Found problems: 4776
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals.
[i]A. Tereshin[/i]
Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\
ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]
How many $f:\mathbb{R} \rightarrow \mathbb{R}$ are there satisfying $f(x)f(y)f(z)=12f(xyz)-16xyz$ for every real $x,y,z$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \qquad \textbf{(E)}\ \text{None}$
For any two positive integers $n$ and $p$, prove that there are exactly ${{(p+1)}^{n+1}}-{{p}^{n+1}}$ functions
$f:\left\{ 1,2,...,n \right\}\to \left\{ -p,-p+1,-p+2,....,p-1,p \right\}$
such that $\left| f(i)-f(j) \right|\le p$ for all $i,j\in \left\{ 1,2,...,n \right\}$.
Let $ p\ge 2 $ be a fixed natural number, and let the sequence of functions $ \left( f_n\right)_{n\ge 2}:[0,1]\longrightarrow\mathbb{R} $ defined as $ f_n (x)=f_{n-1}\left( f_1 (x)\right) , $ where $ f_1 (x)=\sqrt[p]{1-x^p} . $ Find $ a\in (0,1) $ such that:
[b]a)[/b] exists $ b\ge a $ so that $ f_1:[a,b]\longrightarrow [a,b] $ is bijective.
[b]b)[/b] $ \forall x\in [0,1]\quad\exists y\in [0,1]\quad m\in\mathbb{N}\implies \left| f_m(x)-f_m(y)\right| >a|x-y| $
Find the functions $f(x),\ g(x)$ such that
$f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$
$g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$
Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation
$$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$
for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$.
[i]Proposed by Maximiliano Sánchez[/i]
[b]3.[/b] Is there a real-valued function $Af$, defined on the space of the functions, continuous on $[0,1]$, such that $f(x)\leq g(x) $ and$f(x)\not\equiv g(x) $ inply $Af< Ag$? Is this also true if the functions $f(x)$ are required to be monotonically increasing (rather than continuous) on $[0,1]$? [b](R.4)[/b]
Let $n$ be a positive integer and let $A_n$ respectively $B_n$ be the set of nonnegative integers $k<n$ such that the number of distinct prime factors of $\gcd(n,k)$ is even (respectively odd). Show that $|A_n|=|B_n|$ if $n$ is even and $|A_n|>|B_n|$ if $n$ is odd.
Example: $A_{10} = \left\{ 0,1,3,7,9 \right\}$, $B_{10} = \left\{ 2,4,5,6,8 \right\}$.
Find all real numbers $a$ such that there exists a function $f:\mathbb R\to \mathbb R$ such that the following conditions are simultaneously satisfied: (a) $f(f(x))=xf(x)-ax,\;\forall x\in\mathbb{R};$ (b) $f$ is not a constant function; (c) $f$ takes the value $a$.
A function $f(x)$ defined for $x\ge 0$ satisfies the following conditions:
i. for $x,y\ge 0$, $f(x)f(y)\le x^2f(y/2)+y^2f(x/2)$;
ii. there exists a constant $M$($M>0$), such that $|f(x)|\le M$ when $0\le x\le 1$.
Prove that $f(x)\le x^2$.
Let \[N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor).\] Find the remainder when N is divided by 1000. (Here $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to x, and $\lceil x \rceil$ denotes the least integer that is greater than or equal to x.)
Find all functions $f:\mathbb{Z} \rightarrow \mathbb{R}$ such that $f(1)=\tfrac{5}{2}$ and that \[f(x)f(y)=f(x+y)+f(x-y)\] for all integers $x$ and $y$.
Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity
\[f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \qquad \forall x,y \in \mathbb R^+\]
Where $\alpha,\beta$ are given real numbers.
Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds:
$a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$
$b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$
Prove that for every $n\in \mathbb N$, there exists a set $S$ of $n$ positive integers such that for any two distinct $a,b\in S$, $a-b$ divides $a$ and $b$ but none of the other elements of $S$.
[i]Proposed by Iurie Boreico[/i]
Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.
If $a@b=\dfrac{a^3-b^3}{a-b}$, for how many real values of $a$ does $a@1=0$?
Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$:
\[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]
For all real numbers $a$ and $b$, let \[a\Join b=\dfrac{a+b}{a-b}.\] Compute $1008\Join 1007$.
[i] Proposed by David Altizio [/i]
Does there exist a pair $ (f; g)$ of strictly monotonic functions, both from $ \mathbb{N}$ to $ \mathbb{N}$, such that \[ f(g(g(n))) < g(f(n))\] for every $ n \in\mathbb{N}$?
Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$.
Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$
Let the definite integral $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$.
(1) Find $ I_0,\ I_{\minus{}1},\ I_2$.
(2) Find $ I_1$.
(3) Express $ I_{n\plus{}2}$ in terms of $ I_n$.
(4) Find $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$.
(5) Evaluate the definite integrals $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$ in using the avobe results.
You are not allowed to use the formula of integral for $ \sqrt{x^2\plus{}1}$ directively here.
For a natural number $n$, let $T(n)$ denote the number of ways we can place $n$ objects of weights $1,2,\cdots, n$ on a balance such that the sum of the weights in each pan is the same. Prove that $T(100) > T(99)$.