Found problems: 4776
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$
\[f(x + y)=f(f(x)) + y + 2022\]
Let $n$ be a positive integer. A function $f : \{1, 2, \dots, 2n\} \to \{1, 2, 3, 4, 5\}$ is [i]good[/i] if $f(j+2)$ and $f(j)$ have the same parity for every $j = 1, 2, \dots, 2n-2$.
Prove that the number of good functions is a perfect square.
Given a positive integer $n \geq 3$. Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for any $n$ positive reals $a_1,...,a_n$, the following condition is always satisfied:
$\sum_{i=1}^{n}(a_i-a_{i+1})f(a_i+a_{i+1}) = 0$
where $a_{n+1} = a_1$.
Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$.
Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$
Let $f(x)=\frac{9^x}{9^x+3}$. Compute $\sum_{k} f \biggl( \frac{k}{2002} \biggr)$, where $k$ goes over all integers $k$ between $0$ and $2002$ which are coprime to $2002$.
Let $f(x,y)$ be a function from ordered pairs of positive integers to real numbers
such that
\[ f(1,x) = f(x,1) = \frac{1}{x} \quad\text{and}\quad f(x+1,y+1)f(x,y)-f(x,y+1)f(x+1,y) = 1 \]
for all ordered pairs of positive integers $(x,y)$. If $f(100,100) = \frac{m}{n}$ for two relatively prime positive integers $m,n$, compute $m+n$.
[i]David Yang[/i]
Let $D=\{0,1,\dots,9\}$. A direction function for $D$ is a function $f:D \times D \to \{0,1\}.$
A real $r\in [0,1]$ is compatible with $f$ if it can be written in the form $$r = \sum_{j=1}^{\infty} \frac{d_j}{10^j}$$ with $d_j \in D$ and $f(d_j,d_{j+1})=1$ for every positive integer $j$.
Determine the least integer $k$ such that for any direction fuction $f$, if there are $k$ compatible reals with $f$ then there are infinite reals compatible with $f$.
Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.
Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that
\[f(xf(y)+f(x)) = 2f(x)+xy\]
for every reals $x,y$.
For positive integers $k$ and $n$, we know $k \geq n!$. Prove that $ \phi (k) \geq (n-1)!$
Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is [i]sum-free[/i] if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$, their sum $x+y$ does not belong to $A$. Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$, the image $\{f(a):a\in A\}$ is also sum-free.
[i]Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$, there exists a positive integer $m$ such that $f(m)=n$.[/i]
Is it possible that by using the following transformations:
\[ f(x) \mapsto x^2 \cdot f \left(\frac{1}{x}+1 \right) \ \ \ \text{or} \ \ \ f(x) \mapsto (x-1)^2 \cdot f\left(\frac{1}{x-1} \right)\]
the function $f(x)=x^2+5x+4$ is sent to the function $g(x)=x^2+10x+8$ ?
Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function.
a) Prove that $f$ is continous;
b) Prove that there exists an unique function $g:[0,\infty)\to\mathbb{R}$ such that for all $x\geq 0$ we have \[ f(x+g(x)) = f(g(x)) - g(x) . \]
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$
Let $f: \mathbb R \to \mathbb R$ be a function which is differentiable at $0$. Define another function $g: \mathbb R \to \mathbb R$ as follows:
$$g(x) = \begin{cases}
f(x)\sin\left(\frac 1x\right) ~ &\text{if} ~ x \neq 0 \\
0 &\text{if} ~ x = 0.
\end{cases}$$
Suppose that $g$ is also differentiable at $0$. Prove that \[g'(0) = f'(0) = f(0) = g(0) = 0.\]
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have
\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]
Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that
\[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\]
Prove that $(a_n)_{n\ge0}$ is a geometric sequence.
[i]Lucian Dragomir[/i]
Let $\Phi$ denote the Euler totient function. Prove that for infinitely many $k$ we have $\Phi (2^k+1) < 2^{k-1}$ and that for infinitely many $m$ one has $\Phi (2^m+1) > 2^{m-1}$
The function $f:\mathbb Z \to \mathbb Z$ has the property that for all integers $m$ and $n$
\[f(m)+f(n)+f(f(m^2+n^2))=1.\]
We know that integers $a$ and $b$ exist such that $f(a)-f(b)=3$. Prove that integers $c$ and $d$ can be found such that $f(c)-f(d)=1$.
[i]Proposed by Amirhossein Gorzi[/i]
Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.
Find the differentiable function defined in $x>0$ such that ${\int_1^{f(x)} f^{-1}(t)dt=\frac 13(x^{\frac {3}{2}}-8}).$
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.
[i]Carl Lian and Brian Hamrick.[/i]
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that $$f(xy)+f(x+y)=f(x)f(y)+f(x)+f(y)$$ for all $x,y\in\mathbb{Q}$.
If $ f(x) \equal{} ax \plus{} b$ and $ f^{ \minus{} 1}(x) \equal{} bx \plus{} a$ with $ a$ and $ b$ real, what is the value of $ a \plus{} b$?
$ \textbf{(A)} \minus{} \!2 \qquad \textbf{(B)} \minus{} \!1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$
Find the domain of the pairs of positive real numbers $(a,\ b)$ such that there is a $\theta\ (0<\theta \leq \pi)$ such that $\cos a\theta =\cos b\theta$, then draw the domain on the coordinate plane.
30 points