Find all $t\in \mathbb R$, such that $\int_{0}^{\frac{\pi}{2}}\mid \sin x+t\cos x\mid dx=1$ .

Determine continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $\left( {{a}^{2}}+ab+{{b}^{2}} \right)\int\limits_{a}^{b}{f\left( x \right)dx=3\int\limits_{a}^{b}{{{x}^{2}}f\left( x \right)dx,}}$ for every $a,b\in \mathbb{R}$ .

Calculate with $10\%$ relative error \[\int_{-\infty}^{\infty}\cos(100(x^4-x))dx\]

A right parallelepiped (i.e. a parallelepiped one of whose edges is perpendicular to a face) is given. Its vertices have integral coordinates, and no other points with integral coordinates lie on its faces or edges. Prove that the volume of this parallelepiped is a sum of three perfect squares. [i]Proposed by A. Golovanov[/i]

Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx$.

If $f$ is a continuous real function such that $ f(x-1) + f(x+1) \ge x + f(x) $ for all $x$, what is the minimum possible value of $ \displaystyle\int_{1}^{2005} f(x) \, \mathrm{d}x $?

Find the continuous function $f(x)$ such that : \[\int_0^x f(t)\left(\int_0^t f(t)dt\right)dt=f(x)+\frac 12\]

Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.) [i]Proposed by Robin Park[/i]

Compute $\displaystyle\int_0^1\dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}$.

Compute \[\int_0^\infty \dfrac{e^{-x}\sin(x)}{x}dx\]

In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.

$\omega$ is a complex number such that $\omega^{2013} = 1$ and $\omega^m \neq 1$ for $m=1,2,\ldots,2012$. Find the number of ordered pairs of integers $(a,b)$ with $1 \le a, b \le 2013$ such that \[ \frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3} \] is the root of some polynomial with integer coefficients and leading coefficient $1$. (Such complex numbers are called [i]algebraic integers[/i].) [i]Victor Wang[/i]

Let $f(x)$ be a function satisfying \[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\] Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.

For a function $ f(x) \equal{} 6x(1 \minus{} x)$, suppose that positive constant $ c$ and a linear function $ g(x) \equal{} ax \plus{} b\ (a,\ b: \text{constants}\,\ a > 0)$ satisfy the following 3 conditions: $ c^2\int_0^1 f(x)\ dx \equal{} 1,\ \int_0^1 f(x)\{g(x)\}^2\ dx \equal{} 1,\ \int_0^1 f(x)g(x)\ dx \equal{} 0$. Answer the following questions. (1) Find the constants $ a,\ b,\ c$. (2) For natural number $ n$, let $ I_n \equal{} \int_0^1 x^ne^x\ dx$. Express $ I_{n \plus{} 1}$ in terms of $ I_n$. Then evaluate $ I_1,\ I_2,\ I_3$. (3) Evaluate the definite integrals $ \int_0^1 e^xf(x)\ dx$ and $ \int_0^1 e^xf(x)g(x)\ dx$. (4) For real numbers $ s,\ t$, define $ J \equal{} \int_0^1 \{e^x \minus{} cs \minus{} tg(x)\}^2\ dx$. Find the constants $ A,\ B,\ C,\ D,\ E$ by setting $ J \equal{} As^2 \plus{} Bst \plus{} Ct^2 \plus{} Ds\plus{}Et \plus{} F$. (You don't need to find the constant $ F$). (5) Find the values of $ s,\ t$ for which $ J$ is minimal.

For a constant $a$, denote $C(a)$ the part $x\geq 1$ of the curve $y=\sqrt{x^2-1}+\frac{a}{x}$. (1) Find the maximum value $a_0$ of $a$ such that $C(a)$ is contained to lower part of $y=x$, or $y<x$. (2) For $0<\theta <\frac{\pi}{2}$, find the volume $V(\theta)$ of the solid $V$ obtained by revoloving the figure bounded by $C(a_0)$ and three lines $y=x,\ x=1,\ x=\frac{1}{\cos \theta}$ about the $x$-axis. (3) Find $\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)$. 1992 Tokyo University entrance exam/Science, 2nd exam

a) Compute $$\lim_{n\to\infty}n\int^1_0\left(\frac{1-x}{1+x}\right)^ndx.$$ b) Let $k\ge1$ be an integer. Compute $$\lim_{n\to\infty}n^{k+1}\int^1_0\left(\frac{1-x}{1+x}\right)^nx^kdx.$$

5. Let $a,b,c,d,e$ be positive rational numbers. Consider integral expressions $f(x)=ax^2+bx+c$ $g(x)=dx+e$ Put $\frac{f(n)}{g(n)}$ an integer for all positive integers $n$. Then, show that $f(x)$ is dividible by $g(x)$.

Given a real number $x,$ let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \ldots, n_{70}$ such that $k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$ and $k$ divides $n_{i}$ for all $i$ such that $1 \leq i \leq 70.$ Find the maximum value of $\frac{n_{i}}{k}$ for $1\leq i \leq 70.$

$$\int_0^\frac{\pi}{2}\frac{1}{4-3\cos^2(x)}\,dx$$ [i]Proposed by Connor Gordon[/i]

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

A projectile moves in a resisting medium. The resisting force is a function of the velocity and is directed along the velocity vector. The equation $x=f(t)$ (where $f(t)$ is not constant) gives the horizontal distance in terms of the time $t$. Show that the vertical distance $y$ is given by $$y=-gf(t) \int \frac{dt}{f'(t)} + g \int \frac{f(t)}{f'(t)} \, dt +Af(t)+B$$ where $A$ and $B$ are constants and $g$ is the acceleration due to gravity.

Suppose $a_1,\ldots,a_n$ are non-integral real numbers for $n\geq 2$ such that ${a_1}^k+\ldots+{a_n}^k$ is an integer for all integers $1\leq k\leq n$. Prove that none of $a_1,\ldots,a_n$ is rational.

Compute $\sum_{k=0}^{\infty}\int_{0}^{\frac{\pi}{3}}\sin^{2k} x \, dx$.

Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?