For which of the following values of real number $t$, the equation $x^4-tx+\dfrac 1t = 0$ has no root on the interval $[1,2]$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations \[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

For every complex number $z$ different from 0 and 1 we define the following function \[ f(z) := \sum \frac 1 { \log^4 z } \] where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$. b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]

Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a differentiable function, with continuous derivative, and let \[ s_{n}=\sum_{k=1}^{n}f\left( \frac{k}{n}\right) \] Prove that the sequence $(s_{n+1}-s_{n})_{n\in{\mathbb{N}}^{\ast}}$ converges to $\int_{0}^{1}f(x)\mathrm{d}x$.

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52\] and determine when equality holds.

Let $f(x)$ be a periodic function of period $T > 0$ defined over $\mathbb R$. Its first derivative is continuous on $\mathbb R$. Prove that there exist $x, y \in [0, T )$ such that $x \neq y$ and \[f(x)f'(y)=f'(x)f(y).\]

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 $f(t+h)$.

If $a,b,c$ are nonnegative real numbers, then \[{ 3(a^2+b^2+c^2) \geq (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2 \geq (a+b+c)^2.}\]

Show that if a holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ has the property that the modulus of any of its derivatives (of any order) is everywhere dominated by $ 1, $ then $ |f(z)|\le e^{|\text{Im} (z)|} , $ for all complex numbers $ z. $

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

Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\hat{A}$.

Let $ F$ be a surface of nonzero curvature that can be represented around one of its points $ P$ by a power series and is symmetric around the normal planes parallel to the principal directions at $ P$. Show that the derivative with respect to the arc length of the curvature of an arbitrary normal section at $ P$ vanishes at $ P$. Is it possible to replace the above symmetry condition by a weaker one? [i]A. Moor[/i]

Let $f(x)=\frac{P(x)}{Q(x)}$. Where $P(x), Q(x)$ are two non constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})$ for all infinitely many values of $x$. a. Show that $deg(P) <deg(Q).$ b. Show that $P'(1)=2Q'(1)- deg(Q). Q(1)$ Here $P'(x)$ denotes the derivatives of $P(x)$ as usual

Can the graphs of a polynomial of degree 20 and the function $\displaystyle y={1\over x^{40}}$ have exactly 30 points of intersection? [i]Proposed by K. Kokhas[/i]

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

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

Prove the following inequality: $x_1 + 2x_2 + 3x_3 + ... + nx_n \leq \frac{n(n-1)}{2} + x_1 + x_2 ^2 + x_3 ^3 + ... + x_n ^n$ where $\forall _{x_i} x_i > 0$

If $f$ is a real-valued function of one real variable which has a continuous derivative on the closed interval $[a,b]$ and for which there is no $x\in [a,b]$ such that $f(x)=f'(x)=0$, then prove that there is a function $g$ with continuous first derivative on $[a,b]$ such that $fg'-f'g$ is positive on $[a,b].$

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 $x \in \mathbb R.$

Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$ \[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\] ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

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

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 $\lfloor d\rfloor$.

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]

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

Let $f:\mathbb{R} \to \mathbb{R}$ be a function that is a function that is differentiable $n+1$ times for some positive integer $n$ . The $i^{th}$ derivative of $f$ is denoted by $f^{(i)}$ . Suppose- $f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0$. Prove that $f^{(n+1)}(x)=0$ for some $x \in (0,1)$