Found problems: 4776
Let consider the following function set $$F=\{f\ |\ f:\{1,\ 2,\ \cdots,\ n\}\to \{1,\ 2,\ \cdots,\ n\} \}$$
[list=1]
[*] Find $|F|$
[*] For $n=2k$ prove that $|F|< e{(4k)}^{k}$
[*] Find $n$, if $|F|=540$ and $n=2k$
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Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds :
$$f(ac)+f(bc)-f(c)f(ab) \ge 1$$
Proposed by [i]Mojtaba Zare[/i]
Let us call an integer sequence $\{ a_1,a_2, \dots \}$ nice if there exist a function $f: \mathbb{Z^+} \to \mathbb{Z^+} $ such that
$$a_i \equiv a_j \pmod{n} \iff i\equiv j \pmod{f(n)}$$
for all $i,j,n \in \mathbb{Z^+}$. Find all nice sequences.
Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.
Prove that, for any two bounded functions $g_1,g_2 : \mathbb{R}\to[1,\infty),$ there exist functions $h_1,h_2 : \mathbb{R}\to\mathbb{R}$ such that for every $x\in\mathbb{R},$\[\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).\]
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have
\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]
Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.
An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key.
[b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key.
[b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.
Evaluate $ \int_{\minus{}\frac{1}{2}}^{\frac{1}{2}} \frac{x}{\{(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\plus{}(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.
Suppose that $f:[0,1]\to\mathbb R$ is a continuously differentiable function such that $f(0)=f(1)=0$ and $f(a)=\sqrt3$ for some $a\in(0,1)$. Prove that there exist two tangents to the graph of $f$ that form an equilateral triangle with an appropriate segment of the $x$-axis.
Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$. Compute $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$
Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$.
[i]Proposed by Mahyar Sefidgaran, Iran[/i]
Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1} (x))$. The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$,
where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$.
(We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.)
[i](United Kingdom) Luke Betts[/i]
Find the number of root for $\int_0^{\frac{\pi}{2}} e^x\cos (x+a)\ dx=0$ at $0\leq a <2\pi$
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$
Investigate the boundary of the domain of stability ($\max \text{Re }\lambda_j < 0$) in the space of coefficients of the equation $\dddot{x} + a\ddot{x} + b\dot{x} + cx = 0$.
Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations:
$x^3-4x^2-16x+60=y$;
$y^3-4y^2-16y+60=z$;
$z^3-4z^2-16z+60=x$.
Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.
Let be a function satisfying [url=http://mathworld.wolfram.com/CauchyFunctionalEquation.html]Cauchy's functional equation,[/url] and having the property that it's monotonic on a real interval. Prove that this function is globally monotonic.
[i]Florian Dumitrel[/i]
Function $f: R\to R$ is said periodic , if $f$ is not a constant function and there is a number real positive $p$ with the property of $f (x) = f (x + p)$ for every $x \in R$. The smallest positive real number p which satisfies the condition $f (x) = f (x + p)$ for each $x \in R$ is named period of $f$. Given $a$ and $b$ real positive numbers, show that there are periodic functions $f_1$ and $f_2$, with periods $a$ and $b$ respectively, so that $f_1 (x)\cdot f_2 (x)$ is also a periodic function.
Let $n\geq 2$ be a given integer. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
\[f(x-f(y))=f(x+y^n)+f(f(y)+y^n), \qquad \forall x,y \in \mathbb R.\]
show that thee is no function f definedonthe positive real numbes such that :
$f(y) > (y-x)f(x)^2$
Consider a function $f: \mathbb{R} \to \mathbb{R}$ satisfying for all $x \in \mathbb{R}$:
\[
f(x+1) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}.
\]
Prove that there exists a $b > 0$ such that $f(x + b) = f(x)$ for all $x \in \mathbb{R}$.
Let f be a function that satisfies the following conditions:
$(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$.
$(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions;
$(iii)$ $f(0) = 1$.
$(iv)$ $f(1987) \leq 1988$.
$(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$.
Find $f(1987)$.
[i]Proposed by Australia.[/i]