$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$. Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value. \[\int_a^b (px+q-\ln x)dx\]

Evaluate \[\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx\]

Find the indefinite integral $ \int \frac{x^3}{(x\minus{}1)^3(x\minus{}2)}\ dx$.

Find all Riemann integrable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ which have the property that, for all nonconstant and continuous functions $ g:\mathbb{R}\longrightarrow\mathbb{R}, $ and all real numbers $ a,b $ such that $ a<b, $ the following equality holds. $$ \int_a^b \left( f\circ g \right) (x)dx=\int_a^b \left( g\circ f \right) (x)dx $$ [i]Cosmin Nițu[/i]

Consider the graph of the function $y = (1 -x^2)^3$. Find the set of points $M(x,y)$ through which you can draw at least $6$ lines touching this graph.

Evaluate $\int_0^{\frac{\pi}{2}} \sqrt{1-2\sin 2x+3\cos ^ 2 x}\ dx.$ [i]2011 University of Occupational and Environmental Health/Medicine　entrance exam[/i]

Do there exist $\{x,y\}\in\mathbb{Z}$ satisfying $(2x+1)^{3}+1=y^{4}$?

Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$ \[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\] ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

Which of these has the smallest maxima on positive real numbers? $\textbf{(A)}\ \frac{x^2}{1+x^{12}} \qquad\textbf{(B)}\ \frac{x^3}{1+x^{11}} \qquad\textbf{(C)}\ \frac{x^4}{1+x^{10}} \qquad\textbf{(D)}\ \frac{x^5}{1+x^{9}} \qquad\textbf{(E)}\ \frac{x^6}{1+x^{8}}$

Let $f:[0,1]\to(0,1)$ be a surjective function. [list=a] [*]Prove that $f$ has at least one point of discontinuity. [*]Given that $f$ admits a limit in any point of the interval $[0,1],$ show that is has at least two points of discontinuity. [/list][i]Mihai Piticari and Sorin Rădulescu[/i]

[color=darkred]Let $a,b\in\mathbb{R}$ with $0<a<b$ . Prove that: [list] [b]a)[/b] $2\sqrt {ab}\le\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\le a+b$ , for $x,y,z\in [a,b]\, .$ [b]b)[/b] $\left\{\frac {x+y+z}3+\frac {ab}{\sqrt[3]{xyz}}\, |\, x,y,z\in [a,b]\right\}=[2\sqrt {ab},a+b]\, .$ [/list][/color]

Let $x(t)$ be a solution to the differential equation \[\left(x+x^\prime\right)^2+x\cdot x^{\prime\prime}=\cos t\] with $x(0)=x^\prime(0)=\sqrt{\frac{2}{5}}$. Compute $x\left(\dfrac{\pi}{4}\right)$.

The coordinate of $ P$ at time $ t$, moving on a plane, is expressed by $ x = f(t) = \cos 2t + t\sin 2t,\ y = g(t) = \sin 2t - t\cos 2t$. (1) Find the acceleration vector $ \overrightarrow{\alpha}$ of $ P$ at time $ t$ . (2) Let $ L$ denote the line passing through the point $ P$ for the time $ t%Error. "neqo" is a bad command. $, which is parallel to the acceleration vector $ \overrightarrow{\alpha}$ at the time. Prove that $ L$ always touches to the unit circle with center the origin, then find the point of tangency $ Q$. (3) Prove that $ f(t)$ decreases in the interval $ 0\leq t \leqq \frac {\pi}{2}$. (4) When $ t$ varies in the range $ \frac {\pi}{4}\leq t\leq \frac {\pi}{2}$, find the area $ S$ of the figure formed by moving the line segment $ PQ$.

In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$

Compute $e^A$ where $A$ is defined as \[\int_{3/4}^{4/3}\dfrac{2x^2+x+1}{x^3+x^2+x+1}dx.\]

Find the minimum value of $\int_0^{\pi} (a\sin x+b\sin 2x+c\sin 3x-x)^2\ dx$

Solve the following equation in positive integers: $x^{3} = 2y^{2} + 1 $

Find the area of the region bounded by $C: y=-x^4+8x^3-18x^2+11$ and the tangent line which touches $C$ at distinct two points.

The maximum value of \[ 2\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{44^n} \] over all real numbers $\theta$ can be expressed as a common fraction $\tfrac{p}{q}$. Compute $p + q$.

Compute: \[\lim_{x\to 0}\text{ }\dfrac{x^2}{1-\cos(x)}\]

Let $A,B$ be two fixed points on the curve $y=f(x)$, $f$ is continuous with continuous derivative and the arc $\widehat{AB}$ is concave to the chord $AB$. If $P$ is a point on the arc $\widehat{AB}$ for which $AP+PB$ is maximal, prove that $PA$ and $PB$ are equally inclined to the tangent to the curve $y=f(x)$ at $P$.

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.

Prove that the family of curves $$\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1$$ satisfies $$\frac{dy}{dx}(a^2-b^2)=\left(x+y\frac{dy}{dx}\right)\left(x\frac{dy}{dx}-y\right)$$

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$. (1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded? (2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$? Justify your answer.