Found problems: 4776
Find all functions $f: \mathbb N \to \mathbb N$ such that for all positive integers $m,n$,
[b](i)[/b] $mf(f(m))=\left( f(m) \right)^2$,
[b](ii)[/b] If $\gcd(m,n)=d$, then $f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)$,
[b](iii)[/b] $f(m)=m$ if and only if $m=1$.
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
Suppose that the function $ g : (0,1) \rightarrow \mathbb{R}$ can be uniformly approximated by polynomials with nonnegative coefficients. Prove that $ g$ must be analytic. Is the statement also true for the interval $ (\minus{}1,0)$ instead of $ (0,1)$?
[i]J. Kalina, L. Lempert[/i]
Prove the following inequality.
\[\int_{-1}^1 \frac{e^x+e^{-x}}{e^{e^{e^x}}}dx<e-\frac{1}{e}\]
Own
Determine all strictly increasing functions $ f: R\rightarrow R$ satisfying relationship $ f(x\plus{}f(y))\equal{}f(x\plus{}y)\plus{}2005$
for any real values of x and y.
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+f(f(n))+f(n)=3n.\]
Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0,$ with $g$ nonzero and continuous at $0.$ If $fg$ and $f/g$ are differentiable at $0,$ must $f$ be differentiable at $0?$
In $\triangle ABC$, shown on the right, let $r$ denote the radius of the inscribed circle, and let $r_A$, $r_B$, and $r_C$ denote the radii of the smaller circles tangent to the inscribed circle and to the sides emanating from $A$, $B$, and $C$, respectively. Prove that
$r \leq r_A + r_B + r_C$
Members of "Professionous Riddlous" society have been divided into some groups, and groups are changed in a special way each weekend: In each group, one of the members is specified as the best member, and the best members of all groups separate from their previous group and form a new group. If a group has only one member, that member joins the new group and the previous group will be removed. Suppose that the society has $n$ members at first, and all the members are in one group. Prove that a week will come, after which number of members of each group will be at most $1+\sqrt{2n}$.
Find all positive integer $k$s for which such $f$ exists and unique:
$f(mn)=f(n)f(m)$ for $n, m \in \mathbb{Z^+}$
$f^{n^k}(n)=n$ for all $n \in \mathbb{Z^+}$ for which $f^x (n)$ means the n times operation of function $f$(i.e. $f(f(...f(n))...)$)
Define the function $m$ of the three real variables $x$, $y$, $z$ by $m$($x$,$y$,$z$) = max($x^2$,$y^2$,$z^2$), $x$, $y$, $z$ ∈ $R$.
Determine, with proof, the minimum value of $m$ if $x$,$y$,$z$ vary in $R$ subject to the following restrictions:
$x$ + $y$ + $z$ = 0,
$x^2$ + $y^2$ + $z^2$ = 1.
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $a,b\in \mathbb{R}$, with $a<b$ such that $f(a)=f(b)$, there exist some $c\in (a,b)$ such that $f(a)=f(b)=f(c)$. Prove that $f$ is monotonic over $\mathbb{R}$.
The function $F$ is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continuity is not assumed). Prove that $F$ maps squares into squares.
Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying
\[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\]
for all $x, y\in \mathbb{R}^{+}$.
$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$.
[i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.
Let $n$ be an integer. For pair of integers $0 \leq i,$ $j\leq n$ there exist real number $f(i,j)$ such that:
1) $ f(i,i)=0$ for all integers $0\leq i \leq n$
2) $0\leq f(i,l) \leq 2\max \{ f(i,j), f(j,k), f(k,l) \}$ for all integers $i$, $j$, $k$, $l$ satisfying $0\leq i\leq j\leq k\leq l\leq n$.
Prove that $$f(0,n) \leq 2\sum_{k=1}^{n}f(k-1,k)$$
Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
[i]Proposed by James Rickards, Canada[/i]
For positive integers $ n,$ the numbers $ f(n)$ are defined inductively as follows: $ f(1) \equal{} 1,$ and for every positive integer $ n,$ $ f(n\plus{}1)$ is the greatest integer $ m$ such that there is an arithmetic progression of positive integers $ a_1 < a_2 < \ldots < a_m \equal{} n$ for which
\[ f(a_1) \equal{} f(a_2) \equal{} \ldots \equal{} f(a_m).\]
Prove that there are positive integers $ a$ and $ b$ such that $ f(an\plus{}b) \equal{} n\plus{}2$ for every positive integer $ n.$
For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$.
[list=1]
[*] Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?
[*] Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?[/list]
[i]Brian Hamrick.[/i]
Let $p$ be a prime number. A flea is staying at point $0$ of the real line. At each minute,
the flea has three possibilities: to stay at its position, or to move by $1$ to the left or to the right.
After $p-1$ minutes, it wants to be at $0$ again. Denote by $f(p)$ the number of its strategies to do this
(for example, $f(3) = 3$: it may either stay at $0$ for the entire time, or go to the left and then to the
right, or go to the right and then to the left). Find $f(p)$ modulo $p$.
Let $ f$ be a function such that for all integers $ x$ and $ y$ applies $ f(x\plus{}y) \equal{} f(x) \plus{} f(y) \plus{} 6xy \plus{} 1$ and $ f(x) \equal{} f(\minus{}x)$. Then $ f(3)$ equals
$ \text{(A)}\ 26 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 52 \qquad \text{(D)}\ 53 \qquad \text{(E)}\ 54$
$f(x)=|1-2x|,x\in[0,1]$. Then the number of solutions to $f(f(f(x)))=\frac{1}{2}x$ is________.
Evaluate the following definite integral.
\[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]
(a) Let $p>1$ a real number. Find a real constant $c_p$ for which the following statement holds:
If $f: [-1,1]\rightarrow\mathbb{R}$ is a continuously differentiable function with $f(1)>f(-1)$ and $|f'(y)|\le1 \forall y\in[-1,1]$, then $\exists x\in[-1,1]: f'(x)>0$ so that $\forall y\in[-1,1]: |f(y)-f(x)|\le c_p\sqrt[p]{f'(x)}|y-x|$.
(b) What if $p=1$?