Found problems: 4776
Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions
\begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\
g'&=fg^2h+\frac4{fh},\ g(0)=1,\\
h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*}
Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$
There is a unique differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ satisfying $f(x) + (f(x))^3 = x + x^7$ for all real $x.$ The derivative of $f(x)$ at $x = 2$ can be expressed as a common fraction $a/b.$ Compute $a + b.$
Let $f, g:\mathbb{Z}\rightarrow [0,\infty )$ be two functions such that $f(n)=g(n)=0$ with the exception of finitely many integers $n$. Define $h:\mathbb{Z}\rightarrow [0,\infty )$ by \[h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}.\] Let $p$ and $q$ be two positive reals such that $1/p+1/q=1$. Prove that \[ \sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.\]
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number?
$\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$
Let $ x_1, x_2, \cdots, x_{25} $ real numbers such that $ 0 \le x_i \le i (i=1, 2, \cdots, 25) $. Find the maximum value of
\[x_{1}^{3}+x_{2}^{3}+\cdots +x_{25}^{3} - ( x_1x_2x_3 + x_2x_3x_4 + \cdots x_{25}x_1x_2 ) \]
For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that
\[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \]
is even for every positive integer $n$.
The function $f$ is not defined for $x=0$, but, for all non-zero real numbers $x$, $f(x)+2f\left( \frac1x \right)=3x$. The equation $f(x)=f(-x)$ is satisfied by
$\text{(A)} ~\text{exactly one real number}$
$\text{(B)}~\text{exactly two real numbers}$
$\text{(C)} ~\text{no real numbers}$
$\text{(D)} ~\text{infinitely many, but not all, non-zero real numbers}$
$\text{(E)} ~\text{all non-zero real numbers}$
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a differentiable function, with continuous derivative, and let
\[ s_{n}=\sum_{k=1}^{n}f\left( \frac{k}{n}\right) \]
Prove that the sequence $(s_{n+1}-s_{n})_{n\in{\mathbb{N}}^{\ast}}$ converges to $\int_{0}^{1}f(x)\mathrm{d}x$.
Find all pairs of functions $f; g : R \to R$ such that for all reals $x.y \ne 0$ :
$$f(x + y) = g \left(\frac{1}{x}+\frac{1}{y}\right) \cdot (xy)^{2008}$$
On a examination of $n$ questions a student answers correctly $15$ of the first $20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be?
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ \text{the problem cannot be solved} $
Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate
\[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by Sammy Luo[/i]
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$,
$(1+f(x)f(y))f(x+y)=f(x)+f(y)$.
$AD$ 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 $\angle{A}$.
For a given natural number, $ n\ge 2, $ consider two matrices $ A,B\in\mathcal{M}_n(\mathbb{C}) $ that commute and such that $ A $ is invertible and that the function $ M:\mathbb{C}\longrightarrow\mathbb{C} ,M(x)=\det (A+xB) $ is bounded above or below.
Prove that $ B^n=0. $
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible?
$ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$
Let $ \frac{1}{2}\le a_1, a_2, \ldots, a_n \le 5$ be given real numbers and let $ x_1, x_2, \ldots, x_n$ be real numbers satisfying $ 4x_i^2\minus{} 4a_ix_i \plus{} \left(a_i \minus{} 1\right)^2 \le 0$. Prove that \[ \sqrt{\sum_{i\equal{}1}^n\frac{x_i^2}{n}}\le\sum_{i\equal{}1}^n\frac{x_i}{n}\plus{}1\]
Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$.
For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$:
$ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.
Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
$\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$.
Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that
\[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]
The expression $\dfrac{\dfrac{a}{a+y}+\dfrac{y}{a-y}}{\dfrac{y}{a+y}-\dfrac{a}{a-y}}$, a real, $a\neq 0$, has the value $-1$ for:
$\textbf{(A)}\ \text{all but two real values of }y \qquad
\textbf{(B)}\ \text{only two real values of }y \qquad$
$\textbf{(C)}\ \text{all real values of }y \qquad
\textbf{(D)}\ \text{only one real value of }y \qquad
\textbf{(E)}\ \text{no real values of }y$
Let $\phi = \frac{1+\sqrt5}{2}$. Prove that a positive integer appears in the list $$\lfloor \phi \rfloor , \lfloor 2 \phi \rfloor, \lfloor 3\phi \rfloor ,... , \lfloor n\phi \rfloor , ... $$ if and only if it appears exactly twice in the list
$$\lfloor 1/ \phi \rfloor , \lfloor 2/ \phi \rfloor, \lfloor 3/\phi \rfloor , ... ,\lfloor n/\phi \rfloor , ... $$
Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ satisfying the condition $f(n)-f(n+f(m))=m$ for all $m,n\in \mathbb{Z}$
For real numbers $ a$ and $ b$, define $ a\$b\equal{}(a\minus{}b)^2$. What is $ (x\minus{}y)^2\$(y\minus{}x)^2$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ x^2\plus{}y^2 \qquad
\textbf{(C)}\ 2x^2 \qquad
\textbf{(D)}\ 2y^2 \qquad
\textbf{(E)}\ 4xy$