Suppose a parabola with the axis as the $ y$ axis, concave up and touches the graph $ y\equal{}1\minus{}|x|$. Find the equation of the parabola such that the area of the region surrounded by the parabola and the $ x$ axis is maximal.

A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); filldraw(Circle((0,.866),.5),grey,black); label("1",(0,.866),S); filldraw(Circle((0,0),1),white,black); draw((-.5,.866)--(.5,.866),linetype("4 4")); clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle); draw((-1,0)--(1,0)); label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$ $ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$

Show that the number $r(n)$ of representations of $n$ as a sum of two squares has $\pi$ as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]

Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.

On the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ find the point $T=(x_0,y_0)$ such that the triangle bounded by the axes of the ellipse and the tangent at that point has the least area.

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

Evaluate \[\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx\]

Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$. [i]Calvin Deng.[/i]

If $ a\geq b\geq c\geq d > 0$ such that $ abcd\equal{}1$, then prove that \[ \frac 1{1\plus{}a} \plus{} \frac 1{1\plus{}b} \plus{} \frac 1{1\plus{}c} \geq \frac {3}{1\plus{}\sqrt[3]{abc}}.\]

The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows. \[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\] Evaluate \[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]

Consider the polynomials \[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\] \[g(p) = p^3 + 2p + m.\] Find all integral values of $m$ for which $f$ is divisible by $g$.

$$\int_{-\infty}^\infty \frac{\sin(\pi x)}{x(1+x^2)}\,dx$$ [i]Proposed by Vlad Oleksenko[/i]

Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

How many hours does it take a train traveling at an average rate of $ 40$ mph between stops to travel $ a$ miles it makes $ n$ stops of $ m$ minutes each? $ \textbf{(A)}\ \frac{3a\plus{}2mn}{120} \qquad \textbf{(B)}\ 3a\plus{}2mn \qquad \textbf{(C)}\ \frac{3a\plus{}2mn}{12} \qquad \textbf{(D)}\ \frac{a\plus{}mn}{40} \qquad \textbf{(E)}\ \frac{a\plus{}40mn}{40}$

Evaluate $ \sum_{k\equal{}1}^n \int_0^{\pi} (\sin x\minus{}\cos kx)^2dx.$

Let $f(x)=\int_0^ x \frac{1}{1+t^2}dt.$ For $-1\leq x<1$, find $\cos \left\{2f\left(\sqrt{\frac{1+x}{1-x}}\right)\right\}.$ [i]2011 Ritsumeikan University entrance exam/Science and Technology[/i]

$ \lim_{n\to\infty } \frac{1}{2^n}\left( \left( \frac{a}{a+b}+\frac{b}{b+c} \right)^n +\left( \frac{b}{b+c}+\frac{c}{c+a} \right)^n +\left( \frac{c}{c+a}+\frac{a}{a+b} \right)^n \right) ,\quad a,b,c>0 $

Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$

For each polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ we define it's derivative as this and we show it by $p'(x)$: \[p'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+...+2a_2x+a_1\] a) For each two polynomials $p(x)$ and $q(x)$ prove that:(3 points) \[(p(x)q(x))'=p'(x)q(x)+p(x)q'(x)\] b) Suppose that $p(x)$ is a polynomial with degree $n$ and $x_1,x_2,...,x_n$ are it's zeros. prove that:(3 points) \[\frac{p'(x)}{p(x)}=\sum_{i=1}^{n}\frac{1}{x-x_i}\] c) $p(x)$ is a monic polynomial with degree $n$ and $z_1,z_2,...,z_n$ are it's zeros such that: \[|z_1|=1, \quad \forall i\in\{2,..,n\}:|z_i|\le1\] Prove that $p'(x)$ has at least one zero in the disc with length one with the center $z_1$ in complex plane. (disc with length one with the center $z_1$ in complex plane: $D=\{z \in \mathbb C: |z-z_1|\le1\}$)(20 points)

Calculate $ \lim_{n\to\infty }\sum_{t=1}^n\frac{1}{n+t+\sqrt{n^2+nt}} . $ [i]D.M. Bătinețu[/i] and [i]Neculai Stanciu[/i]

Let $z=z(x,y)$ be implicit function with two variables from $2sin(x+2y-3z)=x+2y-3z$. Find $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$.

Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.

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