Found problems: 4776
2009 International Zhautykov Olympiad, 2
Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality:
\[ x\plus{}af(y)\leq y\plus{}f(f(x))
\]
for all $ x,y\in\mathbb{R}$
2002 APMO, 5
Let ${\bf R}$ denote the set of all real numbers. Find all functions $f$ from ${\bf R}$ to ${\bf R}$ satisfying:
(i) there are only finitely many $s$ in ${\bf R}$ such that $f(s)=0$,
and
(ii) $f(x^4+y)=x^3f(x)+f(f(y))$ for all $x,y$ in ${\bf R}$.
1980 AMC 12/AHSME, 14
If the function $f$ is defined by
\[ f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 , \] satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac 32$, then $c$ is
$\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$
2024 India National Olympiad, 4
A finite set $\mathcal{S}$ of positive integers is called cardinal if $\mathcal{S}$ contains the integer $|\mathcal{S}|$ where $|\mathcal{S}|$ denotes the number of distinct elements in $\mathcal{S}$. Let $f$ be a function from the set of positive integers to itself such that for any cardinal set $\mathcal{S}$, the set $f(\mathcal{S})$ is also cardinal. Here $f(\mathcal{S})$ denotes the set of all integers that can be expressed as $f(a)$ where $a \in \mathcal{S}$. Find all possible values of $f(2024)$
$\quad$
Proposed by Sutanay Bhattacharya
2020 Jozsef Wildt International Math Competition, W43
Let $f_1,f_2$ be nonnegative and concave functions. Then prove that
$$(f_1f_2)^{\frac{2^n-1}{n\cdot2^n}}\left(\frac{\displaystyle\prod_{k=1}^n\left(\sqrt[2^k]{f_1}+\sqrt[2^k]{f_2}\right)}{f_1+f_2}\right)^{\frac1n}$$
is concave.
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
2007 Nicolae Păun, 3
Construct a function $ f:[0,1]\longrightarrow\mathbb{R} $ that is primitivable, bounded, and doesn't touch its bounds.
[i]Dorian Popa[/i]
1995 Vietnam Team Selection Test, 3
Consider the function $ f(x) \equal{} \frac {2x^3 \minus{} 3}{3x^2 \minus{} 1}$.
$ 1.$ Prove that there is a continuous function $ g(x)$ on $ \mathbb{R}$ satisfying $ f(g(x)) \equal{} x$ and $ g(x) > x$ for all real $ x$.
$ 2.$ Show that there exists a real number $ a > 1$ such that the sequence $ \{a_n\}$, $ n \equal{} 1, 2, \ldots$, defined as follows $ a_0 \equal{} a$, $ a_{n \plus{} 1} \equal{} f(a_n)$, $ \forall n\in\mathbb{N}$ is periodic with the smallest period $ 1995$.
2013 Romanian Master of Mathematics, 2
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?
2006 District Olympiad, 1
Let $x,y,z$ be positive real numbers. Prove the following inequality: \[ \frac 1{x^2+yz} + \frac 1{y^2+zx } + \frac 1{z^2+xy} \leq \frac 12 \left( \frac 1{xy} + \frac 1{yz} + \frac 1{zx} \right). \]
2011 VJIMC, Problem 3
Let $p$ and $q$ be complex polynomials with $\deg p>\deg q$ and let $f(z)=\frac{p(z)}{q(z)}$. Suppose that all roots of $p$ lie inside the unit circle $|z|=1$ and that all roots of $q$ lie outside the unit circle. Prove that
$$\max_{|z|=1}|f'(z)|>\frac{\deg p-\deg q}2\max_{|z|=1}|f(z)|.$$
1989 National High School Mathematics League, 10
A positive number, if its fractional part, integeral part, and itself are geometric series, then the number is________.
1986 China Team Selection Test, 2
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.
2016 Peru MO (ONEM), 3
Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ such that
\[f(x + y) + f(x + z) - f(x)f(y + z) \ge 1\]
for all $x,y,z \in \mathbb{R}$
2000 National Olympiad First Round, 8
\[\begin{array}{rcl}
(x+y)^5 &=& z \\
(y+z)^5 &=& x \\
(z+x)^5 &=& y \end{array}\]
How many real triples $(x,y,z)$ are there satisfying above equation system?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None}
$
1982 Putnam, B5
For each $x>e^e$ define a sequence $S_x=u_0,u_1,\ldots$ recursively as follows: $u_0=e$, and for $n\ge0$, $u_{n+1}=\log_{u_n}x$. Prove that $S_x$ converges to a number $g(x)$ and that the function $g$ defined in this way is continuous for $x>e^e$.
2018 USA Team Selection Test, 2
Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$,
\[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\]
[i]Proposed by Yang Liu and Michael Kural[/i]
1973 Putnam, B5
(a) Let $z$ be a solution of the quadratic equation
$$az^2 +bz+c=0$$
and let $n$ be a positive integer. Show that $z$ can be expressed as a rational function of $z^n , a,b,c.$
(b) Using (a) or by any other means, express $x$ as a rational function of $x^{3}$ and $x+\frac{1}{x}.$
2021 AMC 10 Fall, 23
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$
2007 QEDMO 5th, 6
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ that satisfy the equation:
$ f\left(\left(f\left(x\right)\right)^2 \plus{} f\left(y\right)\right) \equal{} xf\left(x\right) \plus{} y$
for any two real numbers $ x$ and $ y$.
2007 District Olympiad, 3
Find all continuous functions $f : \mathbb R \to \mathbb R$ such that:
(a) $\lim_{x \to \infty}f(x)$ exists;
(b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.
2016 Romania Team Selection Test, 2
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$
1975 Putnam, B5
Define $f_{0}(x)=e^x$ and $f_{n+1}(x)=x f_{n}'(x)$. Show that $\sum_{n=0}^{\infty} \frac{f_{n}(1)}{n!}=e^e$.
1984 National High School Mathematics League, 6
If $F(\frac{1-x}{1+x})=x$, then
$\text{(A)}F(-2-x)=-2-F(x)\qquad\text{(B)}F(-x)=F(\frac{1+x}{1-x})$
$\text{(C)}F(\frac{1}{x})=F(x)\qquad\text{(D)}F(F(-x))=-x$
2003 Pan African, 3
Find all functions $f: R\to R$ such that:
\[ f(x^2)-f(y^2)=(x+y)(f(x)-f(y)), x,y \in R \]
2011-2012 SDML (High School), 1
The function $f$ is defined by $f\left(x\right)=x^2+3x$. Find the product of all solutions of the equation $f\left(2x-1\right)=6$.