Prove the following inequality. \[ \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})\]

Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and b) $ I$ contains of a polynomial with constant term $ 1$. Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$. [i]Gy. Szekeres[/i]

For $a>0$, denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$. Find the maximum area of $S(a)$.

Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{e^{\frac{1}{\cos ^ 2 x}}\sin x}{\cos ^ 3 x}\ dx$.

Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.

[i]The following problem is open in the sense that the answer to part (b) is not currently known. A proof of part (a) will be awarded 5 points. Up to 7 additional points may be awarded for progress on part (b).[/i] Let $p(x)$ be a polynomial of degree $d$ with coefficients belonging to the set of rational numbers $\mathbb{Q}$. Suppose that, for each $1 \le k \le d-1$, $p(x)$ and its $k$th derivative $p^{(k)}(x)$ have a common root in $\mathbb{Q}$; that is, there exists $r_k \in \mathbb{Q}$ such that $p(r_k) = p^{(k)}(r_k) = 0$. (a) Prove that if $d$ is prime then there exist constants $a, b, c \in \mathbb{Q}$ such that \[ p(x) = c(ax + b)^d. \] (b) For which integers $d \ge 2$ does the conclusion of part (a) hold?

Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$

Let $ p$ be a prime and let \[ l_k(x,y)\equal{}a_kx\plus{}b_ky \;(k\equal{}1,2,...,p^2)\ .\] be homogeneous linear polynomials with integral coefficients. Suppose that for every pair $ (\xi,\eta)$ of integers, not both divisible by $ p$, the values $ l_k(\xi,\eta), \;1\leq k\leq p^2 $, represent every residue class $ \textrm{mod} \;p$ exactly $ p$ times. Prove that the set of pairs $ \{(a_k,b_k): 1\leq k \leq p^2 \}$ is identical $ \textrm{mod} \;p$ with the set $ \{(m,n): 0\leq m,n \leq p\minus{}1 \}.$

Prove that for each positive integer $n$, there exist $n$ consecutive positive integers none of which is an integral power of a prime number.

This problem is easy but nobody solved it. point $A$ moves in a line with speed $v$ and $B$ moves also with speed $v'$ that at every time the direction of move of $B$ goes from $A$.We know $v \geq v'$.If we know the point of beginning of path of $A$, then $B$ must be where at first that $B$ can catch $A$.

Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.

Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.

In a triangle $ABC$ with $BC = CA + \frac 12 AB$, point $P$ is given on side $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2 \angle CPA.$

Let $a$ be a positive constant. Evaluate the following definite integrals $A,\ B$. \[A=\int_0^{\pi} e^{-ax}\sin ^ 2 x\ dx,\ B=\int_0^{\pi} e^{-ax}\cos ^ 2 x\ dx\]. [i]1998 Shinsyu University entrance exam/Textile Science[/i]

Find all twice differentiable functions $f(x)$ such that $f^{\prime \prime}(x)=0$, $f(0)=19$, and $f(1)=99$.

Let $f: \mathbb{R}\to\mathbb{R}$ is a function such that $f( \cot x ) = \cos 2x+\sin 2x$ for all $0 < x < \pi$. Define $g(x) = f(x) f(1-x)$ for $-1 \leq x \leq 1$. Find the maximum and minimum values of $g$ on the closed interval $[-1, 1].$

Find all differentiable functions $f, g:[0,\infty) \to \mathbb{R}$ and the real constant $k\geq 0$ such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and $f(0)=k, f'(0)=-k^2/3$ and also $f(x)\neq 0$ for all $x\geq 0$.\\ \\ [i] (Nora Gavrea)[/i]

If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.

A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$. Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

Evaluate \[\int_0^{\pi} xp^x\cos qx\ dx,\ \int_0^{\pi} xp^x\sin qx\ dx\ (p>0,\ p\neq 1,\ q\in{\mathbb{N^{+}}})\] Own

Evaluate $\int_{\frac 12}^1 \sqrt{1-x^2}\ dx.$

A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according to their kinetic energies after a given time $t$, from least to greatest. [asy] size(225); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); filldraw(circle((0,0),1),gray(.7)); draw((0,-1)--(2,-1),EndArrow); label("$\vec{F}$",(1, -1),S); label("Disk",(-1,0),W); filldraw(circle((5,0),1),gray(.7)); filldraw(circle((5,0),0.75),white); draw((5,-1)--(7,-1),EndArrow); label("$\vec{F}$",(6, -1),S); label("Hoop",(6,0),E); filldraw(circle((10,0),1),gray(.5)); draw((10,-1)--(12,-1),EndArrow); label("$\vec{F}$",(11, -1),S); label("Sphere",(11,0),E); [/asy] $ \textbf{(A)} \ \text{disk, hoop, sphere}$ $\textbf{(B)}\ \text{sphere, disk, hoop}$ $\textbf{(C)}\ \text{hoop, sphere, disk}$ $\textbf{(D)}\ \text{disk, sphere, hoop}$ $\textbf{(E)}\ \text{hoop, disk, sphere} $

Answer the following questions. (1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$. (2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$. (3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.

Let $f:[0,1]\to\mathbb R$ be a continuously differentiable function such that $f(x_0)=0$ for some $x_0\in[0,1]$. Prove that $$\int^1_0f(x)^2dx\le4\int^1_0f’(x)^2dx.$$