Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

$a>0$ $f:[-a,a]\rightarrow R$ such that $f''$ exist and Riemann-integrable suppose $f(a)=f(-a)$ $ f'(-a)=f'(a)=a^2$ Prove that $6a^3\leq \int_{-a}^{a}{f''(x)}^2dx$ Study equality case ? Radu Miculescu

Let $a, b, c$, and $d$ be real numbers. Suppose that $\lfloor na\rfloor +\lfloor nb\rfloor =\lfloor nc\rfloor +\lfloor nd\rfloor $ for all positive integers $n$. Show that at least one of $a+b$, $a-c$, $a-d$ is an integer.

Let $n$ be a given integer greater than two, and let $S = \{1, 2,...,n\}$. Suppose the function $f : S^k \to S$ has the property that $f(a) \ne f(b)$ for every pair $a$ and $b$ of elements in $S^k$ with $a$ and $b$ differ in all components. Prove that $f$ is a function of one of its elements.

Prove that the function \[ f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}\] (where the positive value of the square root is taken) is monotonically decreasing in the interval $ 0<\nu<1$. [P. Turan]

$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$. [i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.

$\indent$ For a positive integer $n$, let $S_n$ be the set of bijective functions from $\{1,2,\dots ,n\}$ to itself. For a pair of positive integers $(a,b)$ such that $1 \leq a <b \leq n$, and for a permutation $\sigma \in S_n$, we say the pair $(a,b)$ is [i][u]expanding[/u][/i] for $\sigma$ if $|\sigma (a)- \sigma(b)| \geq |a-b|$ $\indent$ [b](a)[/b] Is it true that for all integers $n > 1$, there exists $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for permutation $\sigma$ is less than $1000n\sqrt n$ ? $\indent$ [b](b)[/b] Does there exist a positive integer $n>1$ and a permutation $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for the permutation $\sigma$ is less than $\frac{n\sqrt n}{1000}$?

\[\begin{array}{rcl} (x+y)^5 &=& z \\ (y+z)^5 &=& x \\ (z+x)^5 &=& y \end{array}\] How many real triples $(x,y,z)$ are there satisfying above equation system? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None} $

The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50. [i]Proposed by A. Khabrov[/i]

Does there exist a surjective function $f:R \to R$ such that the expression $f(x + y) - f(x) - f(y)$ takes exactly two values $0$ and $1$ for various real $x$ and $y$ ? (E. Barabanov)

Let $f : \mathbb{N} \longrightarrow \mathbb{N}$ be a function such that $(x + y)f (x) \le x^2 + f (xy) + 110$, for all $x, y$ in $\mathbb{N}$. Determine the minimum and maximum values of $f (23) + f (2011)$.

Find the folllowing limit. \[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]

Let $f$ be a function with the following properties: 1) $f(n)$ is defined for every positive integer $n$; 2) $f(n)$ is an integer; 3) $f(2)=2$; 4) $f(mn)=f(m)f(n)$ for all $m$ and $n$; 5) $f(m)>f(n)$ whenever $m>n$. Prove that $f(n)=n$.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x^3)-f(y^3)=(x^2+xy+y^2)(f(x)-f(y))$.

Let $n$ be a given integer with $n$ greater than $7$ , and let $\mathcal{P}$ be a convex polygon with $n$ sides. Any set of $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $\mathcal{P}$ into $n-2$ triangles. A triangle in the triangulation of $\mathcal{P}$ is an interior triangle if all of its sides are diagonals of $\mathcal{P}$. Express, in terms of $n$, the number of triangulations of $\mathcal{P}$ with exactly two interior triangles, in closed form.