Found problems: 884
2019 Romania National Olympiad, 3
$\textbf{a)}$ Prove that there exists a differentiable function $f:(0, \infty) \to (0, \infty)$ such that $f(f'(x)) = x, \: \forall x>0.$
$\textbf{b)}$ Prove that there is no differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f'(x)) = x, \: \forall x \in \mathbb{R}.$
1963 Miklós Schweitzer, 9
Let $ f(t)$ be a continuous function on the interval $ 0 \leq t \leq 1$, and define the two sets of points \[ A_t\equal{}\{(t,0): t\in[0,1]\} , B_t\equal{}\{(f(t),1): t\in [0,1]\}.\] Show that the union of all segments $ \overline{A_tB_t}$ is Lebesgue-measurable, and find the minimum of its measure with respect to all functions $ f$. [A. Csaszar]
2021 Romania National Olympiad, 3
Let $f :\mathbb R \to\mathbb R$ a function $ n \geq 2$ times differentiable so that:
$ \lim_{x \to \infty} f(x) = l \in \mathbb R$ and $ \lim_{x \to \infty} f^{(n)}(x) = 0$.
Prove that: $ \lim_{x \to \infty} f^{(k)}(x) = 0 $ for all $ k \in \{1, 2, \dots, n - 1\} $, where $f^{(k)}$ is the $ k $ - th derivative of $f$.
2007 Gheorghe Vranceanu, 4
Let be a sequence $ \left( a_n \right)_{n\geqslant 1} $ of real numbers defined recursively as
$$ a_n=2007+1004n^2-a_{n-1}-a_{n-2}-\cdots -a_2-a_1. $$ Calculate:
$$ \lim_{n\to\infty} \frac{1}{n}\int_1^{a_n} e^{1/\ln t} dt $$
1995 Miklós Schweitzer, 9
A serpentine is a sequence of points $P_1 , ..., P_m$ in a plane, not necessarily all different, such that the distance between $P_i$ and $P_{i+1}$ is at least 1, and the segments $P_i P_{i +1}$ are alternately horizontal and vertical. Construct a compact set in which there is a sequence of serpentines with arbitrary long lengths but there is no closed serpentine ($P_m = P_i$ for some i < m).
2024 Romania National Olympiad, 4
Let $f,g:\mathbb{R}\to\mathbb{R}$ be functions with $g(x)=2f(x)+f(x^2),$ for all $x \in \mathbb{R}.$
a) Prove that, if $f$ is bounded in a neighbourhood of the origin and $g$ is continuous in the origin, then $f$ is continuous in the origin.
b) Provide an example of function $f$, discontinuous in the origin, for which the function $g$ is continuous in the origin.
1973 Miklós Schweitzer, 5
Verify that for every $ x > 0$, \[ \frac{\Gamma'(x\plus{}1)}{\Gamma (x\plus{}1)} > \log x.\]
[i]P. Medgyessy[/i]
2006 Romania Team Selection Test, 2
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
MIPT Undergraduate Contest 2019, 1.5 & 2.5
Prove the inequality
$$\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2$$
for any sequence of real numbers $x_0, x_1, ..., x_n$ for which $x_0 = x_n = 0.$
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]
2023 District Olympiad, P3
Let $f:[a,b]\to[a,b]$ be a continuous function. It is known that there exist $\alpha,\beta\in (a,b)$ such that $f(\alpha)=a$ and $f(\beta)=b$. Prove that the function $f\circ f$ has at least three fixed points.
2006 District Olympiad, 1
Let $f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty)$ be $n$ continuous functions, $n\geq 1$, and let $\sigma$ be a permutation of the set $\{1,2,\ldots, n\}$. Prove that \[ \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx. \]
2007 VJIMC, Problem 3
A function $f:[0,\infty)\to\mathbb R\setminus\{0\}$ is called [i]slowly changing[/i] if for any $t>1$ the limit $\lim_{x\to\infty}\frac{f(tx)}{f(x)}$ exists and is equal to $1$. Is it true that every slowly changing function has for sufficiently large $x$ a constant sign (i.e., is it true that for every slowly changing $f$ there exists an $N$ such that for every $x,y>N$ we have $f(x)f(y)>0$?)
2021 IMC, 4
Let $f:\mathbb{R}\to \mathbb{R}$ be a function. Suppose that for every $\varepsilon >0$ , there exists a function $g:\mathbb{R}\to (0,\infty)$ such that for every pair $(x,y)$ of real numbers,
if $|x-y|<\text{min}\{g(x),g(y)\}$, then $|f(x)-f(y)|<\varepsilon$
Prove that $f$ is pointwise limit of a squence of continuous $\mathbb{R}\to \mathbb{R}$ functions i.e., there is a squence $h_1,h_2,...,$ of continuous $\mathbb{R}\to \mathbb{R}$ such that $\lim_{n\to \infty}h_n(x)=f(x)$ for every $x\in \mathbb{R}$
2019 AMC 10, 24
Define a sequence recursively by $x_0=5$ and
\[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\]
for all nonnegative integers $n.$ Let $m$ be the least positive integer such that
\[x_m\leq 4+\frac{1}{2^{20}}.\] In which of the following intervals does $m$ lie?
$\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]$
1982 Putnam, A2
For positive real $x$, let
$$B_n(x)=1^x+2^x+\ldots+n^x.$$Prove or disprove the convergence of
$$\sum_{n=2}^\infty\frac{B_n(\log_n2)}{(n\log_2n)^2}.$$
1962 Miklós Schweitzer, 5
Let $ f$ be a finite real function of one variable. Let $ \overline{D}f$ and $ \underline{D}f$ be its upper and lower derivatives, respectively, that is, \[ \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}\] ,
\[ \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}.\] Show that $ \overline{D}f$ and $ \underline{D}f$ are Borel-measurable functions. [A. Csaszar]
1949 Miklós Schweitzer, 2
Compute
$ \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx$ .
2012 USAMO, 6
For integer $n\geq2$, let $x_1, x_2, \ldots, x_n$ be real numbers satisfying \[x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.\]For each subset $A\subseteq\{1, 2, \ldots, n\}$, define\[S_A=\sum_{i\in A}x_i.\](If $A$ is the empty set, then $S_A=0$.)
Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$. For which choices of $x_1, x_2, \ldots, x_n, \lambda$ does equality hold?
2005 District Olympiad, 2
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $a,b\in \mathbb{R}$, with $a<b$ such that $f(a)=f(b)$, there exist some $c\in (a,b)$ such that $f(a)=f(b)=f(c)$. Prove that $f$ is monotonic over $\mathbb{R}$.
2013 Romania National Olympiad, 4
a) Consider\[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] a differentiable and convex function .Show that $f\left( x \right)\le x$, for every $x\ge 0$, than ${f}'\left( x \right)\le 1$ ,for every $x\ge 0$
b) Determine \[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] differentiable and convex functions which have the property that $f\left( 0 \right)=0\,$, and ${f}'\left( x \right)f\left( f\left( x \right) \right)=x$, for every $x\ge 0$
2009 IberoAmerican Olympiad For University Students, 4
Given two positive integers $m,n$, we say that a function $f : [0,m] \to \mathbb{R}$ is $(m,n)$-[i]slippery[/i] if it has the following properties:
i) $f$ is continuous;
ii) $f(0) = 0$, $f(m) = n$;
iii) If $t_1, t_2\in [0,m]$ with $t_1 < t_2$ are such that $t_2-t_1\in \mathbb{Z}$ and $f(t_2)-f(t_1)\in\mathbb{Z}$, then $t_2-t_1 \in \{0,m\}$.
Find all the possible values for $m, n$ such that there is a function $f$ that is $(m,n)$-slippery.
2021 IMO Shortlist, A5
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$
2000 VJIMC, Problem 1
Is there a countable set $Y$ and an uncountable family $\mathcal F$ of its subsets such that for every two distinct $A,B\in\mathcal F$, their intersection $A\cap B$ is finite?
2014 Paenza, 6
(a) Show that if $f:[-1,1]\to \mathbb{R}$ is a convex and $C^2$ function such that $f(1),f(-1)\geq 0$, then:
\[\min_{x\in[-1,1]} \{f(x)\} \geq - \int_{-1}^1 f''\]
(b) Let $B\subset \mathbb{R}^2$ the closed ball with center $0$ and radius $1$. Show that if $f: B \to \mathbb{R}$ is a convex and $C^2$ function and $f\geq 0$ in $\partial B$, then:
\[f(0)\geq -\frac{1}{\sqrt{\pi}} \left( \int_{B} (f_{xx}f_{yy}-f_{xy}^2) \right)^{1/2}\]