Let $f(x)=\frac{\sin x}{x}$, for $x>0$, and let $n$ be a positive integer. Prove that $|f^{(n)}(x)|<\frac{1}{n+1}$, where $f^{(n)}$ denotes the $n^{\mathrm{th}}$ derivative of $f$. (Proposed by Alexander Bolbot, State University, Novosibirsk)

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function. Prove that there exists $w \in [-1,1]$ such that \[ \frac{f'''(w)}{6} = \frac{f(1)}{2}-\frac{f(-1)}{2}-f'(0). \]

Given the polynomial $$P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}.$$ Express the numerical value of its derivative of order $325$ for $x = 0$.

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

Define mapping $F : \mathbb{R}^4\rightarrow \mathbb{R}^4$ as $F(x,\ y,\ z,\ w)=(xy,\ y,\ z,\ w)$ and let mapping $f : S^3\rightarrow \mathbb{R}^4$ be restriction of $F$ to 3 dimensional ball $S^3=\{(x,\ y,\ z,\ w)\in{\mathbb{R}^4} | x^2+y^2+z^2+w^2=1\}$. Find the rank of $df_p$, or the differentiation of $f$ at every point $p$ in $S^3$.

Suppose the function $f(x)-f(2x)$ has derivative $5$ at $x=1$ and derivative $7$ at $x=2$. Find the derivative of $f(x)-f(4x)$ at $x=1$.

Consider the following function \[g(x)=(\alpha+|x|)^{2}e^{(5-|x|)^{2}}\] [b]i)[/b] Find all the values of $\alpha$ for which $g(x)$ is continuous for all $x\in\mathbb{R}$ [b]ii)[/b]Find all the values of $\alpha$ for which $g(x)$ is differentiable for all $x\in\mathbb{R}$.

A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way. How much time passes until the cloud collapses fully? (The constant $0.5427$ is actually $\sqrt{\frac{3 \pi}{32}}$.) $\textbf{(A) } \frac{0.5427}{r_0^2 \sqrt{G \rho_0}}\\ \\ \textbf{(B) } \frac{0.5427}{r_0 \sqrt{G \rho_0}}\\ \\ \textbf{(C) } \frac{0.5427}{\sqrt{r_0} \sqrt{G \rho_0}}\\ \\ \textbf{(D) } \frac{0.5427}{\sqrt{G \rho_0}}\\ \\ \textbf{(E) } \frac{0.5427}{\sqrt{G \rho_0}}r_0$

If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\] then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.

Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$. [i]Proposed by Evan Chen[/i]

The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules: [list] [*] $D(1) = 0$; [*] $D(p)=1$ for all primes $p$; [*] $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$. [/list] Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$.

For each positive integer $n$, let $f_{n}(\vartheta)=\sin(\vartheta)\cdot \sin(2\vartheta) \cdot \sin(4\vartheta)\cdots \sin(2^{n}\vartheta)$. For each real $\vartheta$ and all $n$, prove that \[|f_{n}(\vartheta)| \leq \frac{2}{\sqrt{3}}|f_{n}(\frac{\pi}{3})| \]

A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called [i]basic [/i]if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.

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 $f(x)$?

Let $ f$ be a function from the non-negative integers to the non-negative integers such that $ f(nm) \equal{} n f(m) \plus{} m f(n), f(10) \equal{} 19, f(12) \equal{} 52,$ and $ f(15) \equal{} 26.$ What is $ f(8)$? A. 12 B. 24 C. 36 D. 48 E. 60

If $ a\geq b\geq c\geq d > 0$ such that $ abcd\equal{}1$, then prove that \[ \frac 1{1\plus{}a} \plus{} \frac 1{1\plus{}b} \plus{} \frac 1{1\plus{}c} \geq \frac {3}{1\plus{}\sqrt[3]{abc}}.\]

The $2^{nd}$ order differentiable function $f:\mathbb R \longrightarrow \mathbb R$ is in such a way that for every $x\in \mathbb R$ we have $f''(x)+f(x)=0$. [b]a)[/b] Prove that if in addition, $f(0)=f'(0)=0$, then $f\equiv 0$. [b]b)[/b] Use the previous part to show that there exist $a,b\in \mathbb R$ such that $f(x)=a\sin x+b\cos x$.

The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,\[ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. \]Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have\[ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. \]

A cat is trying to catch a mouse in the non-negative quadrant \[N=\{(x_1,x_2)\in \mathbb{R}^2: x_1,x_2\geq 0\}.\] At time $t=0$ the cat is at $(1,1)$ and the mouse is at $(0,0)$. The cat moves with speed $\sqrt{2}$ such that the position $c(t)=(c_1(t),c_2(t))$ is continuous, and differentiable except at finitely many points; while the mouse moves with speed $1$ such that its position $m(t)=(m_1(t),m_2(t))$ is also continuous, and differentiable except at finitely many points. Thus $c(0)=(1,1)$ and $m(0)=(0,0)$; $c(t)$ and $m(t)$ are continuous functions of $t$ such that $c(t),m(t)\in N$ for all $t\geq 0$; the derivatives $c'(t)=(c'_1(t),c'_2(t))$ and $m'(t)=(m'_1(t),m'_2(t))$ each exist for all but finitely many $t$ and \[(c'_1(t)^2+(c'_2(t))^2=2 \qquad (m'_1(t)^2+(m'_2(t))^2=1,\] whenever the respective derivative exists. At each time $t$ the cat knows both the mouse's position $m(t)$ and velocity $m'(t)$. Show that, no matter how the mouse moves, the cat can catch it by time $t=1$; that is, show that the cat can move such that $c(\tau)=m(\tau)$ for some $\tau\in[0,1]$.

Find the twice-differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that have the property that $$ f'(x)+F(x)=2f(x)+x^2/2, $$ for any real numbers $ x; $ where $ F $ is a primitive of $ f. $ [i]Carmen Botea[/i]

Prove that \[ \frac{1}{a^2+2} + \frac{1}{b^2+2} + \frac{1}{c^2+2} \le \frac{1}{6ab+c^2} + \frac{1}{6bc+a^2} + \frac{1}{6ca+b^2} \] for all positive real numbers $a$, $b$ and $c$ satisfying $a^2+b^2+c^2=1$.

Let $I \subset \mathbb{R}$ be a nondegenerate interval and $f:I \to \mathbb{R}$ a differentiable function. We denote $J= \left\{ \frac{f(b)-f(a)}{b-a} : a,b \in I, a<b \right\}.$ Prove that: $a)$ $J$ is an interval; $b)$ $J \subset f'(I),$ and the set $f'(I) \setminus J$ contains at most two elements; $c)$ Using parts $a)$ and $b),$ deduce that $f'$ has the intermediate value property.

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 $\forall _{x_i} x_i > 0$

A polynomial $P(x)$ of some degree $d$ satisfies $P(n) = n^3 + 10n^2 - 12$ and $P'(n) = 3n^2 + 20n - 1$ for $n = -2, -1, 0, 1, 2.$ Also, $P$ has $d$ distinct (not necessarily real) roots $r_1, r_2, \ldots, r_d.$ The value of \[ \sum_{k=1}^{d}\frac{1}{4 - r_k^2} \] can be expressed as a common fraction $\tfrac{p}{q}.$ What is the value of $p + q?$

Let $A=(a_{ij})\in \mathcal{M}_p(\mathbb{C})$ such that $a_{12}=a_{23}=\ldots=a_{p-1,p}=1$ and $a_{ij}=0$ for any other entry. a)Prove that $A^{p-1}\neq O_p$ and $A^p=O_p$. b)If $X\in \mathcal{M}_{p}(\mathbb{C})$ and $AX=XA$, prove that there exist $a_1,a_2,\ldots,a_p\in \mathbb{C}$ such that: \[X=\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_p \\ 0 & a_1 & a_2 & \ldots & a_{p-1} \\ 0 & 0 & a_1 & \ldots & a_{p-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & a_1 \end{array} \right)\] c)If there exist $B,C\in \mathcal{M}_p(\mathbb{C})$ such that $(I_p+A)^n=B^n+C^n,\ (\forall)n\in \mathbb{N}^*$, prove that $B=O_p$ or $C=O_p$.