Let $x$ be a real number with $0<x<\pi $.Prove that, for all natural number $n$ ,\[sinx+\frac{sin3x}{3}+\frac{sin5x}{5}+\cdots+\frac{sin(2n-1)x}{2n-1}>0.\]

[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by \[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\] Determine the following two sets: [list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$ [b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]

$\omega$ 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]

\[\int_0^{\frac \pi 4} \cot(x)\sqrt{\sin(x)}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

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 $P(x)=Q(x)^2$.

Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

Let $f$, $g$, $h : [a, b] \to \mathbb{R}$, three integrable functions such that:$$\int \limits_a^b fgdx=\int \limits_a^bghdx=\int \limits_a^bhfdx=\int \limits_a^bg^2dx\int \limits_a^bh^2dx=1$$Then$$\int \limits_a^bg^2dx=\int \limits_a^bh^2dx=1$$

Find the minimum value of $\int_{-1}^1 \sqrt{|t-x|}\ dt$

Find the positive value of $ a$ such that the curve $ C_1: x \equal{} \sqrt {2y^2 \plus{} \frac {25}{2}}$ tangent to the parabola $ C_2: y \equal{} ax^2$, then find the equation of the tangent line of $ C_1$ at the point of tangency.

Let $S$ be the domain in the coordinate plane determined by two inequalities: \[y\geq \frac 12x^2,\ \ \frac{x^2}{4}+4y^2\leq \frac 18.\] Denote by $V_1$ the volume of the solid by a rotation of $S$ about the $x$-axis and by $V_2$, by a rotation of $S$ about the $y$-axis. (1) Find the values of $V_1,\ V_2$. (2) Compare the size of the value of $\frac{V_2}{V_1}$ and 1.

Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$

Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

Let $ A\subset \mathbb C$ be a closed and countable set. Prove that if the analytic function $ f: \mathbb C\backslash A\longrightarrow \mathbb C$ is bounded, then $ f$ is constant.

On the $x$-$y$ plane, 3 half-lines $y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)$ intersect with the circle with the center the origin $O$, radius $r\geq 1$ at $A,\ B,\ C$ respectively. Note that $\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}$. If the area of quadrilateral $OABC$ is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .$ [i]2011 Waseda University of Education entrance exam/Science[/i]

Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at the rate of $4\dfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant rate of $3\dfrac{1}{4}$ miles per hour for the first hour, at $3\dfrac{3}{4}$ miles per hour for the second hour, and so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, then $x$ is: $\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 2$

Prove that the Polynomial $ f(x) \equal{} x^{4} \plus{} 26x^{3} \plus{} 56x^{2} \plus{} 78x \plus{} 1989$ can't be expressed as a product $ f(x) \equal{} p(x)q(x)$ , where $ p(x)$ and $ q(x)$ are both polynomial with integral coefficients and with degree at least $ 1$.

Let $ T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx$. Evaluate $ e^T$.

For any $k\in\mathbb{Z}^+$ , prove that:- $2(\sqrt{k+1}-\sqrt{k})<\frac{1}{\sqrt{k}}<2(\sqrt{k}-\sqrt{k-1})$ Also compute integral part of $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{10000}}$.

The number of intengral points $(x,y)$ that fit $(|x|-1)^2+(|y|-1)^2<2$ is $\text{(A)}16\qquad\text{(B)}17\qquad\text{(C)}18\qquad\text{(D)}25$

In the $xy$-plane, for $a>1$ denote by $S(a)$ the area of the figure bounded by the curve $y=(a-x)\ln x$ and the $x$-axis. Find the value of integer $n$ for which $\lim_{a\rightarrow \infty} \frac{S(a)}{a^n\ln a}$ is non-zero real number.

If $ p$ and $ q$ are primes and $ x^2 \minus{} px \plus{} q \equal{} 0$ has distinct positive integral roots, then which of the following statements are true? $ \text{I. The difference of the roots is odd.}$ $ \text{II. At least one root is prime.}$ $ \text{III. } p^2 \minus{} q \text{ is prime.}$ $ \text{IV. } p \plus{} q \text{ is prime.}$ $ \textbf{(A)}\ \text{I only} \qquad \textbf{(B)}\ \text{II only} \qquad \textbf{(C)}\ \text{II and III only} \qquad$ $ \textbf{(D)}\ \text{I, II and IV only} \qquad \textbf{(E)}\ \text{All are true.}$

Consider a function $ f(x)\equal{}xe^{\minus{}x^3}$ defined on any real numbers. (1) Examine the variation and convexity of $ f(x)$ to draw the garph of $ f(x)$. (2) For a positive number $ C$, let $ D_1$ be the region bounded by $ y\equal{}f(x)$, the $ x$-axis and $ x\equal{}C$. Denote $ V_1(C)$ the volume obtained by rotation of $ D_1$ about the $ x$-axis. Find $ \lim_{C\rightarrow \infty} V_1(C)$. (3) Let $ M$ be the maximum value of $ y\equal{}f(x)$ for $ x\geq 0$. Denote $ D_2$ the region bounded by $ y\equal{}f(x)$, the $ y$-axis and $ y\equal{}M$. Find the volume $ V_2$ obtained by rotation of $ D_2$ about the $ y$-axis.

Let $a,\ b$ be positive real numbers with $a<b$. Define the definite integrals $I_1,\ I_2,\ I_3$ by $I_1=\int_a^b \sin\ (x^2)\ dx,\ I_2=\int_a^b \frac{\cos\ (x^2)}{x^2}\ dx,\ I_3=\int_a^b \frac{\sin\ (x^2)}{x^4}\ dx$. (1) Find the value of $I_1+\frac 12I_2$ in terms of $a,\ b$. (2) Find the value of $I_2-\frac 32I_3$ in terms of $a,\ b$. (3) For a positive integer $n$, define $K_n=\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}} \sin\ (x^2)\ dx+\frac 34\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}}\frac{\sin\ (x^2)}{x^4}\ dx$. Find the value of $\lim_{n\to\infty} 2n\pi \sqrt{2n\pi} K_n$. [i]2011 Tokyo University of Science entrance exam/Information Sciences, Applied Chemistry, Mechanical Enginerring, Civil Enginerring[/i]

Let be a sequence $ \left( a_n \right)_{n\geqslant 1} $ of real numbers defined recursively as $$ a_n=2007+1004n^2-a_{n-1}-a_{n-2}-\cdots -a_2-a_1. $$ Calculate: $$ \lim_{n\to\infty} \frac{1}{n}\int_1^{a_n} e^{1/\ln t} dt $$