Found problems: 4776
Find all reals $x$ satisfying $0 \le x \le 5$ and
$\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.
Let $f$ be a function on defined on $|x|<1$ such that $f\left (\tfrac1{10}\right )$ is rational and $f(x)= \sum_{i=1}^{\infty} a_i x^i $ where $a_i\in{\{0,1,2,3,4,5,6,7,8,9\}}$. Prove that $f$ can be written as $f(x)= \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials with integer coefficients.
Let $M(t)$ be measurable and locally bounded function, that is,
\[M(t) \le C_{a,b}, \quad \forall 0 \le a \le t \le b<\infty\]
with some constant $C_{a,b}$, from $[0,\infty)$ to $[0,\infty)$ such that
\[M(t) \le 1+\int_0^t M(t-s)(1+t)^{-1}s^{-1/2} ds, \quad \forall t \ge 0.\]
Show that
\[M(t) \le 10+2\sqrt{5}, \quad \forall t \ge 0.\]
Let $Z^+$ be positive integers set. $f:\mathbb{Z^+}\to\mathbb{Z^+}$ is a function and
we show $ f \circ f \circ ...\circ f $ with $f_l$ for all $l\in \mathbb{Z^+}$ where $f$ is repeated $l$ times. Find all $f:\mathbb{Z^+}\to\mathbb{Z^+}$ functions such that $$ (n-1)^{2020}< \prod _{l=1}^{2020} {f_l}(n)< n^{2020}+n^{2019} $$
for all $n\in \mathbb{Z^+}$
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]
for all $x,y\in\mathbb{R}$.
[i]Proposed by Carl Schildkraut[/i]
Let $g : \mathbb{N} \to \mathbb{N}$ be a function satisfying:
[list]
[*] $g(xy) = g(x)g(y)$ for all $x, y \in \mathbb{N}$,
[*] $g(g(x)) = x$ for all $x \in \mathbb{N}$, and
[*] $g(x) \neq x$ for $2 \leq x \leq 2018$.
[/list]
Find the minimum possible value of $g(2)$.
Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions:
(i) $f(1, 1) =1$,
(ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and
(iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$.
Find $f(1990, 31).$
Show that for any function $ f: (0,\plus{}\infty)\to (0,\plus{}\infty)$ there exist real numbers $ x>0$ and $ y>0$ such that:
$ f(x\plus{}y)<yf(f(x)).$
[b]Dan Schwarz[/b]
Ket $f(x) = x^{2} +ax + b$. If for all nonzero real $x$
$$f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right)$$
and the roots of $f(x) = 0$ are integers, what is the value of $a^{2}+b^{2}$?
Let $f:[0,1]\rightarrow R$ be a continuous function and n be an integer number,n>0.Prove that $\int_0^1f(x)dx \le (n+1)*\int_0^1 x^n*f(x)dx $
We define $N$ as the set of natural numbers $n<10^6$ with the following property:
There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$.
Find $|N|$.
Find the number of functions $f(x)$ from $\{1,2,3,4,5\}$ to $\{1,2,3,4,5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1,2,3,4,5\}$.
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
[i]Victor Wang.[/i]
Let $f(x)=\frac{\sqrt{2+\sqrt2}x+\sqrt{2-\sqrt2}}{-\sqrt{2-\sqrt2}x+\sqrt{2+\sqrt2}}$. Find $\underbrace{f(f(\cdots f}_{1987\text{ times}}(x)\cdots))$.
Suppose that $ f$ is a real-valued function for which \[ f(xy)+f(y-x)\geq f(y+x)\] for all real numbers $ x$ and $ y$.
a) Give a non-constant polynomial that satisfies the condition.
b) Prove that $ f(x)\geq 0$ for all real $ x$.
Given a sphere of unit radius with the big circle (i.e of unit radius) that will be called "equator". We shall use the words "pole", "parallel","meridian" as self-explanatory.
a) Let $g(x)$, where $x$ is a point on the sphere, be the distance from this point to the equator plane. Prove that $g(x)$ has the property if $x_1, x_2, x_3$ are the ends of the pairwise orthogonal radiuses, then $$g(x_1)^2 + g(x_2)^2 + g(x_3)^2 = 1 \,\,\,\, (*)$$
Let function $f(x)$ be an arbitrary nonnegative function on a sphere that satisfies (*) property.
b) Let $x_1$ and $x_2$ points be on the same meridian between the north pole and equator, and $x_1$ is closer to the pole than $x_2$. Prove that $f(x_1) > f(x_2)$.
c) Let $y_1$ be closer to the pole than $y_2$. Prove that $f(y_1) > f(y_2)$.
d) Let $z_1$ and $z_2$ be on the same parallel. Prove that $f(z_1) = f(z_2)$.
e) Prove that for all $x , f(x) = g(x)$.
Find all real numbers $x$ satisfying the equation
\[\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.\]
[b]Problem.[/b] Let $f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ be a function defined as $$f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.$$ Prove that for any positive integer $m,$ the sequence $$m,f(m),f(f(m)),f(f(f(m))),\ldots$$ contains a perfect square.
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.
For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows:
- $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$;
- $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$.
Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.
Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$.
[i]Proposed by Evan Chen[/i]
Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.
Let $ f: [0,1]\rightarrow \mathbb{R}$ a derivable function such that $ f(0)\equal{}f(1)$, $ \int_0^1f(x)dx\equal{}0$ and $ f^{\prime}(x) \neq 1\ ,\ (\forall)x\in [0,1]$.
i)Prove that the function $ g: [0,1]\rightarrow \mathbb{R}\ ,\ g(x)\equal{}f(x)\minus{}x$ is strictly decreasing.
ii)Prove that for each integer number $ n\ge 1$, we have:
$ \left|\sum_{k\equal{}0}^{n\minus{}1}f\left(\frac{k}{n}\right)\right|<\frac{1}{2}$
Let $f$ be the function defined by $f(x) = 4x(1 - x)$. Let $n$ be a positive integer. Prove that there exist distinct real numbers $x_1$, $x_2$, $\ldots\,$, $x_n$ such that $x_{i + 1} = f(x_i)$ for each integer $i$ with $1 \le i \le n - 1$, and such that $x_1 = f(x_n)$.
Consider functions $f$ satisfying the following four conditions:
(1) $f$ is real-valued and defined for all real numbers.
(2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$.
(3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$.
(4) We have $f(2)=4$.
Prove that:
a) There is a function $f$ with $f(3)=9$ satisfying the four conditions.
b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$.