The positive function $p(n)$ is defined as the number of ways that the positive integer $n$ can be written as a sum of positive integers. Show that, for all positive integers $n \ge 2$, \[2^{\lfloor \sqrt{n}\rfloor}< p(n) < n^{3 \lfloor\sqrt{n}\rfloor }.\]

Let $ f:[0,1]\longrightarrow\mathbb{R} $ a continuous function. Prove that $$ \int_0^1 f^2\left( x^2 \right) dx\ge \frac{3}{4}\left( \int_0^1 f(x)dx \right)^2 , $$ and find the circumstances under which equality happens.

Let $\alpha , \beta , \gamma , \delta$ be positive numbers such that for all $x$, $\sin{\alpha x}+\sin {\beta x}=\sin {\gamma x}+\sin {\delta x}$. Prove that $\alpha =\gamma$ or $\alpha=\delta$.

Evaluate \[\int_0^1 \frac{(x^2+x+1)^3\{\ln (x^2+x+1)+2\}}{(x^2+x+1)^3}(2x+1)e^{x^2+x+1}dx.\]

Determine all non-negative integral pairs $ (x, y)$ for which \[ (xy \minus{} 7)^2 \equal{} x^2 \plus{} y^2.\]

Let $P$ be a polygon (not necessarily convex) with $n$ vertices, such that all its sides and diagonals are less or equal with 1 in length. Prove that the area of the polygon is less than $\dfrac {\sqrt 3} 2$.

Let $ f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $ f'$ on $ [0,1]$. Prove that if $ f\left( \frac 12\right) \equal{} 0$, then \[ \int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.\]

Find continuous functions $x(t),\ y(t)$ such that $\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds$ $\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds$

How many two-digit positive integers $ N$ have the property that the sum of $ N$ and the number obtained by reversing the order of the digits of $ N$ is a perfect square? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Find the infinite integrals as follows. (1) 2013 Hiroshima City University entrance exam/Informatic Science $\int \frac{x^2}{2-x^2}dx$ (2) 2013 Kanseigakuin University entrance exam/Science and Technology $\int x^4\ln x\ dx$ (3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

Compute $\displaystyle\int_0^{\pi/3}x\tan^2(x)dx$.

Real numbers $x,y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x,y$, and $z$ are within $1$ unit of each other is greater than $\tfrac{1}{2}$. What is the smallest possible value of $n$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane. Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.

Let $f$ and $g$ be two continuous, distinct functions from $[0,1] \rightarrow (0,+\infty)$ such that $\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$ Let $y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq 0$, natural. Prove that $(y_n)$ is an increasing and divergent sequence.

(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality. \[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \] (2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$

On the $xy$ plane, find the area of the figure bounded by the graphs of $y=x$ and $y=\left|\ \frac34 x^2-3\ \right |-2$. [i]2011 Kyoto University entrance exam/Science, Problem 3[/i]

Evaluate \[\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)\]

In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$. [asy] Label f; xaxis(0,60,blue); yaxis(0,60,blue); real f(real x) { return (x^2)/60; } draw(graph(f,0,53),red); label("$y=x^2$",(30,15),E); real f(real x) { return (x^2)/25; } draw(graph(f,0,38),red); label("$y=2x^2$",(37,37^2/25),E); real f(real x) { return (x^2)/10; } draw(graph(f,0,25),red); label("$y=f(x)$",(24,576/10),W); label("$O(0,0)$",(0,0),S); dot((20,400/25)); dot((20,400/60)); label("$P$",(20,400/25),E); label("$B$",(20,400/60),SE); dot(((4000/25)^(0.5),400/25)); label("$A$",((4000/25)^(0.5),400/25),W); draw((20,400/25)..((4000/25)^(0.5),400/25)); draw((20,400/25)..(20,400/60)); [/asy]

Let $S$ be the sum of all $x$ such that $1\leq x\leq 99$ and \[\{x^2\}=\{x\}^2.\] Compute $\lfloor S\rfloor$.

Evaluate $ \int_0^{\frac {\pi}{2}} \frac {dx}{(1 \plus{} \cos x)^2}$.

Evaluate \[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]

Evaluate $ \int_0^{\frac {\pi}{2}} \frac {(\sin x)^{\cos x}}{(\cos x)^{\sin x} \plus{} (\sin x)^{\cos x}} dx$.

For $ f(x)\equal{}e^{\frac{x}{2}}\cos \frac{x}{2}$, evaluate $ \sum_{n\equal{}0}^{\infty} \int_{\minus{}\pi}^{\pi}f(x)f(x\minus{}2n\pi)dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.