Found problems: 4776
A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying
$\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.
The parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ and vertex $ (h,k)$ is reflected about the line $ y \equal{} k$. This results in the parabola with equation $ y \equal{} dx^2 \plus{} ex \plus{} f$. Which of the following equals $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f$?
$ \textbf{(A)} \ 2b \qquad \textbf{(B)} \ 2c \qquad \textbf{(C)} \ 2a \plus{} 2b \qquad \textbf{(D)} \ 2h \qquad \textbf{(E)} \ 2k$
Find all functions $ f,g: \mathbb Q \to \mathbb Q$ such that for all rational numbers $ x,y$ we have
\[ f(f(x) \plus{} g(y) ) \equal{} g(f(x)) \plus{} y .
\]
sppose that $\sigma_k:\mathbb N \longrightarrow \mathbb R$ is a function such that $\sigma_k(n)=\sum_{d|n}d^k$. $\rho_k:\mathbb N \longrightarrow \mathbb R$ is a function such that $\rho_k\ast \sigma_k=\delta$. find a formula for $\rho_k$.($\frac{100}{6}$ points)
[b]a.)[/b] Let $a,b$ be real numbers. Define sequence $x_k$ and $y_k$ such that
\[x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, \quad y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots \]
Prove that
\[x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}\]
where $\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}$
[b]b.)[/b] Let $u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l} $. For positive integer $m,$ denote the remainder of $u_k$ divided by $2^m$ as $z_{m,k}$. Prove that $z_{m,k},$ $k = 0,1,2, \ldots$ is a periodic function, and find the smallest period.
A real continuous function has the property that its evaluation at any point is nilpotent under composition with itself.
Prove that this function is $ 0. $
[i]Vasile Pop[/i]
Find all functions $ f:[0,1]\longrightarrow [0,1] $ that are bijective, continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow\mathbb{R} , $ the following equality holds.
$$ \int_0^1 g\left( f(x) \right) dx =\int_0^1 g(x) dx $$
Determine all functions $f : \mathbb{R} - [0,1] \to \mathbb{R}$ such that \[ f(x) + f \left( \dfrac{1}{1-x} \right) = \dfrac{2(1-2x)}{x(1-x)} . \]
Find all non decreasing function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ and $m,n \in \mathbb{N}_0$ such that $m+n \neq 0$ there exist $m',n' \in \mathbb{N}_0$ such that $m'+n'=m+n+1$ and $$f(f^m(x)+f^n(y))=f^{m'}(x)+f^{n'}(y)$$ . Note : $f^0(x)=x$ and $f^{n}(x)=f(f^{n-1}(x))$ for all $n \in \mathbb{N}$ . [hide=original]Find all non decreasing functions $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$
$$ f(x+f(y))=f(x)+f(y) \text{ or } f(f(x))+y$$ .[/hide]
Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.
Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \) and \( 0 < f'(t) \leq 1 \) for all \( t \in [0, 1] \). Show that:
\[
\left( \int_0^1 f(t) \, dt \right)^2 \geq \int_0^1 f(t)^3 \, dt.
\]
Consider the parametric function $ f_k:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=x+k\lfloor x \rfloor . $
[b]a)[/b] For which integer values of $ k $ the above function is injective?
[b]b)[/b] For which integer values of $ k $ the above function is surjective?
[b]c)[/b] Given two natural numbers $ n,m, $ create two bijective functions:
$$ \phi : f_m (\mathbb{R} )\cap [0,\infty )\longrightarrow f_n(\mathbb{R})\cap [0,\infty ) $$
$$ \psi : \left(\mathbb{R}\setminus f_m (\mathbb{R})\right)\cap [0,\infty )\longrightarrow\left(\mathbb{R}\setminus f_n (\mathbb{R})\right)\cap [0,\infty ) $$
[i]Cristinel Mortici[/i]
A function $f:\mathbb{N} \to \mathbb{N}$ is given. If $a,b$ are coprime, then $f(ab)=f(a)f(b)$. Also, if $m,k$ are primes (not necessarily different), then $$f(m+k-3)=f(m)+f(k)-f(3).$$ Find all possible values of $f(11)$.
If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].
Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that
\[f(x+y)\ge f(x)+yf(f(x)) \]
for every $x,y\in (0,\infty )$.
For nonzero real numbers $x$ and $y$, define $x\circ y = \tfrac{xy}{x+y}$. Compute \[2^1\circ \left(2^2\circ \left(2^3\circ\cdots\circ\left(2^{2016}\circ 2^{2017}\right)\right)\right).\]
Prove that there is no bijective function $f : \left\{1,2,3,\ldots \right\}\rightarrow \left\{0,1,2,3,\ldots \right\}$ such that $f(mn)=f(m)+f(n)+3f(m)f(n)$.
Find all functions $f : \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+}$ that satisfy $f(mf(n)) = n+f(2015m)$ for all $m,n \in \mathbb{Z}_{+}$.
Let $ A $ be a nonempty set of real numbers, and let be two functions $ f,g:A\longrightarrow A $ having the following properties:
$ \text{(i)} f $ is increasing
$ \text{(ii)} f-g $ is nonpositive everywhere
$ \text{(iii)} f(A)\subset g(A) $
[b]a)[/b] Prove that $ f=g $ if $ A $ is the set of all nonnegative integers.
[b]b)[/b] Is true that $ f=g $ if $ A $ is the set of all integers?
[i]Dorel Miheț[/i]
Given an integer $n\ge 2$, prove that
\[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\].
[hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]
Find all completely multipiclative functions $f:\mathbb{Z}\rightarrow \mathbb{Z}_{\geqslant 0}$ such that for any $a,b\in \mathbb{Z}$ and $b\neq 0$, there exist integers $q,r$ such that
$$a=bq+r$$
and
$$f(r)<f(b)$$
Proposed by Navid Safaei
Find all functions $f: \mathbb{Z}^{2}\to \mathbb{R}^{+}$ such that for all $i, j \in \mathbb{Z}$: \[f(i,j)=\frac{f(i+1, j)+f(i,j+1)+f(i-1,j)+f(i,j-1)}{4}.\]
Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows:
(1). $ \dfrac{1}{2} \in \mathbb{H}$,
(2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1\plus{}x} \in \mathbb{H}$ and also $ \dfrac{x}{1\plus{}x} \in \mathbb{H}$.
Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.
Let $f(u)$ be a continuous function and, for any real number $u$, let $[u]$ denote the greatest integer less than or equal to $u$. Show that for any $x>1$,
\[\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du\]
The set of all numbers x for which \[x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}}\] is a rational number is the set of all:
$\textbf{(A)}\ \text{ integers } x \qquad
\textbf{(B)}\ \text{ rational } x \qquad
\textbf{(C)}\ \text{ real } x\qquad
\textbf{(D)}\ x \text{ for which } \sqrt{x^2+1} \text{ is rational} \qquad
\textbf{(E)}\ x \text{ for which } x+\sqrt{x^2+1} \text{ is rational }$