Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$ satisfy, $$f\left(x+yf(x+y)\right)=y^2+f(x)f(y)$$ [i]Proposed by Dorlir Ahmeti, Kosovo[/i]

Let $f : N \to [0, \infty)$ be a function satisfying the following conditions: a) $f(4)=2$ b) $\frac{1}{f( 0 ) + f( 1)} + \frac{1}{f( 1 ) + f( 2 )} + ... + \frac{1}{f (n ) + f(n + 1) }= f ( n + 1)$ for all integers $n \ge 0$. Find $f(n)$ in closed form.

Find all non-constant polynoms $ f\in\mathbb{Q} [X] $ that don't have any real roots in the interval $ [0,1] $ and for which there exists a function $ \xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right) $ such that $ h_x(x)\neq 0 $ and $ \int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} , $ for all $ x\in [0,1] . $

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

Does there exist a piecewise linear function $f$ defined on the segment [$-1,1]$ (including the ends) such that $f(f(x)) = -x$ for all x? (A function is called piecewise linear if its graph is the union of a finite set of points and intervals; it may be discontinuous).

There are real numbers $a$ and $b$ such that for every positive number $x$, we have the identity \[ \tan^{-1} \bigl( \frac{1}{x} - \frac{x}{8} \bigr) + \tan^{-1}(ax) + \tan^{-1}(bx) = \frac{\pi}{2} \, . \] (Throughout this equation, $\tan^{-1}$ means the inverse tangent function, sometimes written $\arctan$.) What is the value of $a^2 + b^2$?

Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.

Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m\in \mathbb{Z}$: \[f(f(m))=m+1.\]

A function $f$ is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real $x$. If $x = 0$ is a root of $f(x) = 0$, what is the least number of roots $f(x) = 0$ must have in the interval $-1000 \le x \le 1000$?

Let functions $f, g: \mathbb{R}^+ \to \mathbb{R}^+$ satisfy the following: \[ f(g(x)y + f(x)) = (y+2015)f(x) \] for every $x,y \in \mathbb{R}^+$. (a) Prove that $g(x) = \frac{f(x)}{2015}$ for every $x \in \mathbb{R}^+. $ (b) State an example of function that satisfy the equation above and $f(x), g(x) \ge 1$ for every $x \in \mathbb{R}^+$.

In the Euclidean three-dimensional space, we call [i]folding[/i] of a sphere $S$ every partition of $S \setminus \{x,y\}$ into disjoint circles, where $x$ and $y$ are two points of $S$. A folding of $S$ is called [b]linear[/b] if the circles of the [i]folding[/i] are obtained by the intersection of $S$ with a family of parallel planes or with a family of planes containing a straight line $D$ exterior to $S$. Is every [i]folding[/i] of a sphere $S$ [b]linear[/b]?

Give an example of a sequence of continuous functions on $\mathbb R$ converging pointwise to $0$ which is not uniformly convergent on any nonempty open set.

[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 03[/b] In each square of a board drawn into squares of $2^n$ rows and $n$ columns $(n\geq 1)$ are written a 1 or a -1, in such a way that the rows of the board constitute all the possible sequences of length $n$ that they are possible to be formed with numbers 1 and -1. Next, some of the numbers are replaced by zeros. Prove that it is possible to choose some of the rows of the board (It could be a row only) so that in the chosen rows, is fulfilled that the sum of the numbers in each column is zero. ---- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88511]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$. [asy] size(200); pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C); draw(A--B--C--A--D^^C--F^^B--E); pair point=P; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$P$", P, dir(0));[/asy]

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which satisfies the following: [list][*] $f(m)=m$, for all $m\in\mathbb{Z}$;[*] $f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}$, for all $a, b, c, d\in\mathbb{Z}$ such that $|ad-bc|=1$, $c>0$ and $d>0$;[*] $f$ is monotonically increasing.[/list] (a) Prove that the function $f$ is unique. (b) Find $f(\frac{\sqrt{5}-1}{2})$.