A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3\minus{}ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$

Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.

Find the real number $a$ such that the integral $$\int_a^{a+8}e^{-x}e^{-x^2}dx$$ attain its maximum.

Prove the following inequality. \[\int_{-1}^1 \frac{e^x+e^{-x}}{e^{e^{e^x}}}dx<e-\frac{1}{e}\] Own

Problem 3. Let $f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right)$ an increasing function .Prove that: (a) $\int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0.$ (b) Exist $a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right]$ such that $\int_{0}^{a}{f\left( x \right)\sin x\ dx=}\int_{0}^{a}{f\left( x \right)\cos x\ dx}.$

For a positive constant $a$, find the minimum value of $f(x)=\int_0^{\frac{\pi}{2}} |\sin t-ax\cos t|dt.$

Let $f (x)$ be a polynomial with integer coefficients. Prove that if $f (x)= 1979$ for four different integer values of $x$, then $f (x)$ cannot be equal to $2\times 1979$ for any integral value of $x$.

Denote by $C,\ l$ the graphs of the cubic function $C: y=x^3-3x^2+2x$, the line $l: y=ax$. (1) Find the range of $a$ such that $C$ and $l$ have intersection point other than the origin. (2) Denote $S(a)$ by the area bounded by $C$ and $l$. If $a$ move in the range found in (1), then find the value of $a$ for which $S(a)$ is minimized. 50 points

Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?

Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.

\[\int_0^1 \frac{(x+1)\log(x)}{x^3-1}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

Prove that the integral part of the decimal representation of the number $(3+\sqrt{5})^n$ is odd, for every positive integer $n.$

Let $\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f$ continuous $\}$ and $n$ an integer, $n\geq 2$. Find the smallest real constant $c$ such that for any $f\in \mathcal F$ the following inequality takes place \[ \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx. \]

Let $0<a<b$.Prove the following inequaliy. \[\frac{1}{b-a}\int_a^b \left(\ln \frac{b}{x}\right)^2 dx<2\]

Find all differentiable functions $f, g:[0,\infty) \to \mathbb{R}$ and the real constant $k\geq 0$ such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and $f(0)=k, f'(0)=-k^2/3$ and also $f(x)\neq 0$ for all $x\geq 0$.\\ \\ [i] (Nora Gavrea)[/i]

On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$ (1) Draw the graph. (2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$. (3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$. 2010 Tokyo Institute of Technology entrance exam, Second Exam.

For a complex number $z \neq 3$,$4$, let $F(z)$ denote the real part of $\tfrac{1}{(3-z)(4-z)}$. If \[ \int_0^1 F \left( \frac{\cos 2 \pi t + i \sin 2 \pi t}{5} \right) \; dt = \frac mn \] for relatively prime positive integers $m$ and $n$, find $100m+n$. [i]Proposed by Evan Chen[/i]

Let $p>1,\frac1p+\frac1q=1$ and $r>1$. If $u(x,y),v(x,y)>0$, and $f(x,y),g(x,y)$ are continuous functions on $[a,b]\times[c,d]$, then prove $$\left(\frac{\left(\int^b_a\int^d_c(f(x,y)+g(x,y))^rdxdy\right)^{1/r}}{(u(x,y)+v(x,y))^{1/q}}\right)^p\le\left(\frac{\left(\int^b_a\int^d_cf(x,y)^rdxdy\right)^{1/r}}{u(x,y)^{1/q}}\right)^p+\left(\frac{\left(\int^b_a\int^d_cg(x,y)^rdxdy\right)^{1/r}}{v(x,y)^{1/q}}\right)^p,$$ with equality if and only if either $$\left(\lVert f(x,y)\rVert^r_r,\lVert g(x,y)\rVert^r_r\right)=\alpha\left(\lVert u(x,y)\rVert^r_r,\lVert v(x,y)\rVert^r_r\right)$$ for some $\alpha>0$ or $\lVert f(x,y)\rVert^r_r=\lVert g(x,y)\rVert^r_r=0$. [i]Proposed by Chang-Jian Zhao[/i]

Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

Let $a,\ b,\ c$ be positive real numbers such that $b^2>ac.$ Evaluate \[\int_0^{\infty} \frac{dx}{ax^4+2bx^2+c}.\] [i]1981 Tokyo University, Master Course[/i]

Show that there do not exist polynomials $p(x)$ and $q(x)$ each having integer coefficients and of degree greater than or equal to 1 such that \[ p(x)q(x) = x^5 +2x +1 . \]

The rectangle with vertices $(0,0)$, $(0,3)$, $(2,0)$ and $(2,3)$ is rotated clockwise through a right angle about the point $(2,0)$, then about $(5,0)$, then about $(7,0$), and finally about $(10,0)$. The net effect is to translate it a distance $10$ along the $x$-axis. The point initially at $(1,1)$ traces out a curve. Find the area under this curve (in other words, the area of the region bounded by the curve, the $x$-axis and the lines parallel to the $y$-axis through $(1,0)$ and $(11,0)$).

For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.

Show that for nonnegative integers $m$ and $n$, $\frac{\dbinom{m}{0}}{n+1}-\frac{\dbinom{m}{1}}{n+2}+...+(-1)^m\frac{\dbinom{m}{m}}{n+m+1}$ $=\frac{\dbinom{n}{0}}{m+1}-\frac{\dbinom{n}{1}}{m+2}+...+(-1)^n\frac{\dbinom{n}{n}}{m+n+1}$.

In the $ x \minus{} y$ plane, let $ l$ be the tangent line at the point $ A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)$ on the ellipse $ \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a)$. Let denote $ S$ be the area of the figure bounded by $ l,$ the $ x$ axis and the ellipse. (1) Find the equation of $ l$. (2) Express $ S$ in terms of $ a,\ b$. (3) Find the maximum value of $ S$ with the constraint $ a^2 \plus{} 3b^2 \equal{} 4$.