Found problems: 4776
2010 Miklós Schweitzer, 8
Let $ D \subset \mathbb {R} ^ {2} $ be a finite Lebesgue measure of a connected open set and $ u: D \rightarrow \mathbb {R} $ a harmonic function. Show that it is either a constant $ u $ or for almost every $ p \in D $
$$
f ^ {\prime} (t) = (\operatorname {grad} u) (f (t)), \quad f (0) = p
$$has no initial value problem(differentiable everywhere) solution to $ f:[0,\infty) \rightarrow D $.
1995 All-Russian Olympiad, 2
Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions whose graphs both have an axis of symmetry.
[i]D. Tereshin[/i]
2010 N.N. Mihăileanu Individual, 2
Let be a sequence of functions $ \left( f_n \right)_{n\ge 2}:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ defined, for each $ n\ge 2, $ as
$$ f_n(x)=2nx^{2+n} -2(n+2)x^{1+n} +(2+n)x +1. $$
[b]a)[/b] Prove that $ f_n $ has an unique local maxima $ x_n, $ for any $ n\ge 2. $
[b]b)[/b] Show that $ 1=\lim_{n\to\infty } x_n. $
[i]Cătălin Zîrnă[/i]
1987 Traian Lălescu, 1.1
Let $ a\in\mathbb{R}. $ Prove the following proposition:
$$ \left( x,y\in\mathbb{R}\implies x^4+y^4+axy+2\ge 0 \right)\iff |a|\le 4. $$
1999 Korea - Final Round, 1
If the equation:
$f(\frac{x-3}{x+1}) + f(\frac{3+x}{1-x}) = x$
holds true for all real x but $\pm 1$, find $f(x)$.
2007 IberoAmerican Olympiad For University Students, 3
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds:
$\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$,
where $T$ is the period of $f(x)$.
2014 Harvard-MIT Mathematics Tournament, 10
For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$. Find the smallest possible value of $m$.
2008 Philippine MO, 4
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{2008^{2x}}{2008+2008^{2x}}$. Prove that
\[\begin{aligned}
f\left(\frac{1}{2007}\right)+f\left(\frac{2}{2007}\right)+\cdots+f\left(\frac{2005}{2007}\right)+f\left(\frac{2006}{2007}\right)=1003.
\end{aligned}\]
2011 Today's Calculation Of Integral, 763
Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$
2003 Costa Rica - Final Round, 3
If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.
2002 Romania National Olympiad, 2
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has limits at any point and has no local extrema. Show that:
$a)$ $f$ is continuous;
$b)$ $f$ is strictly monotone.
2005 Germany Team Selection Test, 2
For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.
2016 Romania National Olympiad, 2
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying the conditions:
$$ \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\ f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , $$
for all real numbers $ x,y,t $ with $ t\in [0,1] . $
Prove that:
[b]a)[/b] $ f(b)+f(c)\le f(a)+f(d) , $ for any real numbers $ a,b,c,d $ such that $ a\le b\le c\le d $ and $ d-c=b-a. $
[b]b)[/b] for any natural number $ n\ge 3 $ and any $ n $ real numbers $ x_1,x_2,\ldots ,x_n, $ the following inequality holds.
$$ f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right) $$
2005 Czech-Polish-Slovak Match, 1
Let $n$ be a given positive integer. Solve the system
\[x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,\]
\[x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}\]
in the set of nonnegative real numbers.
2005 Taiwan National Olympiad, 3
$a_1, a_2, ..., a_{95}$ are positive reals. Show that
$\displaystyle \sum_{k=1}^{95}{a_k} \le 94+ \prod_{k=1}^{95}{\max{\{1,a_k\}}}$
2014 AIME Problems, 6
The graphs of $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.
2008 Harvard-MIT Mathematics Tournament, 7
Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.
2009 Indonesia TST, 3
Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that
\[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y)
\]
for all $ x,y \in \mathbb{R}$.
2023 Korea - Final Round, 2
Function $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies the following condition.
(Condition) For each positive real number $x$, there exists a positive real number $y$ such that $(x + f(y))(y + f(x)) \leq 4$, and the number of $y$ is finite.
Prove $f(x) > f(y)$ for any positive real numbers $x < y$. ($\mathbb{R^+}$ is a set for all positive real numbers.)
2014 National Olympiad First Round, 27
Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. For how many positive integers $k$ less than $2014$, it is $f(k)=8$?
$
\textbf{(A)}\ 45
\qquad\textbf{(B)}\ 120
\qquad\textbf{(C)}\ 165
\qquad\textbf{(D)}\ 180
\qquad\textbf{(E)}\ 215
$
1995 Cono Sur Olympiad, 3
Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function).
1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$.
2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$
1984 Miklós Schweitzer, 9
[b]9.[/b] Let $X_0, X_1, \dots $ be independent, indentically distributed, nondegenerate random variables, and let $0<\alpha <1$ be a real number. Assume that the series
$\sum_{k=1}^{\infty} \alpha^{k} X_k$
is convergent with probability one. Prove that the distribution function of the sum is continuous. ([b]P. 23[/b])
[T. F. Móri]
1989 IMO Longlists, 54
Let $ n \equal{} 2k \minus{} 1$ where $ k \geq 6$ is an integer. Let $ T$ be the set of all $ n\minus{}$tuples $ (x_1, x_2, \ldots, x_n)$ where $ x_i \in \{0,1\}$ $ \forall i \equal{} \{1,2, \ldots, n\}$ For $ x \equal{} (x_1, x_2, \ldots, x_n) \in T$ and $ y \equal{} (y_1, y_2, \ldots, y_n) \in T$ let $ d(x,y)$ denote the number of integers $ j$ with $ 1 \leq j \leq n$ such that $ x_i \neq x_j$, in particular $ d(x,x) \equal{} 0.$ Suppose that there exists a subset $ S$ of $ T$ with $ 2^k$ elements that has the following property: Given any element $ x \in T,$ there is a unique element $ y \in S$ with $ d(x, y) \leq 3.$ Prove that $ n \equal{} 23.$
2015 CIIM, Problem 5
There are $n$ people seated on a circular table that have seats numerated from 1 to $n$ clockwise. Let $k$ be a fix integer with $2 \leq k \leq n$. The people can change their seats. There are two types of moves permitted:
1. Each person moves to the next seat clockwise.
2. Only the ones in seats 1 and $k$ exchange their seats.
Determine, in function of $n$ and $k$, the number of possible configurations of people in the table that can be attain by using a sequence of permitted moves.
2007 IMAC Arhimede, 2
Let $ABCD$ be a parallelogram that is not rhombus. We draw the symmetrical half-line of $(DC$ with respect to line $BD$. Similarly we draw the symmetrical half- line of $(AB$ with respect to $AC$. These half- lines intersect each other in $P$. If $\frac{AP}{DP}= q$ find the value of $\frac{AC}{BD}$ in function of $q$.