Found problems: 348
For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows.
Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx.
(1) Find $f(\sqrt{3})$
(2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$
(3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.
Find the minimum value of $a\ (0<a<1)$ for which the following definite integral is minimized.
\[ \int_0^{\pi} |\sin x-ax|\ dx \]
Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] [i]Zarathustra (Zeb) Brady.[/i]
Given the polynomial $$P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}.$$
Express the numerical value of its derivative of order $325$ for $x = 0$.
Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$.
Show that
\[ \frac{(b+c)(a^4-b^2c^2)}{ab+2bc+ca}+\frac{(c+a)(b^4-c^2a^2)}{bc+2ca+ab}+\frac{(a+b)(c^4-a^2b^2)}{ca+2ab+bc} \geq 0 \]
for all positive real numbers $a, \: b , \: c.$
It is known that for $0\le x \le 1$ the square trinomial $f (x)$ satisfies the condition $|f(x) | \le 1$. Show that $| f '(0) | \le 8.$
Prove for $ 0 < k \leq 1$ and $ a_i \in \mathbb{R}^\plus{},$ $ i \equal{} 1,2 \ldots, n$ the following inequality holds:
\[ \left( \frac{a_1}{a_2 \plus{} \ldots \plus{} a_n} \right)^k \plus{} \ldots \plus{} \left( \frac{a_n}{a_1 \plus{} \ldots \plus{} a_{n\minus{}1}} \right)^k \geq \frac{n}{(n\minus{}1)^k}.\]
[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]
Find the value of $ a$ for which the area of the figure surrounded by $ y \equal{} e^{ \minus{} x}$ and $ y \equal{} ax \plus{} 3\ (a < 0)$ is minimized.
Find the value of $x+y$ for which the expression
\[\frac{6x^2}{y^6} + \frac{6y^2}{x^6}+9x^2y^2+\frac{4}{x^6y^6}\]
is minimized.
For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$
Consider an arc of a planar curve such that the total curvature of the arc is less than $ \pi$. Suppose, further, that the curvature and its derivative with respect to the arc length exist at every point of the arc and the latter nowhere equals zero. Let the osculating circles belonging to the endpoints of the arc and one of these points be given. Determine the possible positions of the other endpoint.
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$
Find all pairs of twice differentiable functions $f,g \colon \mathbb{R} \to \mathbb{R}$, with their second derivative being continuous, such that the following holds for all $x,y \in \mathbb{R}$: \[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]
Sum the series \[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]
Do either (1) or (2):
(1) Show that any solution $f(t)$ of the functional equation
$$f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1$$
for $x,y\in \mathbb{R}$ satisfies
$$f''(t)= \pm c^{2} f(t)$$
for a constant $c$, assuming the existence and continuity of the second derivative.
Deduce that $f(t)$ is one of the functions
$$ \pm \cos ct, \;\;\; \pm \cosh ct.$$
(2) Let $(a_{i})_{i=1,...,n}$ and $(b_{i})_{i=1,...,n}$ be real numbers. Define an $(n+1)\times (n+1)$-matrix $A=(c_{ij})$ by
$$ c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\;
c_{in+1}=b_{i-1}\; \text{for}\; i>1.$$
The polynomial $p(x)$ is defined by the equation $\det A=0$. Let $f$ be a polynomial and replace $(b_{i})$ with $(f(b_{i}))$. Then $\det A=0$ defines another polynomial $q(x)$. Prove that $f(p(x))-q(x)$ is a multiple of
$$\prod_{i=1}^{n} (x-a_{i}).$$
Find the maximal positive integer $n$, so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$.
A sequence is called an [i]arithmetic progression of the first order[/i] if the differences of the successive terms are constant. It is called an [i]arithmetic progression of the second order[/i] if the differences of the successive terms form an arithmetic progression of the first order. In general, for $k\geq 2$, a sequence is called an [i]arithmetic progression of the $k$-th order[/i] if the differences of the successive terms form an arithmetic progression of the $(k-1)$-th order.
The numbers
\[4,6,13,27,50,84\]
are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the $n$-th term of this progression.
The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:
[list]
[*] $D(1) = 0$;
[*] $D(p)=1$ for all primes $p$;
[*] $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$.
[/list]
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$.
Points $A$, $B$, and $C$ lie in that order on line $\ell$ such that $AB=3$ and $BC=2$. Point $H$ is such that $CH$ is perpendicular to $\ell$. Determine the length $CH$ such that $\angle AHB$ is as large as possible.
Calculate the $100$th derivative of the function
\[\frac{1}{x^2+3x+2}\]
at $x=0$ with $10\%$ relative error.
Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let
\[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \]
Prove that
\[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \]
where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$
Find the twice-differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that have the property that
$$ f'(x)+F(x)=2f(x)+x^2/2, $$
for any real numbers $ x; $ where $ F $ is a primitive of $ f. $
[i]Carmen Botea[/i]