Found problems: 4776
Compute \[\lim_{h\to 0}\dfrac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^4}.\]
There is a connected network with $ 2008$ computers, in which any of the two cycles don't have any common vertex. A hacker and a administrator are playing a game in this network. On the $ 1st$ move hacker selects one computer and hacks it, on the $ 2nd$ move administrator selects another computer and protects it. Then on every $ 2k\plus{}1th$ move hacker hacks one more computer(if he can) which wasn't protected by the administrator and is directly connected (with an edge) to a computer which was hacked by the hacker before and on every $ 2k\plus{}2th$ move administrator protects one more computer(if he can) which wasn't hacked by the hacker and is directly connected (with an edge) to a computer which was protected by the administrator before for every $ k>0$. If both of them can't make move, the game ends. Determine the maximum number of computers which the hacker can guarantee to hack at the end of the game.
Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1$ and
\[ f \left( x \plus{} \frac{13}{42} \right) \plus{} f(x) \equal{} f \left( x \plus{} \frac{1}{6} \right) \plus{} f \left( x \plus{} \frac{1}{7} \right).\]
Prove that $ f$ is a periodic function (that is, there exists a non-zero real number $ c$ such $ f(x\plus{}c) \equal{} f(x)$ for all $ x \in \mathbb{R}$).
Let $f:[0,1]\to\mathbb{R}$ be a continuous function and let $\{a_n\}_n$, $\{b_n\}_n$ be sequences of reals such that
\[ \lim_{n\to\infty} \int^1_0 | f(x) - a_nx - b_n | dx = 0 . \]
Prove that:
a) The sequences $\{a_n\}_n$, $\{b_n\}_n$ are convergent;
b) The function $f$ is linear.
Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.
Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property:
$$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.
Consider a function real-valued function with $C^{\infty}$-class on $\mathbb{R}$ such that:
(a) $f(0)=\frac{df}{dx}(0)=0,\ \frac{d^2f}{dx^2}(0)\neq 0.$
(b) For $x\neq 0,\ f(x)>0.$
Judge whether the following integrals $(i),\ (ii)$ converge or diverge, justify your answer.
$(i)$ \[\int\int_{|x_1|^2+|x_2|^2\leq 1} \frac{dx_1dx_2}{f(x_1)+f(x_2)}.\]
$(ii)$ \[\int\int_{|x_1|^2+|x_2|^2+|x_3|^2\leq 1} \frac{dx_1dx_2dx_3}{f(x_1)+f(x_2)+f(x_3)}.\]
[i]2010 Kyoto University, Master Course in Mathematics[/i]
Find the area of the domain expressed by the following system inequalities.
\[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]
In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$.
[asy]
Label f;
xaxis(0,60,blue);
yaxis(0,60,blue);
real f(real x)
{
return (x^2)/60;
}
draw(graph(f,0,53),red);
label("$y=x^2$",(30,15),E);
real f(real x)
{
return (x^2)/25;
}
draw(graph(f,0,38),red);
label("$y=2x^2$",(37,37^2/25),E);
real f(real x)
{
return (x^2)/10;
}
draw(graph(f,0,25),red);
label("$y=f(x)$",(24,576/10),W);
label("$O(0,0)$",(0,0),S);
dot((20,400/25));
dot((20,400/60));
label("$P$",(20,400/25),E);
label("$B$",(20,400/60),SE);
dot(((4000/25)^(0.5),400/25));
label("$A$",((4000/25)^(0.5),400/25),W);
draw((20,400/25)..((4000/25)^(0.5),400/25));
draw((20,400/25)..(20,400/60));
[/asy]
Let $f : \mathbb R \to \mathbb R$ be a continuous function. Suppose that the restriction of $f$ to the set of irrational numbers is injective. What can we say about $f$? Answer the analogous question if $f$ is restricted to rationals.
If $f(x)=\cos(x)-1+\frac{x^2}2$, then
$\textbf{(A)}~f(x)\text{ is an increasing function on the real line}$
$\textbf{(B)}~f(x)\text{ is a decreasing function on the real line}$
$\textbf{(C)}~f(x)\text{ is increasing on }-\infty<x\le0\text{ and decreasing on }0\le x<\infty$
$\textbf{(D)}~f(x)\text{ is decreasing on }-\infty<x\le0\text{ and increasing on }0\le x<\infty$
Let $P$ be a nonempty collection of subsets of $\{1,\dots,n\}$ such that:
(i) if $S,S'\in P,$ then $S\cup S'\in P$ and $S\cap S'\in P,$ and
(ii) if $S\in P$ and $S\ne\emptyset,$ then there is a subset $T\subset S$ such that $T\in P$ and $T$ contains exactly one fewer element than $S.$
Suppose that $f:P\to\mathbb{R}$ is a function such that $f(\emptyset)=0$ and \[f(S\cup S')= f(S)+f(S')-f(S\cap S')\text{ for all }S,S'\in P.\] Must there exist real numbers $f_1,\dots,f_n$ such that \[f(S)=\sum_{i\in S}f_i\] for every $S\in P?$
Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that
\[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0.
\]
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.
For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \]
Find all natural numbers $ n $ such that $ s(n) = 2010 $
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$.
Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.
$f$ be a twice differentiable real valued function satisfying
\[ f(x)+f^{\prime\prime}(x)=-xg(x)f^{\prime}(x) \]
where $g(x)\ge 0$ for all real $x$. Show that $|f(x)|$ is bounded.
$ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$ satisfy the given conditions
$ a)$ $ f(0,0)\equal{}1$ , $ f(0,1)\equal{}1$ ,
$ b)$ $ \forall k \notin \left\{0,1\right\}$ $ f(0,k)\equal{}0$ and
$ c)$ $ \forall n \geq 1$ and $ k$ , $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$
find the sum $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$
Let $ f(x) \equal{} (1 \plus{} x)\cdot\sqrt{(2 \plus{} x^2)}\cdot\sqrt[3]{(3 \plus{} x^3)}$. Determine $ f'(1)$.
Let $E$ be a point on side $[AD]$ of rhombus $ABCD$. Lines $AB$ and $CE$ meet at $F$, lines $BE$ and $DF$ meet at $G$. If $m(\widehat{DAB}) = 60^\circ $, what is$m(\widehat{DGB})$?
$
\textbf{a)}\ 45^\circ
\qquad\textbf{b)}\ 50^\circ
\qquad\textbf{c)}\ 60^\circ
\qquad\textbf{d)}\ 65^\circ
\qquad\textbf{e)}\ 75^\circ
$
Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$.
\[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\]
Let $w\in \mathbb{C}\setminus \mathbb{R}$, $|w|\neq 1$. Prove that $f\colon \mathbb{C} \to \mathbb{C}$, given by $f(z)= z+w\overline{z}$, is a bijection, and find its inverse.
For a given integer $n \ge 2$, find the maximum possible value of $V_n = \sin x_1 \cos x_2 +\sin x_2 \cos x_3 +...+\sin x_n \cos x_1$, where $x_1,x_2,...,x_n$ are real numbers.
On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1$, $y_2$, $\ldots$, $y_{24}$ on the number line such that for all $k$ such that $1\leq k\leq 12$, we have that $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\textstyle\sum_{n=1}^{24}(|y_n-1|+|y_n+1|).\]