Found problems: 4776
Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions
\[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\]
for all $x \in \mathbb{Q^+}.$
For how many integers $1\leq n\leq 9999$ is there a solution to the congruence \[\phi(n)\equiv 2\,\,\,\pmod{12},\] where $\phi(n)$ is the Euler phi-function?
Let be a continuous function $ f:[0,1]\longrightarrow\left( 0,\infty \right) $ having the property that, for any natural number $ n\ge 2, $ there exist $ n-1 $ real numbers $ 0<t_1<t_2<\cdots <t_{n-1}<1, $ such that
$$ \int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt. $$
Calculate $ \lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} . $
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for every pair of real numbers $x$ and $y$,
\[f(x+y^2)=f(x)+|yf(y)|.\]
An infinite sequence of real numbers $a_1,a_2,a_3,\dots$ is called $\emph{spooky}$ if $a_1=1$ and for all integers $n>1$,
\[\begin{array}{c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c}
na_1&+&(n-1)a_2&+&(n-2)a_3&+&\dots&+&2a_{n-1}&+&a_n&<&0,\\
n^2a_1&+&(n-1)^2a_2&+&(n-2)^2a_3&+&\dots&+&2^2a_{n-1}&+&a_n&>&0.
\end{array}\]Given any spooky sequence $a_1,a_2,a_3,\dots$, prove that
\[2013^3a_1+2012^3a_2+2011^3a_3+\cdots+2^3a_{2012}+a_{2013}<12345.\]
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$, such that for any real numbers $x,y$ with $y\neq 0$ we have $$f(f(x)+y)f\left(\frac{1}{y}\right)=xf\left(\frac{1}{y}\right) + 1.$$
[i]Proposed by Marius Cerlat[/i]
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuously differentiable function. Prove that
\[\left\vert f(1)-\int_0^1 f(x) dx\right\vert \le \frac{1}{2} \max_{x \in [0,1]} \vert f'(x)\vert.\]
Let $k,n$ and $r$ be positive integers.
(a) Let $Q(x)=x^k+a_1x^{k+1}+\cdots+a_nx^{k+n}$ be a polynomial with real coefficients. Show that the function $\frac{Q(x)}{x^k}$ is strictly positive for all real $x$ satisfying
$$0<|x|<\frac1{1+\sum\limits_{i=1}^n |a_i|}$$
(b) Let $P(x)=b_0+b_1x+\cdots+b_rx^r$ be a non zero polynomial with real coefficients. Let $m$ be the smallest number such that $b_m \neq 0$. Prove that the graph of $y=P(x)$ cuts the $x$-axis at the origin (i.e., $P$ changes signs at $x=0$) if and only if $m$ is an odd integer.
Find all functions $f\colon R \to R$ such that
\[f\left(x^{2}+yf(x)\right) = f(x)^{2}+xf(y)\]
for all reals $x,y$.
Let $M$ be the set of all functions $f$ with the following properties:
[b](i)[/b] $f$ is defined for all real numbers and takes only real values.
[b](ii)[/b] For all $x, y \in \mathbb R$ the following equality holds: $f(x)f(y) = f(x + y) + f(x - y).$
[b](iii)[/b] $f(0) \neq 0.$
Determine all functions $f \in M$ such that
[b](a)[/b] $f(1)=\frac 52$,
[b](b)[/b] $f(1)= \sqrt 3$.
Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$.
[i](Theresia Eisenkoelbl)[/i]
Let ${\mathbb Q}^{+}$ be the set of positive rational numbers. Construct a function $f:{\mathbb Q}^{+}\rightarrow{\mathbb Q}^{+}$ such that \[f(xf(y)) = \frac{f(x)}{y}\] for all $x, y \in{\mathbb Q}^{+}$.
In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d$ are positive integers and $d$ is not divisible by the square of any prime number. Find $m+n+d.$
Prove that for any $a$ the function $y (x) = \cos (\cos x) + a \cdot \sin (\sin x)$ is periodic.
Find its smallest period in terms of $a$.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any $(x, y)$ real numbers we have
$f(xf(y)+f(x))+f(y^2)=f(x)+yf(x+y)$
Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.
Let $I\subset \mathbb{R}$ be an interval and $f(x)$ a continuous real-valued function on $I$. Let $y_1$ and $y_2$ be linearly independent solutions of $y''=f(x)y$ taking positive values on $I$. Show that for some positive number $k$ the function $k\cdot\sqrt{y_1 y_2}$ is a solution of $y''+\frac{1}{y^{3}}=f(x)y$.
Given a positive, monotone function $ F(x)$ on $ (0, \infty)$ such that $ F(x)/x$ is monotone nondecreasing and $ F(x)/x^{1+d}$ is monotone nonincreasing for some positive $ d$, let $ \lambda_n >0$ and $ a_n \geq 0 , \;n \geq 1$. Prove that if \[ \sum_{n=1}^{\infty} \lambda_n F \left( a_n \sum _{k=1}^n \frac{\lambda_k}{\lambda_n} \right) < \infty,\] or \[ \sum_{n=1}^{\infty} \lambda_n F \left( \sum _{k=1}^n a_k \frac{\lambda_k}{\lambda_n} \right) < \infty,\] then $ \sum_{n=1}^ {\infty} a_n$ is convergent.
[i]L. Leindler[/i]
In $ \triangle ABC, \ \cos(2A \minus{} B) \plus{} \sin(A\plus{}B) \equal{} 2$ and $ AB\equal{}4.$ What is $ BC?$
$ \textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ \sqrt{3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{3}$
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$.
(For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)
Let \(\mathbb{Z}^2\) denote the set of points in the Euclidean plane with integer coordinates. Find all functions \(f : \mathbb{Z}^2 \to [0,1]\) such that for any point \(P\), the value assigned to \(P\) is the average of all the values assigned to points in \(\mathbb{Z}^2\) whose Euclidean distance from \(P\) is exactly 2020.
Let $k\ge2$ be an integer. The function $f:\mathbb N\to\mathbb N$ is defined by
$$f(n)=n+\left\lfloor\sqrt[k]{n+\sqrt[k]n}\right\rfloor.$$Determine the set of values taken by the function $f$.
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 $12$?
$\textbf{(A) }0\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }10$
Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that:
a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$.
b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.
Suppose that for every $n$ the number $m(n)$ is chosen such that $m(n)\ln(m(n))=n-\frac 12$. Show that $b_n$ is asymptotic to the following expression where $b_n$ is the $n-$th Bell number, that is the number of ways to partition $\{1,2,\ldots,n\}$:
\[ \frac{m(n)^ne^{m(n)-n-\frac 12}}{\sqrt{\ln n}}. \]
Two functions $f(n)$ and $g(n)$ are asymptotic to each other if $\lim_{n\rightarrow \infty}\frac{f(n)}{g(n)}=1$.