Consider a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that for any third degree polynomial function $P:[0,1]\to [0,1]$, we have \[\int_0^1f(P(x))dx=0\] Prove that $f(x)=0,\ (\forall)x\in [0,1]$. [i]Mihai Piticari[/i]

Suppose a function $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x+y)|\geqslant|f(x)+f(y)|$ for all real numbers $x$ and $y$. Prove that equality always holds. Is the conclusion valid if the sign of the inequality is reversed?

Let $C$ be a real number, and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function such that $$ \lim_{x \to \infty} f(x)=C, \;\; \; \lim_{x \to \infty} f'''(x)=0.$$ Prove that $$ \lim_{x \to \infty} f'(x) =0 \;\; \text{and} \;\; \lim_{x \to \infty} f''(x)=0.$$

Let $f(x,y)$ and $g(x,y)$ be real valued functions defined for every $x,y \in \{1,2,..,2000\}$. If there exist $X,Y \subset \{1,2,..,2000\}$ such that $s(X)=s(Y)=1000$ and $x\notin X$ and $y\notin Y$ implies that $f(x,y)=g(x,y)$ than, what is the maximum number of $(x,y)$ couples where $f(x,y)\neq g(x,y)$.

Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that \[f(a)\cdot f^\prime(a)\cdot f^{\prime\prime}(a)\cdot f^{\prime\prime\prime}(a)\geq 0.\]

Let $a,b,c,d\in\mathbb{Z}_{\ge 0}$, $d\ne 0$ and the function $f:\mathbb{Z}_{\ge 0}\to\mathbb Z_{\ge 0}$ defined by \[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\] Prove that the following are equivalent: [list=1] [*] $f$ is surjective; [*] $c=0$, $b<d$ and $0<a\le d$. [/list] [i]Tiberiu Trif[/i]

Let $f, g : \mathbb R \to \mathbb R$ be two functions such that $g\circ f : \mathbb R \to \mathbb R$ is an injective function. Prove that $f$ is also injective.

Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-[i]limited[/i] if $ |\sigma(i)\minus{}i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k\plus{}1}.$

$f: \mathbb R\longrightarrow\mathbb R^{+}$ is a non-decreasing function. Prove that there is a point $a\in\mathbb R$ that \[f(a+\frac1{f(a)})<2f(a)\]

Let $ \{\phi_n(x) \}$ be a sequence of functions belonging to $ L^2(0,1)$ and having norm less that $ 1$ such that for any subsequence $ \{\phi_{n_k}(x) \}$ the measure of the set \[ \{x \in (0,1) : \;|\frac{1}{\sqrt{N}} \sum _{k=1}^N \phi_{n_k}(x)| \geq y\ \}\] tends to $ 0$ as $ y$ and $ N$ tend to infinity. Prove that $ \phi_n$ tends to $ 0$ weakly in the function space $ L^2(0,1).$ [i]F. Moricz[/i]

Let $ f:[2,\infty )\rightarrow\mathbb{R} $ be a differentiable function satisfying $ f(2)=0 $ and $$ \frac{df}{dx}=\frac{2}{x^2+f^4{x}} , $$ for any $ x\in [2,\infty ) . $ Show that there exists $ \lim_{x\to\infty } f(x) $ and is at most $ \ln 3. $ [i]Gabriel Daniilescu[/i]

It is given function $f(x)=3x-2$ $a)$ Find $g(x)$ if $f(2x-g(x))=-3(1+2m)x+34$ $b)$ Solve the equation: $g(x)=4(m-1)x-4(m+1)$, $m \in \mathbb{R}$

Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$. Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$

Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define \[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\] Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$ (Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)

Let $f$ be a real function defined in the interval $[0, +\infty )$ and suppose that there exist two functions $f', f''$ in the interval $[0, +\infty )$ such that \[f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0.\] Let $g$ be a function for which \[g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}.\] Prove that $g$ is bounded.

Let $ f$ be a real-valued function on the plane such that for every square $ ABCD$ in the plane, $ f(A)\plus{}f(B)\plus{}f(C)\plus{}f(D)\equal{}0.$ Does it follow that $ f(P)\equal{}0$ for all points $ P$ in the plane?

It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$. Find the value of $f(\cos x)$.

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Suppose that $V$ is a finite dimensional vector space over the real numbers equipped with an inner product and $S:V\times V \longrightarrow \mathbb R$ is a skew symmetric function that is linear for each variable when others are kept fixed. Prove there exists a linear transformation $T:V \longrightarrow V$ such that $\forall u,v \in V: S(u,v)=<u,T(v)>$. We know that there always exists $v\in V$ such that $W=<v,T(v)>$ is invariant under $T$. (it means $T(W)\subseteq W$). Prove that if $W$ is invariant under $T$ then the following subspace is also invariant under $T$: $W^{\perp}=\{v\in V:\forall u\in W <v,u>=0\}$. Prove that if dimension of $V$ is more than $3$, then there exist a two dimensional subspace $W$ of $V$ such that the volume defined on it by function $S$ is zero!!!! (This is the way that we can define a two dimensional volume for each subspace $V$. This can be done for volumes of higher dimensions.)

Let $f: [0,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $$f(x) = \int_{x - 1}^xf(t)\mathrm{d}t$$ for all $x \geq 1$. Show that $f$ has bounded variation on $[1,\infty)$, i.e. $$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$

For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that \[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]

Find all (weakly) increasing $f\colon \mathbb{R}\to \mathbb{R}$ for which \[f(f(x)+y)=f(f(y)+x)\] holds for all $x, y\in \mathbb{R}$.

Calculate the first term of the asymptotic expression as $k\to\infty$ of the integral \[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{\sqrt{1+x^{2n}}}dx\]

Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contribues $4''$ or $10''$ or $19''$ to the total height of the tower. How many differnt tower heights can be achieved using all 94 of the bricks?

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.