Let $a$ be a positive constant. Find the value of $\ln a$ such that \[\frac{\int_1^e \ln (ax)\ dx}{\int_1^e x\ dx}=\int_1^e \frac{\ln (ax)}{x}\ dx.\]

Let $f:[0,1]\rightarrow \mathbb{R}$ a differentiable function such that $f(0)=f(1)=0$ and $|f'(x)|\le 1,\ \forall x\in [0,1]$. Prove that: \[\left|\int_0 ^1f(t)dt\right|<\frac{1}{4}\]

For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.

Let $ f(x)$ be a continuous function defined in the interval $ 0\leq x\leq 1.$ Prove that $ \int_0^1 xf(x)f(1\minus{}x)\ dx\leq \frac{1}{4}\int_0^1 \{f(x)^2\plus{}f(1\minus{}x)^2\}\ dx.$

The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is $ \textbf{(A)}\ 100 \qquad\textbf{(B)}\ 125 \qquad\textbf{(C)}\ 123 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ \text{none of these} $

Evaluate \[\int_0^{\pi} \left(1+\sum_{k=1}^n k\cos kx\right)^2dx\ \ (n=1,\ 2,\ \cdots).\] [i]2011 Doshisya University entrance exam/Life Medical Sciences[/i]

Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$

Evaluate \[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]

If \[ \sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq \] for relatively prime positive integers $p,q$, find $p+q$. [i]Proposed by Michael Kural[/i]

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function . Suppose \[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\] $\forall x\in \mathbb{R}$ and all $t>0$ . Then show that there exists a constant $c$ such that $f(x)=cx\ \forall x$

A polygon has each side integral and each pair of adjacent sides perpendicular (it is not necessarily convex). Show that if it can be covered by non-overlapping $2 x 1$ dominos, then at least one of its sides has even length.

Prove the following inequality. \[\frac{e-1}{n+1}\leqq\int^e_1(\log x)^n dx\leqq\frac{(n+1)e+1}{(n+1)(n+2)}\ (n=1,2,\cdot\cdot\cdot) \] 1994 Kyoto University entrance exam/Science

Prove that for any $ 2n$ real numbers $ a_{1}$, $ a_{2}$, ..., $ a_{n}$, $ b_{1}$, $ b_{2}$, ..., $ b_{n}$, we have $ \sum_{i < j}{\left|a_{i}\minus{}a_{j}\right|}\plus{}\sum_{i < j}{\left|b_{i}\minus{}b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}\minus{}b_{j}\right|}$.

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$ Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$

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 $1\leq i \leq 70.$

For what $\lambda$ does the equation $$ \int_{0}^{1} \min(x,y) f(y)\; dy =\lambda f(x)$$ have continuous solutions which do not vanish identically in $(0,1)?$ What are these solutions?

$p(x) $ is a real polynomial of degree $3$. Find necessary and sufficient conditions on its coefficients in order that $p(n)$ is integral for every integer $n$.

When the parabola which has the axis parallel to $y$ -axis and passes through the origin touch to the rectangular hyperbola $xy=1$ in the first quadrant moves, prove that the area of the figure sorrounded by the parabola and the $x$-axis is constant.

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Prove that $$\int_0^{\pi/2}f(\sin(2x))\sin x\, dx = \int_0^{\pi/2} f(\cos^2 x)\cos x\, dx.$$

We will say that an octagon is integral if its is equiangular, its vertices are lattice points (i.e., points with integer coordinates), and its area is an integer. For example, the figure on the right shows an integral octagon of area $21$. Determine, with proof, the smallest positive integer $K$ so that for every positive integer $k\geq K$, there is an integral octagon of area $k$. [asy] size(200); defaultpen(linewidth(0.8)); draw((-1/2,0)--(17/2,0)^^(0,-1/2)--(0,15/2)); for(int i=1;i<=6;++i){ draw((0,i)--(17/2,i),linetype("4 4")); } for(int i=1;i<=8;++i){ draw((i,0)--(i,15/2),linetype("4 4")); } draw((2,1)--(1,2)--(1,3)--(4,6)--(5,6)--(7,4)--(7,3)--(5,1)--cycle,linewidth(1)); label("$1$",(1,0),S); label("$2$",(2,0),S); label("$x$",(17/2,0),SE); label("$1$",(0,1),W); label("$2$",(0,2),W); label("$y$",(0,15/2),NW); [/asy]

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.

Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have \[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.

Find the constants $ a,\ b,\ c$ such that a function $ f(x)\equal{}a\sin x\plus{}b\cos x\plus{}c$ satisfies the following equation for any real numbers $ x$. \[ 5\sin x\plus{}3\cos x\plus{}1\plus{}\int_0^{\frac{\pi}{2}} (\sin x\plus{}\cos t)f(t)\ dt\equal{}f(x).\]

Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that \[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \] where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.