Suppose that $ f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $ \int_0^1f(x)\,dx\equal{}0.$ Prove that for every $ \alpha\in(0,1),$ \[ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|\]

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$. Show that \[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]

Let \[f(x)=\int_0^1 |t-x|t \, dt\] for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?

3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and: $\frac{EF}{FD}=\frac{AD}{DB}$ Prove that $CF \bot AE$

Let $f : \mathbb R \to \mathbb R$ be a function such that \[f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R\] Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$

If $a, b, c, d$ are positive numbers satisfying $a^3 + b^3 +3ab = c + d = 1,$ prove that \[\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.\]

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic function and $F:\mathbb{R}\to\mathbb{R}$ given by \[F(x)=\int_0^xf(t)\ \text{d}t.\] Prove that if $F$ has a finite derivative, then $f$ is continuous. [i]Dorin Andrica & Mihai Piticari[/i]

Prove that $\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}+\sqrt[3]{n+2}\rfloor =\lfloor \sqrt[3]{27n+26}\rfloor$ for all positive integers $n$.

Find all real numbers $x, y$ such that: $y+3\sqrt{x+2}=\frac{23}2+y^2-\sqrt{49-16x}$

A smooth function $f:I\to \mathbb{R}$ is said to be [i]totally convex[/i] if $(-1)^k f^{(k)}(t) > 0$ for all $t\in I$ and every integer $k>0$ (here $I$ is an open interval). Prove that every totally convex function $f:(0,+\infty)\to \mathbb{R}$ is real analytic. [b]Note[/b]: A function $f:I\to \mathbb{R}$ is said to be [i]smooth[/i] if for every positive integer $k$ the derivative of order $k$ of $f$ is well defined and continuous over $\mathbb{R}$. A smooth function $f:I\to \mathbb{R}$ is said to be [i]real analytic[/i] if for every $t\in I$ there exists $\epsilon> 0$ such that for all real numbers $h$ with $|h|<\epsilon$ the Taylor series \[\sum_{k\geq 0}\frac{f^{(k)}(t)}{k!}h^k\] converges and is equal to $f(t+h)$.

Take the following differential equation: \[3(3+x^2)\frac{dx}{dt}=2(1+x^2)^2e^{-t^2}\] If $x(0)\leq 1$, prove that there exists $M>0$ such that $|x(t)|<M$ for all $t\geq 0$.

Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$. Evaluate $\int_0^2 g(t)dt$. [i]2010 Kumamoto University entrance exam/Science[/i]

For every natural number $n\geq{2}$ consider the following affirmation $P_n$: "Consider a polynomial $P(X)$ (of degree $n$) with real coefficients. If its derivative $P'(X)$ has $n-1$ distinct real roots, then there is a real number $C$ such that the equation $P(x)=C$ has $n$ real,distinct roots." Are $P_4$ and $P_5$ both true? Justify your answer.