Let be a real number $ c $ and a differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that $$ f(c)\neq \frac{1}{b-a}\int_a^b f(x)dx, $$ for any real numbers $ a\neq b. $ Prove that $ f'(c)=0. $ [i]Florin Rotaru[/i]

Suppose $A\subseteq \mathbb R^m$ is closed and non-empty. Let $f:A\to A$ is a lipchitz function with constant less than 1. (ie there exist $c<1$ that $|f(x)-f(y)|<c|x-y|,\ \forall x,y \in A)$. Prove that there exists a unique point $x\in A$ such that $f(x)=x$.

Define $ n_a!$ for $ n$ and $ a$ positive to be \[ n_a ! \equal{} n (n\minus{}a)(n\minus{}2a)(n\minus{}3a)...(n\minus{}ka)\] where $ k$ is the greatest integer for which $ n>ka$. Then the quotient $ 72_8!/18_2!$ is equal to $ \textbf{(A)}\ 4^5 \qquad \textbf{(B)}\ 4^6 \qquad \textbf{(C)}\ 4^8 \qquad \textbf{(D)}\ 4^9 \qquad \textbf{(E)}\ 4^{12}$

Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If \[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\] then \[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]

Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]

In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$

The function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(x^2)f^{\prime\prime}(x)=f^\prime (x)f^\prime (x^2)$ for all real $x$. Given that $f(1)=1$ and $f^{\prime\prime\prime}(1)=8$, determine $f^\prime (1)+f^{\prime\prime}(1)$.

Find the number of ordered pairs of positive integers $(a,b)$ with $a+b$ prime, $1\leq a, b \leq 100$, and $\frac{ab+1}{a+b}$ is an integer. [i]Author: Alex Zhu[/i]

In a triangle $ ABC$ right-angled at $ C$ , the median through $ B$ bisects the angle between $ BA$ and the bisector of $ \angle B$. Prove that \[ \frac{5}{2} < \frac{AB}{BC} < 3\]

Find all functions $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions. (a) $ f(x) \ge 0 $ for all $ x \in \mathbb{R} $. (b) For $ a, b, c, d \in \mathbb{R} $ with $ ab + bc + cd = 0 $, equality $ f(a-b) + f(c-d) = f(a) + f(b+c) + f(d) $ holds.

Find all functions $f$ from the set of real numbers to itself satisfying \[ f(x(1+y)) = f(x)(1 + f(y)) \] for all real numbers $x, y$.

In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$. [asy] Label f; xaxis(0,60,blue); yaxis(0,60,blue); real f(real x) { return (x^2)/60; } draw(graph(f,0,53),red); label("$y=x^2$",(30,15),E); real f(real x) { return (x^2)/25; } draw(graph(f,0,38),red); label("$y=2x^2$",(37,37^2/25),E); real f(real x) { return (x^2)/10; } draw(graph(f,0,25),red); label("$y=f(x)$",(24,576/10),W); label("$O(0,0)$",(0,0),S); dot((20,400/25)); dot((20,400/60)); label("$P$",(20,400/25),E); label("$B$",(20,400/60),SE); dot(((4000/25)^(0.5),400/25)); label("$A$",((4000/25)^(0.5),400/25),W); draw((20,400/25)..((4000/25)^(0.5),400/25)); draw((20,400/25)..(20,400/60)); [/asy]

For $x\geq 0$, Prove that $\int_0^x (t-t^2)\sin ^{2002} t \,dt<\frac{1}{2004\cdot 2005}$

Assuming that the roots of $x^3 + p \cdot x^2 + q \cdot x + r = 0$ are real and positive, find a relation between $p,q$ and $r$ which gives a necessary condition for the roots to be exactly the cosines of the three angles of a triangle.

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

For an arbitrary ordinal number $\alpha$ let $H(\alpha)$ denote the set of functions $f\colon \alpha \rightarrow \{ -1,0,1\}$ that map all but finitely many elements of $\alpha$ to $0$. Order $H(\alpha)$ according to the last difference, that is, for $f, g\in H(\alpha)$ let $f\prec g$ if $f(\beta) < g(\beta)$ holds for the maximum ordinal number $\beta < \alpha$ with $f(\beta) \neq g(\beta)$. Prove that the ordered set $(H(\alpha), \prec)$ is scattered (i.e. it doesn't contain a subset isomorphic to the set of rational numbers with the usual order), and that any scattered order type can be embedded into some $(H(\alpha), \prec)$.

Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

The function $ f$ has the property that for each real number $ x$ in its domain, $ 1/x$ is also in its domain and \[ f(x) \plus{} f\left(\frac {1}{x}\right) \equal{} x. \]What is the largest set of real numbers that can be in the domain of $ f$? $ \textbf{(A) } \{ x | x\ne 0\} \qquad \textbf{(B) } \{ x | x < 0\} \qquad \textbf{(C) }\{ x | x > 0\}\\ \textbf{(D) } \{ x | x\ne \minus{} 1 \text{ and } x\ne 0 \text{ and } x\ne 1\} \qquad \textbf{(E) } \{ \minus{} 1,1\}$

Define two functions $f(t)=\frac 12\left(t+\frac{1}{t}\right),\ g(t)=t^2-2\ln t$. When real number $t$ moves in the range of $t>0$, denote by $C$ the curve by which the point $(f(t),\ g(t))$ draws on the $xy$-plane. Let $a>1$, find the area of the part bounded by the line $x=\frac 12\left(a+\frac{1}{a}\right)$ and the curve $C$.

In a rhombus, $ ABCD$, line segments are drawn within the rhombus, parallel to diagonal $ BD$, and terminated in the sides of the rhombus. A graph is drawn showing the length of a segment as a function of its distance from vertex $ A$. The graph is: $ \textbf{(A)}\ \text{A straight line passing through the origin.} \\ \textbf{(B)}\ \text{A straight line cutting across the upper right quadrant.} \\ \textbf{(C)}\ \text{Two line segments forming an upright V.} \\ \textbf{(D)}\ \text{Two line segments forming an inverted V.} \\ \textbf{(E)}\ \text{None of these.}$

Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function.

Points $A_1, B_1, C_1$ inside an acute-angled triangle $ABC$ are selected on the altitudes from $A, B, C$ respectively so that the sum of the areas of triangles $ABC_1, BCA_1$, and $CAB_1$ is equal to the area of triangle $ABC$. Prove that the circumcircle of triangle $A_1B_1C_1$ passes through the orthocenter $H$ of triangle $ABC$.

The sequence of functions $f_0,f_1,f_2,...$ is given by the conditions: $f_0(x) = |x|$ for all $x \in R$ $f_{n+1}(x) = |f_n(x)-2|$ for $n=0,1,2,...$ and all $x \in R$. For each positive integer $n$, solve the equation $f_n(x)=1$.

In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2+10$, $10+2$, and $3+2+2+5$ are three such ways.

Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.