Found problems: 884
2012 Pre-Preparation Course Examination, 3
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?
1963 Miklós Schweitzer, 6
Show that if $ f(x)$ is a real-valued, continuous function on the half-line $ 0\leq x < \infty$, and \[ \int_0^{\infty} f^2(x)dx
<\infty\] then the function \[ g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt\] satisfies \[ \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx.\] [B. Szokefalvi-Nagy]
1998 Miklós Schweitzer, 4
For any measurable set $H \subset R$ , we define the sequence $a_n(H)$ by the formula:
$$a_n(H) = \lambda \bigg([0,1] \setminus \bigcup_{k = n}^{2n} (H + \log_2 k) \bigg)$$
where $\lambda$ denotes the Lebesgue measure and $\log_2$ denotes the binary logarithm. Prove that there is a measurable, 1-periodic, positive measure set $H \subset R$ , such that the sequence $a_n( H )$ does not belong to any space $l_p$ ($1 \leq p < \infty$).
[hide=not sure about this part]For what numbers $1 \leq p <\infty$ is it true that whenever H is 1-periodic, positive measure, the sequence $a_n( H )$ belongs to the space $l_p$?[/hide]
2021 SEEMOUS, Problem 4
For $p \in \mathbb{R}$, let $(a_n)_{n \ge 1}$ be the sequence defined by
\[ a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx. \]
Determine all possible values of $p$ for which the series $\sum_{n=1}^\infty a_n$ converges.
2000 VJIMC, Problem 3
Let $a_1,a_2,\ldots$ be a bounded sequence of reals. Is it true that the fact
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=c$$implies $b=c$?
2020 CIIM, 6
For a set $A$, we define $A + A = \{a + b: a, b \in A \}$. Determine whether there exists a set $A$ of positive integers such that $$\sum_{a \in A} \frac{1}{a} = +\infty \quad \text{and} \quad \lim_{n \rightarrow +\infty} \frac{|(A+A) \cap \{1,2,\cdots,n \}|}{n}=0.$$
[hide=Note]Google translated from [url=http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales]http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales[/url][/hide]
2001 Miklós Schweitzer, 2
Let $\alpha \leq -2$ be an integer. Prove that for every pair $(\beta_0, \beta_1)$ of integers there exists a uniquely determined sequence $0\leq q_0, \ldots, q_k<\alpha ^ 2 - \alpha$ of integers, such that $q_k\neq 0$ if $(\beta_0, \beta 1)\neq (0,0)$ and
$$\beta_i=\sum_{j=0}^k q_j(\alpha - i)^j,\text{ for }i=0,1$$
2005 Gheorghe Vranceanu, 4
Let be a sequence of real numbers $ \left( x_n \right)_{n\geqslant 0} $ with $ x_0\neq 0,1 $ and defined as $ x_{n+1}=x_n+x_n^{-1/x_0} . $
[b]a)[/b] Show that the sequence $ \left( x_n\cdot n^{-\frac{x_0}{1+x_0}} \right)_{n\geqslant 0} $ is convergent.
[b]b)[/b] Prove that $ \inf_{x_0\neq 0,1} \lim_{n\to\infty } x_n\cdot n^{-\frac{x_0}{1+x_0}} =1. $
2016 Brazil Undergrad MO, 6
Let it \(C,D > 0\). We call a function \(f:\mathbb{R} \rightarrow \mathbb{R}\) [i]pretty[/i] if \(f\) is a \(C^2\)-class, \(|x^3f(x)| \leq C\) and \(|xf''(x)| \leq D\).
[list='i']
[*] Show that if \(f\) is pretty, then, given \(\epsilon \geq 0\), there is a \(x_0 \geq 0\) such that for every \(x\) with \(|x| \geq x_0\), we have \(|x^2f'(x)| < \sqrt{2CD}+\epsilon\).
[*] Show that if \(0 < E < \sqrt{2CD}\) then there is a pretty function \(f\) such that for every \(x_0 \geq 0\) there is a \(x > x_0\) such that \(|x^2f'(x)| > E\).
[/list]
1980 IMO, 4
Given a real number $x>1$, prove that there exists a real number $y >0$ such that
\[\lim_{n \to \infty} \underbrace{\sqrt{y+\sqrt {y + \cdots+\sqrt y}}}_{n \text{ roots}}=x.\]
1987 Traian Lălescu, 1.2
Let $ I $ be a real interval, and $ f:I\longrightarrow\mathbb{R} $ be a continuous function. Prove that $ f $ is monotone if and only if $ \min(\left( f(a),f(b)\right) \le\frac{1}{b-a}\int_a^b f(x)dx \le\max\left( f(a),f(b) \right) , $ for any distinct $ a,b\in I. $
1979 Miklós Schweitzer, 10
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]
1997 VJIMC, Problem 2
Let $\alpha\in(0,1]$ be a given real number and let a real sequence $\{a_n\}^\infty_{n=1}$ satisfy the inequality
$$a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldots$$Prove that if $\{a_n\}$ is bounded, then it must be convergent.
2012 Online Math Open Problems, 22
Let $c_1,c_2,\ldots,c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1,a_2,\ldots,a_{6030}$, there exist 6030 real numbers $\{b_1,b_2,\ldots,b_{6030}\}$ such that \[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\] and \[b_n = \sum_{d\mid n} c_d a_{n/d}\] for $n=1,2,\ldots,6030$. Find $c_{6030}$.
[i]Victor Wang.[/i]
2004 Miklós Schweitzer, 8
Prove that for any $0<\delta <2\pi$ there exists a number $m>1$ such that for any positive integer $n$ and unimodular complex numbers $z_1,\ldots, z_n$ with $z_1^v+\dots+z_n^v=0$ for all integer exponents $1\le v\le m$, any arc of length $\delta$ of the unit circle contains at least one of the numbers $z_1,\ldots, z_n$.
2007 Gheorghe Vranceanu, 1
Let $ M $ denote the set of the primitives of a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $
[b]ii)[/b] Show that $ M $ along with the operation $ *:M^2\longrightarrow M $ defined as $ F*G=F+G(2007) $ form a commutative group.
[b]iii)[/b] Show that $ M $ is isomorphic with the additive group of real numbers.
2004 Alexandru Myller, 1
[b]a)[/b] Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of real numbers having the property that $ \left| x_{n+1} -x_n \right|\leqslant 1/2^n, $ for any $ n\geqslant 1. $
Show that $ \left( x_n \right)_{n\ge 1} $ is convergent.
[b]b)[/b] Create a sequence $ \left( y_n \right)_{n\ge 1} $ of real numbers that has the following properties:
$ \text{(i) } \lim_{n\to\infty } \left( y_{n+1} -y_n \right) = 0 $
$ \text{(ii) } $ is bounded
$ \text{(iii) } $ is divergent
[i]Eugen Popa[/i]
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]
2006 Victor Vâlcovici, 2
Let be a differentiable function $ f:[0,1]\longrightarrow\mathbb{R} $ whose derivative has a positive Lipschitz constant $ L. $ Show that
[b]a)[/b] $ x,y\in [0,1]\implies | f(x)-f(y)-f'(y)(x-y) |\le L\cdot (x-y)^2 $
[b]b)[/b] $ \lim_{n\to\infty } \left( n\int_0^1 f(x)dx-\sum_{i=1}^nf\left( \frac{2i-1}{2n} \right) \right) =0. $
2018 District Olympiad, 3
Show that a continuous function $f : \mathbb{R} \to \mathbb{R}$ is increasing if and only if
\[(c - b)\int_a^b f(x)\, \text{d}x \le (b - a) \int_b^c f(x) \, \text{d}x,\]
for any real numbers $a < b < c$.
2003 Romania National Olympiad, 2
Let be an odd natural number $ n\ge 3. $ Find all continuous functions $ f:[0,1]\longrightarrow\mathbb{R} $ that satisfy the following equalities.
$$ \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\} $$
[i]Titu Andreescu[/i]
2005 Unirea, 3
$a_1=b_1=1$
$a_{n+1}=b_n+\frac{1}{n}$
$b_{n+1}=a_n-\frac{1}{n}$
Prove that $a_n$, $b_n$ is not convergent, but $a_nb_n$ is convergent
Laurentin Panaitopol
MIPT Undergraduate Contest 2019, 1.2
Does there exist a strictly increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$ that takes on only irrational values?
1987 Traian Lălescu, 1.4
[b]a)[/b] Determine all sequences of real numbers $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ that satisfy $ x_{n+2}+x_{n+1}=x_n, $ for any nonnegative integer $ n. $
[b]b)[/b] If $ y_k>0 $ and $ y_k^k=y_k+k, $ for all naturals $ k, $ calculate $ \lim_{n\to\infty }\frac{\ln n}{n\left( x_n-1\right)} . $
2012 Pre-Preparation Course Examination, 5
The $2^{nd}$ order differentiable function $f:\mathbb R \longrightarrow \mathbb R$ is in such a way that for every $x\in \mathbb R$ we have $f''(x)+f(x)=0$.
[b]a)[/b] Prove that if in addition, $f(0)=f'(0)=0$, then $f\equiv 0$.
[b]b)[/b] Use the previous part to show that there exist $a,b\in \mathbb R$ such that $f(x)=a\sin x+b\cos x$.