Find all the functions $f: \mathbb R \rightarrow \mathbb R$ satisfying : $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y \in \mathbb R$

The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, it intersects the graph $y=f(x)$ at precisely one point, which is $\frac{1}{\sqrt{m}}$ units from the origin. Let $a$ be the unique real number for which $f$ takes on its maximum value at $x=a$ (you may assume that such an $a$ exists). Find $\int_{0}^{a}f(x) \, dx$.

Find all function $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that: \[f\left(\frac{f(x)}{x}+y\right)=1+f(y), \quad \forall x,y \in \mathbb R^+.\]

Given positive numbers $a,b,c$, find the minimum of the function $f(x) = \sqrt{a^2 +x^2} +\sqrt{(b-x)^2 +c^2}$.

Consider a countinuous function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R}_{>0} $ that verifies the following conditions: $ \text{(1)} x f(f(x))=(f(x))^2,\quad\forall x\in\mathbb{R}_{>0} $ $ \text{(2)} \lim_{\stackrel{x\to 0}{x>0}} \frac{f(x)}{x}\in\mathbb{R}\cup\{ \pm\infty \} $ [b]a)[/b] Show that $ f $ is bijective. [b]b)[/b] Prove that the sequences $ \left( (\underbrace{f\circ f\circ\cdots \circ f}_{\text{n times}} ) (x) \right)_{n\ge 1} ,\left( (\underbrace{f^{-1}\circ f^{-1}\circ\cdots \circ f^{-1}}_{\text{n times}} ) (x) \right)_{n\ge 1} $ are both arithmetic progressions, for any fixed $ x\in\mathbb{R}_{>0} . $ [b]c)[/b] Determine the function $ f. $ [i]Nelu Chichirim[/i]

Prove that in Euclidean ring $ R$ the quotient and remainder are always uniquely determined if and only if $ R$ is a polynomial ring over some field and the value of the norm is a strictly monotone function of the degree of the polynomial. (To be precise, there are two trivial cases: $ R$ can also be a field or the null ring.) [i]E. Fried[/i]

If the real function $f(x)=\cos x+\sum_{i=1}^{n}\cos(a_ix)$ is periodic, prove that $a_i,i\in\{1,2,...,n\}$, are rational numbers.

For constant $a$, find the differentiable function $f(x)$ satisfying $\int_0^x (e^{-x}-ae^{-t})f(t)dt=0$.

Define a function $f(n)$ from the positive integers to the positive integers such that $f(f(n))$ is the number of positive integer divisors of $n$. Prove that if $p$ is a prime, then $f(p)$ is prime.

Let $f:\mathbb{R}\to \mathbb{R}$ a function which transforms any closed bounded interval in a closed bounded interval and any open bounded interval in an open bounded interval. Prove that $f$ is continuous. [i]Mihai Piticari[/i]

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfies the followings. (Note that $\mathbb{R}$ stands for the set of all real numbers) (1) For each real numbers $x$, $y$, the equality $f(x+f(x)+xy) = 2f(x)+xf(y)$ holds. (2) For every real number $z$, there exists $x$ such that $f(x) = z$.

Find all functions $f: Q^{+}\rightarrow R$ such that $f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$ for all $x,y\in Q^{+}$

Inside or on the faces of a tetrahedron with five edges of length $2$ and one edge of lenght $1$, there is a point $P$ having distances $a, b, c, d$ to the four faces of the tetrahedron. Determine the locus of all points $P$ such that $a+b+c+d$ is minimal and the locus of all points $P$ such that $a+b+c+d$ is maximal.

Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find \[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]

Determine the highest power of $1980$ which divides \[\frac{(1980n)!}{(n!)^{1980}}.\]

Let $f: \mathbb R \to \mathbb R$ be a function which is differentiable at $0$. Define another function $g: \mathbb R \to \mathbb R$ as follows: $$g(x) = \begin{cases} f(x)\sin\left(\frac 1x\right) ~ &\text{if} ~ x \neq 0 \\ 0 &\text{if} ~ x = 0. \end{cases}$$ Suppose that $g$ is also differentiable at $0$. Prove that \[g'(0) = f'(0) = f(0) = g(0) = 0.\]

Compute \[ \displaystyle\sum_{n=0}^{\infty} \dfrac {n}{n^4 + n^2 + 1}. \]

Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true: $$f(x+y+f(y))=f(x)+2y$$ [I]proposed by F. Petrov[/i]

If $\sin x+\cos x=1/5$ and $0\le x<\pi$, then $\tan x$ is $\textbf{(A) }-\frac{4}{3}\qquad\textbf{(B) }-\frac{3}{4}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{3}\qquad$ $\textbf{(E) }\text{not completely determined by the given information}$

Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: \[ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . \] [i]Radu Gologan[/i]

For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$? $\textbf{(A) }0\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }10$

Consider the functions $ f(x) = ax^{2} + bx + c $ , $ g(x) = cx^{2} + bx + a $, where a, b, c are real numbers. Given that $ |f(-1)| \leq 1 $, $ |f(0)| \leq 1 $, $ |f(1)| \leq 1 $, prove that $ |f(x)| \leq \frac{5}{4} $ and $ |g(x)|  \leq 2 $ for $ -1 \leq  x \leq 1 $.

How many functions $f: \left\{1,2,3\right\} \to \left\{1,2,3 \right\}$ satisfy $f(f(x))=f(f(f(x)))$ for every $ x $?

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that, for all real numbers $x$ and $y$, $$f(x)(x+f(f(y))) = f(x^2)+xf(y)$$

Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$. Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$