Found problems: 4776
Evaluate $\sum_{n=2017}^{2030}\sum_{k=1}^{n}\left\{\frac{\binom{n}{k}}{2017}\right\}$.
[i]Note: $\{x\}=x-\lfloor x\rfloor$ for every real numbers $x$.[/i]
Show that $\phi(n)+\sigma(n) \ge 2n$ for all positive integers $n$.
Let $f : [0,1] \rightarrow \mathbb{R}$ be injective function such that $f(0)+f(1)=1$. Prove that exists $x_1$, $x_2 \in [0,1]$, $x_1 \neq x_2$ such that $2f(x_1)<f(x_2)+\frac{1}{2}$. After that state at least one generalization of this result
Let $ f(x)$ be a real function such that
\[ \lim_{x \rightarrow \plus{}\infty} \frac{f(x)}{e^x}\equal{}1\]
and $ |f''(x)|\leq c|f'(x)|$ for all sufficiently large $ x$. Prove that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f'(x)}{e^x}\equal{}1.\]
[i]P. Erdos[/i]
Prove that the series $\sum_p c_p f(px)$, where the summation is over all primes, unconditionally converges in $L^2[0,1]$ for every $1$-periodic function $f$ whose restriction to $[0,1]$ is in $L^2[0,1]$ if and only if $\sum_p |c_p|<\infty$. ([i]Unconditional convergence[/i] means convergence for all rearrangements.) [G. Halasz]
Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$, determine $ f(9) .$
Let be a number $ x $ and three positive numbers $ a,b,c $ such that $ a^x+b^x=c^x. $
Prove that $ a^y,b^y,c^y $ are the lenghts of the sides of an obtuse triangle if and only if $ y<x<2y. $
Evaluate $ \int_{\frac {\pi}{4}}^{\frac {3}{4}\pi } \cos \frac {1}{\sin \left(\frac {1}{\sin x}\right)}\cdot \cos \left(\frac {1}{\sin x}\right)\cdot \frac {\cos x}{\sin ^ 2 x\cdot \sin ^ 2 \left(\frac {1}{\sin x }\right)}\ dx$
Last Edited, Sorry
kunny
Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\]
for all $x,y\in\mathbb{R}$.
Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties:
(i) $f(1) = 2$;
(ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.
For a sequence $ a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\}$ we have $ a_0 \equal{} 1$ and \[ a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n} \quad \forall n \in \mathbb{N}.\] Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.
For any positive integer $n$, let $f(n)$ denote the $n$- th positive nonsquare integer, i.e., $f(1) = 2, f(2) = 3, f(3) = 5, f(4) = 6$, etc. Prove that $f(n)=n +\{\sqrt{n}\}$ where $\{x\}$ denotes the integer closest to $x$.
(For example, $\{\sqrt{1}\} = 1, \{\sqrt{2}\} = 1, \{\sqrt{3}\} = 2, \{\sqrt{4}\} = 2$.)
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x+yf(x))=f(x)+xf(y) \quad \text{for all}\ x,y \in\mathbb{R}\]
Let $f:\mathbb N\to \mathbb N$. Show that $f(m)+n\mid f(n)+m$ for all positive integers $m\le n$ if and only if $f(m)+n\mid f(n)+m$ for all positive integers $m\ge n$.
[i]Proposed by Carl Schildkraut[/i]
Show that there is no function $f:{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$ such that $f(x+y)>f(x)(1+yf(x))$
for all $x,y\in {{\mathbb{R}}^{+}}$.
Find all real numbers $ x$ such that \[ \sqrt{x\plus{}1}\plus{}\sqrt{x\plus{}3} \equal{} \sqrt{2x\minus{}1}\plus{}\sqrt{2x\plus{}1}.\]
Let $f(m,1)=f(1,n)=1$ for $m\geq 1, n\geq 1$ and let $f(m,n)=f(m-1, n)+ f(m, n-1) +f(m-1 ,n-1)$ for $m>1$ and $n>1$. Also let
$$ S(n)= \sum_{a+b=n} f(a,b) \,\,\;\; a\geq 1 \,\, \text{and} \,\; b\geq 1.$$
Prove that
$$S(n+2) =S(n) +2S(n+1) \,\, \; \text{for} \, \, n \geq 2.$$
For positive integers $k,m$, where $1\le k\le 5$, define the function $f(m,k)$ as
\[f(m,k)=\sum_{i=1}^{5}\left[m\sqrt{\frac{k+1}{i+1}}\right]\]
where $[x]$ denotes the greatest integer not exceeding $x$. Prove that for any positive integer $n$, there exist positive integers $k,m$, where $1\le k\le 5$, such that $f(m,k)=n$.
Let $\mathbb{R}^+$ be the set of all positive real numbers. Show that there is no function $f:\mathbb{R}^+ \to \mathbb{R}^+$ satisfying
\[f(x+y)=f(x)+f(y)+\dfrac{1}{2012}\]
for all positive real numbers $x$ and $y$.
[i]Proposer: Fajar Yuliawan[/i]
Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous function such that
$$ \int_0^1 (x-1)f(x)dx =0. $$
Show that:
[b]a)[/b] There exists $ a\in (0,1) $ such that $ \int_0^a xf(x)dx =0. $
[b]b)[/b] There exists $ b\in (0,1) $ so that $ \int_0^b xf(x)dx=bf(b). $
Let $f:[0,1]\to [0,1]$ be a continuous function. We say $f\equiv 0$ if $f(x)=0$ for all $x\in [0,1]$ and similarly $f\not\equiv 0$ if there exists at least one $x\in [0,1]$ such that $f(x)\neq 0$. Suppose $f\not\equiv 0$, $f \circ f \not\equiv 0$ but $f \circ f \circ f \equiv 0$. Do there exists such an $f$? If yes construct such an function, if no prove it
Determine all functions $f : \mathbb{R}^+\rightarrow\mathbb{R}$ that satisfy the following
$f(1)=2008$, $|{f(x)}| \le x^2+1004^2$, $f\left (x+y+\frac{1}{x}+\frac{1}{y}\right )=f\left (x+\frac{1}{y}\right )+f\left (y+\frac{1}{x}\right ).$
$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.
Suppose a continuous function $f:[-\frac{\pi}{4},\frac{\pi}{4}]\to[-1,1]$ and differentiable on $(-\frac{\pi}{4},\frac{\pi}{4})$. Then, there exists a point $x_0\in (-\frac{\pi}{4},\frac{\pi}{4})$ such that
$$|f'(x_0)|\leq 1+f(x_0)^2$$
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$.
[i]Proposed by Evan Chen[/i]