find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(-f(x)-f(y)) = 1-x-y$ $\quad \forall x,y \in \mathbb{Z}$

Find the index of the singular point $0$ of the vector field \[(xy+yz+xz)\]

Find the greatest real number $M$ for which \[ a^2+b^2+c^2+3abc \geq M(ab+bc+ca) \] for all non-negative real numbers $a,b,c$ satisfying $a+b+c=4.$

Let $f : \mathbb{R} \to \mathbb{R}$ be a function whose second derivative is continuous. Suppose that $f$ and $f''$ are bounded. Show that $f'$ is also bounded.

Let be a sequence of continuous functions $ \left( f_n \right)_{n\ge 1} :[0,1]\longrightarrow\mathbb{R} $ satisfying the following properties: $ \text{a) } $ for any natural $ n $ and $ x\in [1/n,1] ,$ it follows $ \left| f_n(x) \right|\leqslant 1/n. $ $ \text{b) } $ for any natural $ n, $ it follows $ \int_0^1 f_n^2(t)dt\leqslant 1. $ Then, $\lim_{n\to 0} \int_0^1\left| f_n(t) \right| dt=0 $ [i]Cristinel Mortici[/i]

Denote the set of all positive integers by $\mathbb{N}$, and the set of all ordered positive integers by $\mathbb{N}^2$. For all non-negative integers $k$, define [i]good functions of order k[/i] recursively for all non-negative integers $k$, among all functions from $\mathbb{N}^2$ to $\mathbb{N}$ as follows: (i) The functions $f(a,b)=a$ and $f(a,b)=b$ are both good functions of order $0$. (ii) If $f(a,b)$ and $g(a,b)$ are good functions of orders $p$ and $q$, respectively, then $\gcd(f(a,b),g(a,b))$ is a good function of order $p+q$, while $f(a,b)g(a,b)$ is a good function of order $p+q+1$. Prove that, if $f(a,b)$ is a good function of order $k\leq \binom{n}{3}$ for some positive integer $n\geq 3$, then there exist a positive integer $t\leq \binom{n}{2}$ and $t$ pairs of non-negative integers $(x_1,y_1),\ldots,(x_n,y_n)$ such that $$f(a,b)=\gcd(a^{x_1}b^{y_1},\ldots,a^{x_t}b^{y_t})$$ holds for all positive integers $a$ and $b$. [i]Proposed by usjl[/i]

Find all reals $x$ satisfying $0 \le x \le 5$ and $\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.

Find all functions $ f:\mathbb{R}\to\mathbb{R} $ such that $ f(f(x)^2+f(y)) = xf(x)+y $,$ \forall x,y\in R $.

Let $X=\{1,2,\dots,n\},$ and let $k\in X.$ Show that there are exactly $k\cdot n^{n-1}$ functions $f:X\to X$ such that for every $x\in X$ there is a $j\ge 0$ such that $f^{(j)}(x)\le k.$ [Here $f^{(j)}$ denotes the $j$th iterate of $f,$ so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$]

Let $S = \{2, 3, 4, \ldots\}$ denote the set of integers that are greater than or equal to $2$. Does there exist a function $f : S \to S$ such that \[f (a)f (b) = f (a^2 b^2 )\text{ for all }a, b \in S\text{ with }a \ne b?\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals.

Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

Let $A=\{1, 3, 5, 7, 9\}$ and $B=\{2, 4, 6, 8, 10\}$. Let $f$ be a randomly chosen function from the set $A\cup B$ into itself. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probablity that $f$ is a one-to-one function on $A\cup B$ given that it maps $A$ one-to-one into $A\cup B$ and it maps $B$ one-to-one into $A\cup B$. Find $m+n$.

Let $g:[2013,2014]\to\mathbb{R}$ a function that satisfy the following two conditions: i) $g(2013)=g(2014) = 0,$ ii) for any $a,b \in [2013,2014]$ it hold that $g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).$ Prove that $g$ has zeros in any open subinterval $(c,d) \subset[2013,2014].$

Find all real numbers $a$ such that there exists a function $f:\mathbb R\to \mathbb R$ such that the following conditions are simultaneously satisfied: (a) $f(f(x))=xf(x)-ax,\;\forall x\in\mathbb{R};$ (b) $f$ is not a constant function; (c) $f$ takes the value $a$.

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".

A strictly decreasing function $f:(0, \infty)\Rightarrow (0, \infty)$ attaining all positive values and positive numbers $a_1\ne b_1$ are given. Numbers $a_2, b_2, a_3, b_3, ...$ satisfy $$a_{n+1}=a_n+f(b_n),\;\;\;\;\;\;\;b_{n+1}=b_n+f(a_n)$$ for every $n\geq 1$. Prove that there exists a positive integer $n$ satisfying $|a_n-b_n| >2025$.

Let $f:[0, \infty) \to (0, \infty)$ be an increasing function and $g:[0, \infty) \to \mathbb{R}$ be a two times differentiable function such that $g''$ is continuous and $g''(x)+f(x)g(x) = 0, \: \forall x \geq 0.$ $\textbf{a)}$ Provide an example of such functions, with $g \neq 0.$ $\textbf{b)}$ Prove that $g$ is bounded.

Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n)) = f(m)+n.\]

Suppose that $f : \mathbb R^+ \to \mathbb R^+$ is a decreasing function such that \[f(x+y)+f(f(x)+f(y))=f(f(x+f(y))+f(y+f(x)), \quad \forall x,y \in \mathbb R^+.\] Prove that $f(x) = f^{-1}(x).$

Find all function $f:\mathbb{R}\to \mathbb{R}$ such that \[f(x)f(y)=f(xy-1)+yf(x)+xf(y)\] for all $x,y \in \mathbb{R}$

Let us define a function $f:\mathbb N\to\mathbb N_0$ by $f(1)=0$ and, for all $n\in\mathbb N$, $$f(2n)=2f(n)+1,\qquad f(2n+1)=2f(n).$$Given a positive integer $p$, define a sequence $(u_n)$ by $u_0=p$ and $u_{k+1}=f(u_k)$ whenever $u_k\ne0$. (a) Prove that, for each $p\in\mathbb N$, there is a unique integer $v(p)$ such that $u_{v(p)}=0$. (b) Compute $v(1994)$. What is the smallest integer $p>0$ for which $v(p)=v(1994)$. (c) Given an integer $N$, determine the smallest integer $p$ such that $v(p)=N$.

Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that $f(km)+f(kn)-f(k)f(mn)\ge 1$ for all $k,m,n\in\mathbb{N}$.

A quartic funtion $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.