Found problems: 4776
Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that
\[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]
[color=darkred] Let $(R,+,\cdot)$ be a ring and let $f$ be a surjective endomorphism of $R$ such that $[x,f(x)]=0$ for any $x\in R$ , where $[a,b]=ab-ba$ , $a,b\in R$ . Prove that:
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[b]a)[/b] $[x,f(y)]=[f(x),y]$ and $x[x,y]=f(x)[x,y]$ , for any $x,y\in R\ ;$
[b]b)[/b] If $R$ is a division ring and $f$ is different from the identity function, then $R$ is commutative.
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Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$:
$$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$
[i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]
Find all functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that $|f(k)| \leq k$ for all positive integers $k$ and there is a prime number $p>2024$ which satisfies both of the following conditions:
$1)$ For all $a \in \mathbb{N}$ we have $af(a+p) = af(a)+pf(a),$
$2)$ For all $a \in \mathbb{N}$ we have $p|a^{\frac{p+1}{2}}-f(a).$
[i]Authored by Nikola Velov[/i]
For positive numbers $a, b, c$ define $A = \frac{(a + b + c)}{3}$, $G = \sqrt[3]{abc}$, $H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.$ Prove that
\[ \left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.\]
Given fi nitely many open half planes on the Euclidean plane. The boundary lines of these half planes divide the plane into convex domains. Find a polynomial $C(q)$ of degree two so that the following holds: for any $q\ge 1$ integer, if the half planes cover each point of the plane at least $q$ times, then the set of points covered exactly $q$ times is the union of at most $C(q)$ domains.
(translated by L. Erdős)
For constant $a$, find the differentiable function $f(x)$ satisfying $\int_0^x (e^{-x}-ae^{-t})f(t)dt=0$.
Suppose that $ \tau$ is a metrizable topology on a set $ X$ of cardinality less than or equal to continuum. Prove that there exists a separable and metrizable topology on $ X$ that is coarser that $ \tau$.
[i]L. Juhasz[/i]
Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.
The elliptic curve $y^2=x^3+1$ is tangent to a circle centered at $(4,0)$ at the point $(x_0,y_0)$. Determine the sum of all possible values of $x_0$.
[color=blue][size=150]PERU TST IMO - 2006[/size]
Saturday, may 20.[/color]
[b]Question 02[/b]
Find all pairs $(a,b)$ real positive numbers $a$ and $b$ such that:
$[a[bn]]= n-1,$
for all $n$ positive integer.
Note: [x] denotes the integer part of $x$.
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[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88510]Spanish version[/url]
$\text{\LaTeX}{}$ed by carlosbr
Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and
$$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$
Calculate the following indefinite integrals.
[1] $\int \frac{6x+1}{\sqrt{3x^2+x+4}}dx$
[2] $\int \frac{e^x}{e^x+e^{a-x}}dx$
[3] $\int \frac{(\sqrt{x}+1)^3}{\sqrt{x}}dx$
[4] $\int x\ln (x^2-1)dx$
[5] $\int \frac{2(x+2)}{x^2+4x+1}dx$
Let $f$ be a function satisfying $f(x+y+z)=f(x)+f(y)+f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1)=1$, $f(2)=2$. Then $\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals
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[*] 4
[*] 6
[*] 12
[*] 24
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[*] Prove that $$\lim \limits_{n \to \infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)=0$$
[*] Calculate $$\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)$$
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Show that every integer greater than $1$ can be written as a sum of two square-free integers.
Given a positive integer number $k$, define the function $f$ on the set of all positive integer numbers to itself by
\[f(n)=\begin{cases}1, &\text{if }n\le k+1\\ f(f(n-1))+f(n-f(n-1)), &\text{if }n>k+1\end{cases}\]
Show that the preimage of every positive integer number under $f$ is a finite non-empty set of consecutive positive integers.
Does there exist a piecewise linear, continuous, surjective mapping $f: [0,1]\to [0,1]$ such that $f(0)=f(1)=0$, and for all positive integer $n$,
$$2.0001^{(n-10)} <P_n(f)<2.9999^{(n+10)}$$holds, where $P_n(f)$ is the number of points $x$ such that $\underbrace{f(\dotsc f}_n(x)\dotsc )=x$?
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(n))+f(n)=2n+2001 \text{ or }2n+2002.\]
Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that
\[\sqrt[4]{\frac{\sqrt{3}}{a}+6\sqrt{3}b}+\sqrt[4]{\frac{\sqrt{3}}{b}+6\sqrt{3}c}+\sqrt[4]{\frac{\sqrt{3}}{c}+6\sqrt{3}a}\le\frac{1}{abc}\]
When does inequality hold?
Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals.
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
A certain function $f$ has the properties that $f(3x)=3f(x)$ for all positive real values of $x$, and that $f(x)=1-\mid x-2 \mid$ for $1\leq x \leq 3$. Find the smallest $x$ for which $f(x)=f(2001)$.
Let $f(x)$ be a continuous function satisfying $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$
Find the value of $k$ for which $\int_0^{\pi} f(x)dx$ is maximized.
Let $f:\mathbb{R}\rightarrow [0,\infty)$. Prove that $f(x+y)\ge (y+1)f(x),\ (\forall)x\in \mathbb{R}$ if and only if the function $g:\mathbb{R}\rightarrow [0,\infty),\ g(x)=e^{-x}f(x),\ (\forall)x\in \mathbb{R}$ is increasing.