Found problems: 4776
Prove that in any triangle $ABC$ the following inequality holds: \[ \frac{a^{2}}{b+c-a}+\frac{b^{2}}{a+c-b}+\frac{c^{2}}{a+b-c}\geq 3\sqrt{3}R. \]
[i]Laurentiu Panaitopol[/i]
Find the continuous function satisfying the following equation.
\[\int_0^x f(t)dt+\int_0^x tf(x-t)dt=e^{-x}-1.\]
[i]1978 Shibaura Institute of Technology entrance exam[/i]
A function $f$ has its domain equal to the set of integers $0$, $1$, $\ldots$, $11$, and $f(n)\geq 0$ for all such $n$, and $f$ satisfies
[list]
[*]$f(0)=0$
[*]$f(6)=1$
[*]If $x\geq 0$, $y\geq 0$, and $x+y\leq 11$, then $f(x+y)=\tfrac{f(x)+f(y)}{1-f(x)f(y)}$.[/list]
Find $f(2)^2+f(10)^2$.
Find all functions $f:\mathbb R\to \mathbb R$ such that $$xy(f(x+y)-f(x)-f(y))=2f(xy)$$ for all $x,y\in \mathbb R.$
[i]Proposed by USJL[/i]
For every natural $n$ let $\phi (n)$ be the number of coprime numbers $k \in \{1,2,...,n\}$. (Example: $\phi (12) = 4$, because among the numbers $1, 2, ..., 12$ there are only the$ 4$ numbers, $1, 5, 7$ and $11$ coprime to$12.$)
If $k$ is a natural number, then one defines $\phi^k (n)=\underbrace{\strut \phi (\phi ...(\phi (n)) ...)}_{(k \, times \phi)}$ (Example: $\phi^3 (n)=\phi (\phi (\phi (n))) $)
For every whole $n> 2$ let $c(n)$ be the smallest natural number $k$ with $\phi^k (n)= 2$.
Prove that $c (ab) = c (a) + c (b)$ for odd integers $a$ and $b$, both of which are greater than $2$, .
Find out the number of real solutions of $x^2e^{\sin x}=1$
[list=1]
[*] 0
[*] 1
[*] 2
[*] 3
[/list]
For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have
\[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \]
and determine when equality holds.
[i]Wenyu Cao.[/i]
Compute $e^A$ where $A$ is defined as \[\int_{3/4}^{4/3}\dfrac{2x^2+x+1}{x^3+x^2+x+1}dx.\]
Does there exist an entire function $F \colon \mathbb{C}\to \mathbb{C}$ such that $F$ is not zero everywhere, $|F(z)|\leq e^{|z|}$ for all $z\in \mathbb{C}$, $|F(iy)|\leq 1$ for all $y\in \mathbb{R}$, and $F$ has infinitely many real roots.
Let $\mathbb{N}$ be the set of positive integers. Let $f$ be a function defined on $\mathbb{N}$, which satisfies the inequality $f(n + 1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that for any $n$ we have $f(n) = n.$
Let $\phi = \tfrac{1+\sqrt 5}2$ be the positive root of $x^2=x+1$. Define a function $f:\mathbb N\to\mathbb N$ by
\begin{align*}
f(0) &= 1\\
f(2x) &= \lfloor\phi f(x)\rfloor\\
f(2x+1) &= f(2x) + f(x).
\end{align*}
Find the remainder when $f(2007)$ is divided by $2008$.
Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate
\[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by Sammy Luo[/i]
The increasing interval of $f(x)=\log_{\frac{1}{2}}(x^2-2x-3)$ is
$\text{(A)}(-\infty,-1)\qquad\text{(B)}(-\infty,1)\qquad\text{(C)}(1,+\infty)\qquad\text{(D)}(3,+\infty)$
Given a polynomial $P(x,y)$ with real coefficients, suppose that some real function $f:\mathbb R \to \mathbb R$ satisfies
$$P(x,y) = f(x+y)-f(x)-f(y)$$for all $x,y\in\mathbb R$. Show that some polynomial $q$ satisfies
$$P(x,y) = q(x+y)-q(x)-q(y)$$
Let $f(x,y)$ and $g(x,y)$ be real valued functions defined for every $x,y \in \{1,2,..,2000\}$. If there exist $X,Y \subset \{1,2,..,2000\}$ such that $s(X)=s(Y)=1000$ and $x\notin X$ and $y\notin Y$ implies that $f(x,y)=g(x,y)$ than, what is the maximum number of $(x,y)$ couples where $f(x,y)\neq g(x,y)$.
The graphs of $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.
Let $r,n$ be positive integers. For a set $A$, let ${A \choose r}$ denote the family of all $r$-element subsets of $A$. Prove that if $A$ is infinite and $f : {A \choose r} \to {1,2,...,n}$ is any function, then there exists an infinite subset $B$ of $A$ such that $f(X) = f(Y)$ for all $X,Y \in {B \choose r}$.
Let $f(k)$ be the number of integers $n$ satisfying the following conditions:
(i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed;
(ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$.
Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$.
[i]Proposed by Dirk Laurie, South Africa[/i]
Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a function such that
(i) For all $x,y \in \Bbb{R}$, \[ f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y) \]
(ii) For all $x \in [0,1)$, $f(0) \geq f(x)$,
(iii) $-f(-1) = f(1) = 1$.
Find all such functions $f$.
Let $ a$, $ b$, $ c$ be three real numbers. For each positive integer number $ n$, $ a^n \plus{} b^n \plus{} c^n$ is an integer number. Prove that there exist three integers $ p$, $ q$, $ r$ such that $ a$, $ b$, $ c$ are the roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} r \equal{} 0$.
Describe all functions $f: \{ 0,1\}^n \to \{ 0,1\}$ which satisfy the equation
\begin{align*}
& f(f(a_{11},a_{12},\dotsc ,a_{1n}),f(a_{21},a_{22},\dotsc ,a_{2n}),\dotsc ,f(a_{n1},a_{n2},\dotsc ,a_{nn}))\\
& = f(f(a_{11},a_{21},\dotsc ,a_{n1}),f(a_{12},a_{22},\dotsc ,a_{n2}),\dotsc ,f(a_{1n},a_{2n},\dotsc ,a_{nn}))\end{align*}
for arbitrary $a_{ij}\in \{ 0,1\}$ where $i,j\in \{1,2,\dotsc ,n\}.$
Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.
Let $ f$ be a real function defined on the positive half-axis for which $ f(xy)\equal{}xf(y)\plus{}yf(x)$ and $ f(x\plus{}1) \leq f(x)$ hold for every positive $ x$ and $ y$. Show that if $ f(1/2)\equal{}1/2$, then \[ f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x)\] for every $ x\in (0,1)$.
[i]Z. Daroczy, Gy. Maksa[/i]
Find all $f:\mathbb{R} \to \mathbb{R}$ such that:
(a) For every real number $a$ there exist real number $b$:$f(b)=a$
(b) If $x>y$ then $f(x)>f(y)$
(c) $f(f(x))=f(x)+12x.$
Let $a$ be rational and $b,c,d$ are real numbers, and let $f: \mathbb{R}\to [-1.1]$ be a function satisfying $f(x+a+b)-f(x+b)=c[x+2a+[x]-2[x+a]-[b]]+d$ for all $x$. Show that $f$ is periodic.