, 5
Let $ k$ be a positive integer, $ z$ a complex number, and $ \varepsilon <\frac12$ a positive number. Prove that the following inequality holds for infinitely many positive integers $ n$: \[ \mid \sum_{0\leq l \leq \frac{n}{k+1}} \binom{n-kl}{l}z^l \mid \geq (\frac 12-\varepsilon)^n.\]
[i]P. Turan[/i]
1996 IMC, 4
Let $a_{1}=1$, $a_{n}=\frac{1}{n} \sum_{k=1}^{n-1}a_{k}a_{n-k}$ for $n\geq 2$. Show that
i) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}<2^{-\frac{1}{2}}$;
ii) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}$
1999 Romania National Olympiad, 1
Find all continuous functions $ f: \mathbb{R}\to [1,\infty)$ for wich there exists $ a\in\mathbb{R}$ and a positive integer $ k$ such that \[ f(x)f(2x)\cdot...\cdot f(nx)\leq an^k\]
for all real $ x$ and all positive integers $ n$.
[i]author :Radu Gologan[/i]
1986 USAMO, 5
By a partition $\pi$ of an integer $n\ge 1$, we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$, then the partitions $\pi$ are $1+1+1+1$, $1+1+2$, $1+3, 2+2$, and $4$).
For any partition $\pi$, define $A(\pi)$ to be the number of $1$'s which appear in $\pi$, and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$. (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$, then $A(\pi)=2$ and $B(\pi) = 3$).
Prove that, for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$.
Gheorghe Țițeica 2024, P1
Let $a>1$ and $b>1$ be rational numbers. Denote by $\mathcal{F}_{a,b}$ the set of functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $$f(ax)=bf(x), \text{ for all }x\geq 0.$$
a) Prove that the set $\mathcal{F}_{a,b}$ contains both Riemann integrable functions on any interval and functions that are not Riemann integrable on any interval.
b) If $f\in\mathcal{F}_{a,b}$ is Riemann integrable on $[0,\infty)$ and $\int_{\frac{1}{a}}^{a}f(x)dx=1$, calculate $$\int_a^{a^2} f(x)dx\text{ and }\int_0^1 f(x)dx.$$
[i]Vasile Pop[/i]
1996 IMC, 2
Evaluate the definite integral
$$\int_{-\pi}^{\pi}\frac{\sin nx}{(1+2^{x})\sin x} dx,$$
where $n$ is a natural number.
2011 Romania National Olympiad, 2
Let be a continuous function $ f:[0,1]\longrightarrow\left( 0,\infty \right) $ having the property that, for any natural number $ n\ge 2, $ there exist $ n-1 $ real numbers $ 0<t_1<t_2<\cdots <t_{n-1}<1, $ such that
$$ \int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt. $$
Calculate $ \lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} . $
2003 Tuymaada Olympiad, 4
Find all continuous functions $f(x)$ defined for all $x>0$ such that for every $x$, $y > 0$
\[ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) . \]
[i]Proposed by F. Petrov[/i]
2006 Mathematics for Its Sake, 3
Let be two positive real numbers $ a,b, $ and an infinite arithmetic sequence of natural numbers $ \left( x_n \right)_{n\ge 1} . $
Study the convergence of the sequences
$$ \left( \frac{1}{x_n}\sum_{i=1}^n\sqrt[x_i]{b} \right)_{n\ge 1}\text{ and } \left( \left(\sum_{i=1}^n \sqrt[x_i]{a}/\sqrt[x_i]{b} \right)^\frac{x_n}{\ln x_n} \right)_{n\ge 1} , $$
and calculate their limits.
[i]Dumitru Acu[/i]
1970 Miklós Schweitzer, 10
Prove that for every $ \vartheta$, $ 0<\vartheta<1$, there exist a sequence $ \lambda_n$ of positive integers and a series $ \sum_{n=1}^{\infty} a_n$ such that
(i)$ \lambda_{n+1}-\lambda_n > (\lambda_n)^{\vartheta}$,
(ii) $ \lim_{r\rightarrow 1^-} \sum_{n=1}^{\infty} a_nr^{\lambda_n}$ exists,
(iii) $ \sum _{n=1}^{\infty} a_n$ is divergent.
[i]P. Turan[/i]
2012 Romania National Olympiad, 3
[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by
\[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\]
Determine the following two sets:
[list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$
[b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]
2012 Pre-Preparation Course Examination, 6
Suppose that $a_{ij}$ are real numbers in such a way that for each $i$, the series $\sum_{j=1}^{\infty}a_{ij}$ is absolutely convergent. In fact we have a series of absolutely convergent serieses. Also we know that for each bounded sequence $\{b_j\}_j$ we have $\lim_{i\to \infty} \sum_{j=1}^{\infty}a_{ij}b_j=0$. Prove that
$\lim_{i\to \infty}\sum_{j=1}^{\infty}|a_{ij}|=0$.