Found problems: 47
2015 Indonesia MO Shortlist, A4
Determine all functions $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that
\[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every $x,y,z \in \mathbb{R}$
and there exists some real $a$ such that $f(x,a) = f(a,x) $ for every $x \in \mathbb{R}$.
2019 Poland - Second Round, 3
Let $f(t)=t^3+t$. Decide if there exist rational numbers $x, y$ and positive integers $m, n$ such that $xy=3$ and:
\begin{align*}
\underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)).
\end{align*}
2004 Purple Comet Problems, 12
If $f(x, y) = xy + 2x + y + 1$, find $f(f(2, f(3, 4)), 5)$.
2020 LIMIT Category 2, 2
The number of functions $g:\mathbb{R}^4\to\mathbb{R}$ such that, $\forall a,b,c,d,e,f\in\mathbb{R}$ :
(i) $g(1,0,0,1)=1$
(ii) $g(ea,b,ec,d)=eg(a,b,c,d)$
(iii) $g(a+e, b, c+f, d)= g(a,b,c,d)+g(e,b,f,d)$
(iv) $g(a,b,c,d)+g(b,a,d,c)=0$
is :
(A)$1$
(B)$0$
(C)$\text{infinitely many}$
(D)$\text{None of these}$
[Hide=Hint(given in question)]
Think of matrices[/hide]
2022 Indonesia TST, A
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
\[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$.
2021 India National Olympiad, 6
Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions:
[list]
[*] $f$ maps the zero polynomial to itself,
[*] for any non-zero polynomial $P \in \mathbb{R}[x]$, $\text{deg} \, f(P) \le 1+ \text{deg} \, P$, and
[*] for any two polynomials $P, Q \in \mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots.
[/list]
[i]Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha[/i]
2021 ISI Entrance Examination, 4
Let $g:(0,\infty) \rightarrow (0,\infty)$ be a differentiable function whose derivative is continuous, and such that $g(g(x)) = x$ for all $x> 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.
2016 IMC, 1
Let $f : \left[ a, b\right]\rightarrow\mathbb{R}$ be continuous on $\left[ a, b\right]$ and differentiable on $\left( a, b\right)$. Suppose that $f$ has infinitely many zeros, but there is no $x\in \left( a, b\right)$ with $f(x)=f'(x)=0$.
(a) Prove that $f(a)f(b)=0$.
(b) Give an example of such a function on $\left[ 0, 1\right]$.
(Proposed by Alexandr Bolbot, Novosibirsk State University)
2021 Caucasus Mathematical Olympiad, 8
An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions
$f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$).
A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.
1989 Brazil National Olympiad, 3
A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$.
Determine, justifying your answer, the set of all possible values for $f$.
2009 Jozsef Wildt International Math Competition, W. 6
Prove that$$p (n)= 2+ \left (p (1) + \cdots + p\left ( \left [\frac {n}{2} \right ] + \chi_1 (n)\right ) + \left (p'_2(n) + \cdots + p' _{ \left [\frac {n}{2} \right ] - 1}(n)\right )\right )$$for every $n \in \mathbb {N}$ with $n>2$ where $\chi $ denotes the principal character Dirichlet modulo 2, i.e.$$ \chi _1 (n) = \begin{cases} 1 & \text{if } (n,2)=1 \\ 0 &\text{if } (n,2)>1 \end{cases} $$with $p (n) $ we denote number of possible partitions of $n $ and $p' _m(n) $ we denote the number of partitions of $n$ in exactly $m$ sumands.
1984 Austrian-Polish Competition, 8
The functions $f_0,f_1 : (1,\infty) \to (1,\infty)$ are given by $ f_0(x) = 2x$ and$ f_1(x) =\frac{x}{x-1}$. Show that for any real numbers $a, b$ with $1 \le a < b$ there exist a positive integer $k$ and indices $i_1,i_2,...,i_k \in \{0,1\}$ such that $a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b$.
2008 Korea Junior Math Olympiad, 7
Find all pairs of functions $f; g : R \to R$ such that for all reals $x.y \ne 0$ :
$$f(x + y) = g \left(\frac{1}{x}+\frac{1}{y}\right) \cdot (xy)^{2008}$$
Today's calculation of integrals, 876
Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition :
1) $f(-1)\geq f(1).$
2) $x+f(x)$ is non decreasing function.
3) $\int_{-1}^ 1 f(x)\ dx=0.$
Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$
2020-2021 OMMC, 2
The function $f(x)$ is defined on the reals such that
$$f\left(\frac{1-4x}{4-x}\right) = 4-xf(x)$$
for all $x \ne 4$. There exists two distinct real numbers $a, b \ne 4$ such that $f(a) = f(b) = \frac{5}{2}$. $a+b$ can be represented as $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $10p + q$.
MIPT Undergraduate Contest 2019, 2.3
Let $A$ and $B$ be rectangles in the plane and $f : A \rightarrow B$ be a mapping which is uniform on the interior of $A$, maps the boundary of $A$ homeomorphically to the boundary of $B$ by mapping the sides of $A$ to corresponding sides in $B$. Prove that $f$ is an affine transformation.
2006 Tournament of Towns, 2
Do there exist functions $p(x)$ and $q(x)$, such that $p(x)$ is an even function while $p(q(x))$ is an odd function (different from 0)?
[i](3 points)[/i]
2023 USA EGMO Team Selection Test, 2
Consider pairs of functions $(f, g)$ from the set of nonnegative integers to itself such that
[list]
[*] $f(0) + f(1) + f(2) + \cdots + f(42) \le 2022$;
[*] for any integers $a \ge b \ge 0$, we have $g(a+b) \le f(a) + f(b)$.
[/list]
Determine the maximum possible value of $g(0) + g(1) + g(2) + \cdots + g(84)$ over all such pairs of functions.
[i]Evan Chen (adapting from TST3, by Sean Li)[/i]
2021 Brazil Undergrad MO, Problem 2
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ from $C^2$ (id est, $f$ is twice differentiable and $f''$ is continuous.) such that for every real number $t$ we have $f(t)^2=f(t \sqrt{2})$.
1991 Baltic Way, 9
Find the number of real solutions of the equation $a e^x = x^3$, where $a$ is a real parameter.
2002 Junior Balkan Team Selection Tests - Moldova, 12
Let $M$ be an empty set of real numbers. For any $x \in M$ the functions $f: M\to M$ and $g: M\to M$ satisfy the relations $f (g (x)) = g (f (x)) = x$ and $f (x) + g (x) = x$. Show that $- x \in M$ ¸ and $f (-x) = -f (x)$ whatever $x \in M$.
Kvant 2021, M2661
An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions
$f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$).
A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.