If $F(\frac{1-x}{1+x})=x$, then $\text{(A)}F(-2-x)=-2-F(x)\qquad\text{(B)}F(-x)=F(\frac{1+x}{1-x})$ $\text{(C)}F(\frac{1}{x})=F(x)\qquad\text{(D)}F(F(-x))=-x$

For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.

Find all $k>0$ such that there exists a function $f : [0,1]\times[0,1] \to [0,1]$ satisfying the following conditions: $f(f(x,y),z)=f(x,f(y,z))$; $f(x,y) = f(y,x)$; $f(x,1)=x$; $f(zx,zy) = z^{k}f(x,y)$, for any $x,y,z \in [0,1]$

Suppose that $ \sin a \plus{} \sin b \equal{} \sqrt {\frac {5}{3}}$ and $ \cos a \plus{} \cos b \equal{} 1.$ What is $ \cos(a \minus{} b)?$ $ \textbf{(A)}\ \sqrt {\frac {5}{3}} \minus{} 1 \qquad \textbf{(B)}\ \frac {1}{3}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {2}{3}\qquad \textbf{(E)}\ 1$

Let $\displaystyle a,b,c,d \in \mathbb R$ such that $\displaystyle a>c>d>b>1$ and $\displaystyle ab>cd$. Prove that $\displaystyle f : \left[ 0,\infty \right) \to \mathbb R$, defined through \[ \displaystyle f(x) = a^x+b^x-c^x-d^x, \, \forall x \geq 0 , \] is strictly increasing.

[b]interesting function[/b] $S$ is a set with $n$ elements and $P(S)$ is the set of all subsets of $S$ and $f : P(S) \rightarrow \mathbb N$ is a function with these properties: for every subset $A$ of $S$ we have $f(A)=f(S-A)$. for every two subsets of $S$ like $A$ and $B$ we have $max(f(A),f(B))\ge f(A\cup B)$ prove that number of natural numbers like $x$ such that there exists $A\subseteq S$ and $f(A)=x$ is less than $n$. time allowed for this question was 1 hours and 30 minutes.

Let $k\ge 1$ be a positive integer. We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip. Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.

Let $g:(0,\infty) \rightarrow (0,\infty)$ be a differentiable function whose derivative is continuous, and such that $g(g(x)) = x$ for all $x> 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.

$a,b,c \geq-3/4$ and $a+b+c=1$. Show that: $\frac{a}{1+a^{2}}+\frac{b}{1+b^{2}}+\frac{c}{1+c^{2}}\leq \frac{9}{10}$

Let $ a_0$ be a natural number. The sequence $ (a_n)$ is defined by $ a_{n\plus{}1}\equal{}\frac{a_n}{5}$ if $ a_n$ is divisible by $ 5$ and $ a_{n\plus{}1}\equal{}[a_n \sqrt{5}]$ otherwise . Show that the sequence $ a_n$ is increasing starting from some term.

[color=darkred]Determine all functions $ f : [0; \plus{} \infty) \rightarrow [0; \plus{} \infty)$, such that \[ f(x \plus{} y \minus{} z) \plus{} f(2\sqrt {xz}) \plus{} f(2\sqrt {yz}) \equal{} f(x \plus{} y \plus{} z)\] for all $ x,y,z \in [0; \plus{} \infty)$, for which $ x \plus{} y\ge z$.[/color]

Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution $ f(t)$ and any real number $ c,f(t\plus{}c)$ is also a solution.

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying \[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \] for all positive integer $m,n$.

Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$.

$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$ prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$

Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.