The equation $x^3+px+q=0$ has three distinct real roots. Show that $p<0$

Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.

Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\dots,x_n,$ and suppose that (a) $\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0$ (identically) and that (b) $x_1^2+\cdots+x_n^2$ divides $P(x_1,\dots,x_n).$ Show that $P=0$ identically.

As the Kubiks head homeward, away from the beach in the family van, Jerry decides to take a different route away from the beach than the one they took to get there. The route involves lots of twists and turns, prompting Hannah to wonder aloud if Jerry's "shortcut" will save any time at all. Michael offers up a problem as an analogy to his father's meandering: "Suppose dad drives around, making right-angled turns after $\textit{every}$ mile. What is the farthest he could get us from our starting point after driving us $500$ miles assuming that he makes exactly $300$ right turns?" "Sounds almost like an energy efficiency problem," notes Hannah only half jokingly. Hannah is always encouraging her children to think along these lines. Let $d$ be the answer to Michael's problem. Compute $\lfloor d\rfloor$.

([b]4[/b]) Let $ f(x) \equal{} \sin^6\left(\frac {x}{4}\right) \plus{} \cos^6\left(\frac {x}{4}\right)$ for all real numbers $ x$. Determine $ f^{(2008)}(0)$ (i.e., $ f$ differentiated $ 2008$ times and then evaluated at $ x \equal{} 0$).

Let $f :\mathbb R \to\mathbb R$ a function $ n \geq 2$ times differentiable so that: $ \lim_{x \to \infty} f(x) = l \in \mathbb R$ and $ \lim_{x \to \infty} f^{(n)}(x) = 0$. Prove that: $ \lim_{x \to \infty} f^{(k)}(x) = 0 $ for all $ k \in \{1, 2, \dots, n - 1\} $, where $f^{(k)}$ is the $ k $ - th derivative of $f$.

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

Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$.

Let $f:[0,1]\to(0,\infty)$ be a continous function on $[0,1]$ and let $A=\int_0^1 f(t)\mathrm{d}t.$[list=a] [*]Consider the function $F:[0,1]\to[0,A]$ defined by \[F(x)=\int_0^xf(t)\mathrm{d}t.\]Prove that $F(x)$ has an inverse function, which is differentiable. [*]Prove that there exists a unique function $g:[0,1]\to[0,1]$ for which\[\int_0^xf(t)\mathrm{d}t=\int_{g(x)}^1f(t)\mathrm{d}t\]is satisfied for every $x\in [0,1].$ [*]Prove that there exists $c\in[0,1]$ for which\[\lim_{x\to c}\frac{g(x)-c}{x-c}=-1,\]whre $g$ is the function uniquely determined at b. [/list]

Let $f,g,h:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is differentiable, $g$ and $h$ are monotonic, and $f'=f+g+h$. Prove that the set of the points of discontinuity of $g$ coincides with the respective set of $h$.

Let $n$ be a positive integer, $n\geq 2$. For each $t\in \mathbb{R}$, $t\neq k\pi$, $k\in\mathbb{Z}$, we consider the numbers \[ x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) = \sum_{k=1}^n k(n-k)\sin{(tk)}. \] Prove that if $x_n(t) = y_n(t) =0$ if and only if $\tan {\frac {nt}2} = n \tan {\frac t2}$. [i]Constantin Buse[/i]

A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.

Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$. Prove that \[ \tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} . \]

Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let \[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \] Prove that \[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \] where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?

Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.

Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x)=f^\prime(x)f^{\prime\prime}(x)$. What is the leading coefficient of $f(x)$?

Let be a differentiable function $ f:[0,1]\longrightarrow\mathbb{R} $ whose derivative has a positive Lipschitz constant $ L. $ Show that [b]a)[/b] $ x,y\in [0,1]\implies | f(x)-f(y)-f'(y)(x-y) |\le L\cdot (x-y)^2 $ [b]b)[/b] $ \lim_{n\to\infty } \left( n\int_0^1 f(x)dx-\sum_{i=1}^nf\left( \frac{2i-1}{2n} \right) \right) =0. $

The continuous function and twice differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $2007^{2}\cdot f(x)+f''(x)=0$. Prove that there exist two such real numbers $k$ and $l$ such that $f(x)=l\cdot\sin(2007x)+k\cdot\cos(2007x)$.

Evaluate $\dfrac{d}{dx}\left(\sin x - \dfrac{4}{3}\sin^3 x\right)$ when $x=15$.

Let $C$ be a real number, and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function such that $$ \lim_{x \to \infty} f(x)=C, \;\; \; \lim_{x \to \infty} f'''(x)=0.$$ Prove that $$ \lim_{x \to \infty} f'(x) =0 \;\; \text{and} \;\; \lim_{x \to \infty} f''(x)=0.$$

Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$. Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$

Let $f : \left[ 0, 1 \right] \to \mathbb R$ be a continuous function and $g : \left[ 0, 1 \right] \to \left( 0, \infty \right)$. Prove that if $f$ is increasing, then \[\int_{0}^{t}f(x) g(x) \, dx \cdot \int_{0}^{1}g(x) \, dx \leq \int_{0}^{t}g(x) \, dx \cdot \int_{0}^{1}f(x) g(x) \, dx .\]

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 $f(t+h)$.

Find the minimum value of $ \int_0^{\pi} \left(x \minus{} \pi a \minus{} \frac {b}{\pi}\cos x\right)^2dx$.