Found problems: 4776
For each real number $k$, denote by $f(k)$ the larger of the two roots of the quadratic equation
$$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values.
For every real number $x$, let $[x]$ be the greatest integer less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always
$\textbf{(A) }6W\qquad\textbf{(B) }6[W]\qquad\textbf{(C) }6([W]-1)\qquad\textbf{(D) }6([W]+1)\qquad \textbf{(E) }-6[-W]$
Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \]
[i]Ion Rasa[/i]
Let $f, g$ be bijections on $\{1, 2, 3, \dots, 2016\}$. Determine the value of $$\sum_{i=1}^{2016}\sum_{j=1}^{2016}[f(i)-g(j)]^{2559}.$$
Let $f$ be a diff erentiable real-valued function defi ned on the positive real numbers. The tangent lines to the graph of $f$ always meet the $y$-axis 1 unit lower than where they meet the function. If $f(1)=0$, what is $f(2)$?
Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $
[b]a)[/b] Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n. $
[b]b)[/b] If $ n=6, $ find the number of functions $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $ that have the property that $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $ divides $ 36. $
Let $ A$ represent the set of all functions $ f : \mathbb{N} \rightarrow \mathbb{N}$ such that for all $ k \in \overline{1, 2007}$, $ f^{[k]} \neq \mathrm{Id}_{\mathbb{N}}$ and $ f^{[2008]} \equiv \mathrm{Id}_{\mathbb{N}}$.
a) Prove that $ A$ is non-empty.
b) Find, with proof, whether $ A$ is infinite.
c) Prove that all the elements of $ A$ are bijective functions.
(Denote by $ \mathbb{N}$ the set of the nonnegative integers, and by $ f^{[k]}$, the composition of $ f$ with itself $ k$ times.)
For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.$
Find all real-valued functions $f$ defined on the set of all non-zero real numbers such that:
(i) $f(1)=1$,
(ii) $f\left(\frac{1}{x+y}\right)=f\left(\frac{1}{x}\right)+f\left(\frac{1}{y}\right)$ for all non-zero $x,y,x+y$,
(iii) $(x+y)\cdot f(x+y)=xy\cdot f(x)\cdot f(y)$ for all non-zero $x,y,x+y$.
If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$?
$\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$
Let $ \mathbb{R}$ denote the set of all real numbers and $ \mathbb{R}^\plus{}$ the subset of all positive ones. Let $ \alpha$ and $ \beta$ be given elements in $ \mathbb{R},$ not necessarily distinct. Find all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}$ such that
\[ f(x)f(y) \equal{} y^{\alpha} f \left( \frac{x}{2} \right) \plus{} x^{\beta} f \left( \frac{y}{2} \right) \forall x,y \in \mathbb{R}^\plus{}.\]
Let $a, b, c$ be positive real numbers in the interval $[0, 1]$ with $a+b, b+c, c+a \ge 1$. Prove that
\[
1 \le (1-a)^2 + (1-b)^2 + (1-c)^2 +
\frac{2\sqrt{2} abc}{\sqrt{a^2+b^2+c^2}}.
\]
Let $a,b>1$ be two real numbers. Prove that $a>b$ if and only if there exists a function $f: (0,\infty)\to\mathbb{R}$ such that
i) the function $g:\mathbb{R}\to\mathbb{R}$, $g(x)=f(a^x)-x$ is increasing;
ii) the function $h:\mathbb{R}\to\mathbb{R}$, $h(x)=f(b^x)-x$ is decreasing.
Let $f$ be a function that satisfies :
\[ \displaystyle f(x)+2f\left(\frac{x+\frac{2001}2}{x-1}\right) = 4014-x. \]
Find $f(2004)$.
The real numbers $x_1, x_2, ... , x_n$ belong to the interval $(0,1)$ and satisfy $x_1 + x_2 + ... + x_n = m + r$, where $m$ is an integer and $r \in [0,1)$. Show that $x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2$.
Let $f(x)$ be a periodic function with periods $T$ and $1$($0<T<1$).Prove that:
(1)If $T$ is rational,then there exists a prime $p$ such that $\frac{1}{p}$ is also a period of $f$;
(2)If $T$ is irrational,then there exists a strictly decreasing infinite sequence {$a_n$},with $1>a_n>0$ for all positive integer $n$,such that all $a_n$ are periods of $f$.
Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$.
Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$.
Find a closed form for $a_n$.
[i]Proposed by Bobby Shen[/i]
For any real $ a$ define $ f_a : \mathbb{R} \rightarrow \mathbb{R}^2$ by the law $ f_a(t) \equal{} \left( \sin(t), \cos(at) \right)$.
a) Prove that $ f_{\pi}$ is not periodic.
b) Determine the values of the parameter $ a$ for which $ f_a$ is periodic.
[b]Remark[/b]. L. Euler proved in $ 1737$ that $ \pi$ is irrational.
$p(m)$ is the number of distinct prime divisors of a positive integer $m>1$ and $f(m)$ is the $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$ th smallest prime divisor of $m$. Find all positive integers $n$ satisfying the equation: $$f(n^2+2) + f(n^2+5) = 2n-4$$
Is there a function $f(x)$ defined for all $x \in R$ and for all $x, y \in R $ satisfying the inequality
$$|f(x + y) + \sin x + \sin y| < 2?$$
Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that
\[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]
Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.
If $ g(x)\equal{}1\minus{}x^2$ and $ f(g(x)) \equal{} \frac{1\minus{}x^2}{x^2}$ when $ x\neq0$, then $ f(1/2)$ equals
$ \textbf{(A)}\ 3/4 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ \sqrt2/2 \qquad
\textbf{(E)}\ \sqrt2$
Find all such functions $f :\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following holds :
\[ f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) \]
Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$