Found problems: 4776
Let $ f:[0,1]\longrightarrow (0,\infty ) $ be a continuous function and $ \left( b_n \right)_{n\ge 1} $ be a sequence of numbers from the interval $ (0,1) $ that converge to $ 0. $
[b]a)[/b] Demonstrate that for any fixed $ n, $ the equation $ F(x)=b_nF(1)+\left( 1-b_n\right) F(0) $ has an unique solution, namely $ x_n, $ where $ F $ is a primitive of $ f. $
[b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{x_n}{b_n} . $
Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$
Find all function $f,g: \mathbb{Q} \to \mathbb{Q}$ such that
\[\begin{array}{l}
f\left( {g\left( x \right) - g\left( y \right)} \right) = f\left( {g\left( x \right)} \right) - y \\
g\left( {f\left( x \right) - f\left( y \right)} \right) = g\left( {f\left( x \right)} \right) - y \\
\end{array}\]
for all $x,y \in \mathbb{Q}$.
Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation
\[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]
Let $\mathbb{Z}^+$ denote the set of all positive integers. Find all surjective functions $f:\mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ that satisfy all of the following conditions: for all $a,b,c \in \mathbb{Z}^+$,
(i)$f(a,b) \leq a+b$;
(ii)$f(a,f(b,c))=f(f(a,b),c)$
(iii)Both $\binom{f(a,b)}{a}$ and $\binom{f(a,b)}{b}$ are odd numbers.(where $\binom{n}{k}$ denotes the binomial coefficients)
Let $n(n>1)$ be an odd. We define $x_k=(x^{(k)}_1,x^{(k)}_2,\cdots ,x^{(k)}_n)$ as follow:
$x_0=(x^{(0)}_1,x^{(0)}_2,\cdots ,x^{(0)}_n)=(1,0,\cdots ,0,1)$;
$ x^{(k)}_i =\begin{cases}0, \quad x^{(k-1)}_i=x^{(k-1)}_{i+1},\\ 1, \quad x^{(k-1)}_i\not= x^{(k-1)}_{i+1},\end{cases} $
$i=1,2,\cdots ,n$, where $x^{(k-1)}_{n+1}= x^{(k-1)}_1$.
Let $m$ be a positive integer satisfying $x_0=x_m$. Prove that $m$ is divisible by $n$.
Find the smallest number among the following numbers:
$ \textbf{(A) }\sqrt{55}-\sqrt{52}\qquad\textbf{(B) }\sqrt{56}-\sqrt{53}\qquad\textbf{(C) }\sqrt{77}-\sqrt{74}\qquad\textbf{(D) }\sqrt{88}-\sqrt{85}\qquad\textbf{(E) }\sqrt{70}-\sqrt{67} $
Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]
Let $f: \mathbb{N} \to \mathbb{N}$ be an arbitrary function. Prove that there exist two positive integers $x$ and $y$ which satisfy $f(x+y) \le f(2x+f(y))$.
[i](Proposed by David Anghel, Romania)[/i]
Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$
Let $p$ be the [i]potentioral[/i] function, from positive integers to positive integers, defined by $p(1) = 1$ and $p(n + 1) = p(n)$, if $n + 1$ is not a perfect power and $p(n + 1) = (n + 1) \cdot p(n)$, otherwise. Is there a positive integer $N$ such that, for all $n > N,$ $p(n) > 2^n$?
For each positive integer $ n$, evaluate the sum
\[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]
Given any two real numbers $\alpha$ and $\beta , 0 \leq \alpha < \beta \leq 1$, prove that there exists a natural number $m$ such that
\[\alpha < \frac{\phi(m)}{m} < \beta.\]
Let $n\geq 2$ and $\mathcal{M}$ be a subset of $S_n$ with at least two elements, and which is closed under composition. Consider a function $f:\mathcal{M}\rightarrow\mathbb{R}$ which satisfies $$|f(\sigma\tau)-f(\sigma)-f(\tau)|\leq 1,$$ for all $\sigma,\tau\in\mathcal{M}$. Prove that $$\max_{\sigma,\tau\in\mathcal{M}}|f(\sigma)-f(\tau)|\leq 2-\frac{2}{|\mathcal{M}|}.$$
Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.
A function $f(x)$ satisfies $\begin{cases} f(x)=-f''(x)-(4x-2)f'(x)\\ f(0)=a,\ f(1)=b \end{cases}$
Evaluate $\int_0^1 f(x)(x^2-x)\ dx.$
Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies
\[
f(g(x)y + f(x)) = (y+2015)f(x)
\]
for every $x,y \in \mathbb{R^+} $
a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$
b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$
Determine, whether there exist a function $f$ defined on the set of all positive real numbers and taking positive values such that $f(x+y)\geqslant yf(x)+f(f(x))$ for all positive x and y?
Prove that
\[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\]
for all positive real numbers $x,y$ and $z$.
Let $\mathbb{N}^*=\{1,2,\ldots\}$. Find al the functions $f: \mathbb{N}^*\rightarrow \mathbb{N}^*$ such that:
(1) If $x<y$ then $f(x)<f(y)$.
(2) $f\left(yf(x)\right)=x^2f(xy)$ for all $x,y \in\mathbb{N}^*$.
Find all pairs of function $f : Q \rightarrow R$ and $g : Q \rightarrow R,$ for which equations
\begin{align*}
f(x+y) &= f(x) f(y) + g(x) g(y) \\
g(x+y) &= f(x)g(y) + g(x)f(y) + g(x)g(y)
\end{align*}
holds for all rational numbers $x$ and $y.$
[i]Proposed by Gurgen Asatryan, Armenia [/i]
Decide whether exists function $f: \mathbb{N} \rightarrow \mathbb{N}$, such that for each $n \in \mathbb{N}$ is $f(f(n) )= 2n$.
Evaluate $\int_0^1\frac{\ln(x+1)}{x^2+1}\,dx.$
Let $f$ be a function such that $f(0)=1$, $f^\prime (0)=2$, and \[f^{\prime\prime}(t)=4f^\prime(t)-3f(t)+1\] for all $t$. Compute the $4$th derivative of $f$, evaluated at $0$.
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.