Found problems: 4776
Let $n>1$ be an integer and let $f_1$, $f_2$, ..., $f_{n!}$ be the $n!$ permutations of $1$, $2$, ..., $n$. (Each $f_i$ is a bijective function from $\{1,2,...,n\}$ to itself.) For each permutation $f_i$, let us define $S(f_i)=\sum^n_{k=1} |f_i(k)-k|$. Find $\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)$.
Let $\phi(n)$ be the number of positive integers $k<n$ which are relatively prime to $n$. For how many distinct values of $n$ is $\phi(n)$ equal to $12$?
$\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$
$\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$
$\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }8$
$\textbf{(J) }9\hspace{14.2em}\textbf{(K) }10\hspace{13.5em}\textbf{(L) }11$
$\textbf{(M) }12\hspace{13.3em}\textbf{(N) }13$
Find the area of the domain of the system of inequality
\[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]
Let $ A\subset \mathbb C$ be a closed and countable set. Prove that if the analytic function $ f: \mathbb C\backslash A\longrightarrow \mathbb C$ is bounded, then $ f$ is constant.
Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all $x,y,z\in \mathbb{R^+}$:
$$\biggl\{\frac{f(x)}{f(y)}\biggl\}+\biggl\{\frac{f(y)}{f(z)}\biggl\}+ \biggl\{\frac{f(z)}{f(x)}\biggl\}= \biggl\{\frac{x}{y}\biggl\} +\biggl\{\frac{y}{z}\biggl\}+ \biggl\{\frac{z}{x}\biggl\}$$
Note: For any real number $x$, let $\{x\}$ denote the fractional part of $x$, defined as For example, $\{2,7\}=0,7$ .
Let $ f(x)$ be a function with the two properties:
[list=a]
[*] for any two real numbers $ x$ and $ y$, $ f(x \plus{} y) \equal{} x \plus{} f(y)$, and
[*] $ f(0) \equal{} 2$
[/list]
What is the value of $ f(1998)$?
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 1996\qquad
\textbf{(D)}\ 1998\qquad
\textbf{(E)}\ 2000$
For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always:
\[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]
Let $d,p,q$ be fixed positive integers, and $d$ is not a perfect square. $\mathbb{N}$ is the set of all positive integers, and $S=\{m+n\sqrt{d}|m,n \in \mathbb{N}\} \cup \{0\}$. Suppose the function $f: S \to S$ satisfies the following conditions for all $x,y \in S$:
(i) $f((xy)^p)=(f(x)f(y))^p$
(ii)$f((x+y)^q)=(f(x)+f(y))^q$
Find the function $f$.
(a) Build a function $f : \mathbb R \to \mathbb R_+$ with the property $\left( \mathcal P \right)$, i.e. all $x \in \mathbb Q$ are local, strict minimum points.
(b) Build a function $f : \mathbb Q \to \mathbb R_+$ such that every point is a local, strict minimum point and such that $f$ is unbounded on $I \cap \mathbb Q$, where $I$ is a non-degenerate interval.
(c) Let $f: \mathbb R \to \mathbb R_+$ be a function unbounded on every $I \cap \mathbb Q$, where $I$ is a non-degenerate interval. Prove that $f$ doesn't have the property $\left( \mathcal P \right)$.
Positive numbers $ a$, $ b$, $ c$ are such that $ \left(a \plus{} b\right)\left(b \plus{} c\right)\left(c \plus{} a\right) \equal{} 8$. Prove the inequality
\[ \frac {a \plus{} b \plus{} c}{3}\ge\sqrt [27]{\frac {a^3 \plus{} b^3 \plus{} c^3}{3}}
\]
Functions $ f$ and $ g$ are quadratic, $ g(x) \equal{} \minus{} f(100 \minus{} x)$, and the graph of $ g$ contains the vertex of the graph of $ f$. The four $ x$-intercepts on the two graphs have $ x$-coordinates $ x_1$, $ x_2$, $ x_3$, and $ x_4$, in increasing order, and $ x_3 \minus{} x_2 \equal{} 150$. The value of $ x_4 \minus{} x_1$ is $ m \plus{} n\sqrt p$, where $ m$, $ n$, and $ p$ are positive integers, and $ p$ is not divisible by the square of any prime. What is $ m \plus{} n \plus{} p$?
$ \textbf{(A)}\ 602\qquad \textbf{(B)}\ 652\qquad \textbf{(C)}\ 702\qquad \textbf{(D)}\ 752\qquad \textbf{(E)}\ 802$
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
Let $a\in\mathbb{R}-\{0\}$. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$.
[i]Dan Schwartz[/i]
For a fixed integer $k$, determine all polynomials $f(x)$ with integer coefficients such that $f(n)$ divides $(n!)^k$ for every positive integer $n$.
Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y \in \mathbb{Q}$: \[f(x+y)+f(x-y)=2(f(x)+f(y)).\]
Determine all functions $f : \mathbb R \to \mathbb R$ satisfying the following two conditions:
(a) $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$, and
(b) $\lim_{x\to \infty} f(x) = 0$.
Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying
\[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)}
\]
for all $ x \neq y$.
a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares.
b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.
For each positive integer $n$, let $f(n)$ be a positive integer. Show that if $f(n + 1) > f(f(n))$ for every positive integer n, then $f(x) = x$ for all positive integers $x$.
Let $S$ be the set of points inside a given equilateral triangle $ABC$ with side $1$ or on its boundary. For any $M \in S, a_M, b_M, c_M$ denote the distances from $M$ to $BC,CA,AB$, respectively. Define
\[f(M) = a_M^3 (b_M - c_M) + b_M^3(c_M - a_M) + c_M^3(a_M - b_M).\]
[b](a)[/b] Describe the set $\{M \in S | f(M) \geq 0\}$ geometrically.
[b](b)[/b] Find the minimum and maximum values of $f(M)$ as well as the points in which these are attained.
For every set $S$ with $n(\ge3)$ distinct integers, show that there exists a function $f:\{1,2,\dots,n\}\rightarrow S$ satisfying the following two conditions.
(i) $\{ f(1),f(2),\dots,f(n)\} = S$
(ii) $2f(j)\neq f(i)+f(k)$ for all $1\le i<j<k\le n$.
Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)\plus{}1)$ if and only if $ n\equal{}1.$
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 $x,y>0$.
Find all pairs of functions $ f, g: \mathbb R \to \mathbb R$ such that
[list]
(i) if $ x < y$, then $ f(x) < f(y)$;
(ii) $ f(xy) \equal{} g(y)f(x) \plus{} f(y)$ for all $ x, y \in \mathbb R$.
[/list]
Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$