Found problems: 4776
2015 Romania National Olympiad, 2
Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $
[b]a)[/b] Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n. $
[b]b)[/b] If $ n=6, $ find the number of functions $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $ that have the property that $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $ divides $ 36. $
2016 CIIM, Problem 1
Find all functions $f:(0,+\infty) \to (0,+\infty)$ that satisfy
$(i)$ $f(xf(y))=yf(x), \forall x,y > 0,$
$(ii)$ $\displaystyle\lim_{x\to+\infty} f(x) = 0.$
2022 Romania EGMO TST, P4
For every positive integer $N\geq 2$ with prime factorisation $N=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ we define \[f(N):=1+p_1a_1+p_2a_2+\cdots+p_ka_k.\] Let $x_0\geq 2$ be a positive integer. We define the sequence $x_{n+1}=f(x_n)$ for all $n\geq 0.$ Prove that this sequence is eventually periodic and determine its fundamental period.
1979 Kurschak Competition, 2
$f$ is a real-valued function defined on the reals such that $f(x) \le x$ and $f(x + y) \le f(x) + f(y)$ for all $x, y$. Prove that $f(x) = x$ for all $x$.
2006 VJIMC, Problem 3
For a function $f:[0,1]\to\mathbb R$ the secant of $f$ at points $a,b\in[0,1]$, $a<b$, is the line in $\mathbb R^2$ passing through $(a,f(a))$ and $(b,f(b))$. A function is said to intersect its secant at $a,b$ if there exists a point $c\in(a,b)$ such that $(c,f(c))$ lies on the secant of $f$ at $a,b$.
1. Find the set $\mathcal F$ of all continuous functions $f$ such that for any $a,b\in[0,1]$, $a<b$, the function $f$ intersects its secant at $a,b$.
2. Does there exist a continuous function $f\notin\mathcal F$ such that for any rational $a,b\in[0,1],a<b$, the function $f$ intersects its secant at $a,b$?
2008 Grigore Moisil Intercounty, 2
Let $ n\in \mathbb{N^*}$ and $ f: [0,1]\rightarrow \mathbb{R}$ a continuos function with the prop. $ \int_{0}^{1}(1\minus{}x^n)f(x)dx\equal{}0$.
Prove that $ \int_{0}^{1}f^2(x)dx \geq 2(n\plus{}1)\left(\int_{0}^{1}f(x)dx\right)^2$
1963 Miklós Schweitzer, 7
Prove that for every convex function $ f(x)$ defined on the interval $ \minus{}1\leq x \leq 1$ and having absolute value at most $ 1$,
there is a linear function $ h(x)$ such that \[ \int_{\minus{}1}^1|f(x)\minus{}h(x)|dx\leq 4\minus{}\sqrt{8}.\] [L. Fejes-Toth]
2018 Miklós Schweitzer, 7
Describe all functions $f: \{ 0,1\}^n \to \{ 0,1\}$ which satisfy the equation
\begin{align*}
& f(f(a_{11},a_{12},\dotsc ,a_{1n}),f(a_{21},a_{22},\dotsc ,a_{2n}),\dotsc ,f(a_{n1},a_{n2},\dotsc ,a_{nn}))\\
& = f(f(a_{11},a_{21},\dotsc ,a_{n1}),f(a_{12},a_{22},\dotsc ,a_{n2}),\dotsc ,f(a_{1n},a_{2n},\dotsc ,a_{nn}))\end{align*}
for arbitrary $a_{ij}\in \{ 0,1\}$ where $i,j\in \{1,2,\dotsc ,n\}.$
Istek Lyceum Math Olympiad 2016, 1
Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which \[f(x+y)=f(x-y)+f(f(1-xy))\] holds for all real numbers $x$ and $y$
2005 Today's Calculation Of Integral, 8
Calculate the following indefinite integrals.
[1] $\int x(x^2+3)^2 dx$
[2] $\int \ln (x+2) dx$
[3] $\int x\cos x dx$
[4] $\int \frac{dx}{(x+2)^2}dx$
[5] $\int \frac{x-1}{x^2-2x+3}dx$
1997 Pre-Preparation Course Examination, 1
Let $f: \mathbb R \to\mathbb R$ be a function such that $|f(x)| \leq 1$ for all $x \in \mathbb R$ and
\[f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac 17 \right) + f \left( x + \frac 16 \right), \quad \forall x \in \mathbb R.\]
Show that $f$ is a periodic function.
2001 Federal Competition For Advanced Students, Part 2, 1
Find all functions $f :\mathbb R \to \mathbb R$ such that for all real $x, y$
\[f(f(x)^2 + f(y)) = xf(x) + y.\]
2008 Iran MO (3rd Round), 4
=A subset $ S$ of $ \mathbb R^2$ is called an algebraic set if and only if there is a polynomial $ p(x,y)\in\mathbb R[x,y]$ such that
\[ S \equal{} \{(x,y)\in\mathbb R^2|p(x,y) \equal{} 0\}
\]
Are the following subsets of plane an algebraic sets?
1. A square
[img]http://i36.tinypic.com/28uiaep.png[/img]
2. A closed half-circle
[img]http://i37.tinypic.com/155m155.png[/img]
2012 Harvard-MIT Mathematics Tournament, 10
Suppose that there are $16$ variables $\{a_{i,j}\}_{0\leq i,j\leq 3}$, each of which may be $0$ or $1$. For how many settings of the variables $a_{i,j}$ do there exist positive reals $c_{i,j}$ such that the polynomial \[f(x,y)=\sum_{0\leq i,j\leq 3}a_{i,j}c_{i,j}x^iy^j\] $(x,y\in\mathbb{R})$ is bounded below?
2012 ELMO Shortlist, 2
For positive rational $x$, if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$. For example, $f(1)=2$.
a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$, then $f(x)$ divides $|m-n|$.
b) Let $n$ be a positive integer. If all $x$ which satisfy $f(x)=f(2^nx)$ also satisfy $f(x)=2^n-1$, find all possible values of $n$.
[i]Anderson Wang.[/i]
1961 AMC 12/AHSME, 13
The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to:
${{ \textbf{(A)}\ t^3 \qquad\textbf{(B)}\ t^2+t \qquad\textbf{(C)}\ |t^2+t| \qquad\textbf{(D)}\ t\sqrt{t^2+1} }\qquad\textbf{(E)}\ |t|\sqrt{1+t^2} } $
2006 Federal Competition For Advanced Students, Part 2, 2
Find all monotonous functions $ f: \mathbb{R} \to \mathbb{R}$ that satisfy the following functional equation:
\[f(f(x)) \equal{} f( \minus{} f(x)) \equal{} f(x)^2.\]
2006 ISI B.Stat Entrance Exam, 6
(a) Let $f(x)=x-xe^{-\frac1x}, \ \ x>0$. Show that $f(x)$ is an increasing function on $(0,\infty)$, and $\lim_{x\to\infty} f(x)=1$.
(b) Using part (a) or otherwise, draw graphs of $y=x-1, y=x, y=x+1$, and $y=xe^{-\frac{1}{|x|}}$ for $-\infty<x<\infty$ using the same $X$ and $Y$ axes.
1989 China Team Selection Test, 2
$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$.
[i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.
2016 Iran MO (2nd Round), 6
Find all functions $f: \mathbb N \to \mathbb N$ Such that:
1.for all $x,y\in N$:$x+y|f(x)+f(y)$
2.for all $x\geq 1395$:$x^3\geq 2f(x)$
2001 District Olympiad, 3
Let $f:\mathbb{R}\to \mathbb{R}$ a function which transforms any closed bounded interval in a closed bounded interval and any open bounded interval in an open bounded interval.
Prove that $f$ is continuous.
[i]Mihai Piticari[/i]
1993 Swedish Mathematical Competition, 6
For real numbers $a$ and $b$ define $f(x) = \frac{1}{ax+b}$. For which $a$ and $b$ are there three distinct real numbers $x_1,x_2,x_3$ such that $f(x_1) = x_2$, $f(x_2) = x_3$ and $f(x_3) = x_1$?
2015 Moldova Team Selection Test, 1
Find all functions $f : \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+}$ that satisfy $f(mf(n)) = n+f(2015m)$ for all $m,n \in \mathbb{Z}_{+}$.
2020 JBMO TST of France, 3
Let n be a nonzero natural number. We say about a function f ∶ R ⟶ R that is n-positive
if, for any real numbers $x_1, x_2,...,x_n$
with the property that $x_1+x_2+...+x_n = 0$,
the inequality $f(x_1)+f(x_2)+...+f(x_n)=>0$ is true
a) Is it true that any 2020-positive function is also 1010-positive?
b) Is it true that any 1010-positive function is 2020-positive?
2020 Miklós Schweitzer, 6
Does there exist an entire function $F \colon \mathbb{C}\to \mathbb{C}$ such that $F$ is not zero everywhere, $|F(z)|\leq e^{|z|}$ for all $z\in \mathbb{C}$, $|F(iy)|\leq 1$ for all $y\in \mathbb{R}$, and $F$ has infinitely many real roots.