Found problems: 4776
Let $\mathbb{N} =\{1, 2, 3, ...\}$ be the set of positive integers. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a function that gives a positive integer value, to every positive integer. Suppose that $f$ satisfies the following conditions:
$f(1)=1$
$f(a+b+ab)=a+b+f(ab)$
Find the value of $f(2015)$
Proposed by Jose Antonio Gomez Ortega
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$.
If $(a,~b,~c)$ is a triple of real numbers, de fine
[list]
[*] $g(a,~b,~c)=(a+b,~b+c,~a+c)$, and
[*] $g^n(a,~b,~c)=g(g^{n-1}(a,~b,~c))$ for $n\ge 2$[/list]
Suppose that there exists a positive integer $n$ so that $g^n(a,~b,~c)=(a,~b,~c)$ for some $(a,~b,~c)\neq (0,~0,~0)$. Prove that $g^6(a,~b,~c)=(a,~b,~c)$
Find the largest integer $n$ for which $2^n$ divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \][i]Proposed by Evan Chen[/i]
Let $f: [0,1]\rightarrow \mathbb{R}$ continuous. We say that $f$ crosses the axis at $x$ if $f(x)=0$ but $\exists y,z \in [x-\epsilon,x+\epsilon]: f(y)<0<f(z)$ for any $\epsilon$.
(a) Give an example of a function that crosses the axis infinitely often.
(b) Can a continuous function cross the axis uncountably often?
Find all continuous functions $f : \mathbb R \to \mathbb R$ such that:
(a) $\lim_{x \to \infty}f(x)$ exists;
(b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.
Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions:
(i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$;
(ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$;
(iii) there exists a rational number $a>1$ such that $f(a)=a$.
Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$.
[i]Proposed by Bulgaria[/i]
The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous.
(A.I. Plotkin, Leningrad)
Let $a_i > 0$ for $i=1,2,\dots,n$ and suppose $a_1 + a_2 + \dots + a_n = 1$. Prove that for any positive integer $k$,
\[ \left( a_1^k + \frac{1}{a_1^k} \right) \left( a_2^k + \frac{1}{a_2^k} \right) \dots \left( a_n^k + \frac{1}{a_n^k} \right) \ge \left( n^k + \frac{1}{n^k} \right)^n. \]
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f^{(19)}(n)+97f(n)=98n+232.\]
Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)
For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50)=12$ because the prime factorization of $50$ is $ 2 \cdot 5^2 $, and $ 2 + 5 + 5 = 12 $. What is the range of the function $r$, $ \{ r(n) : n \ \text{is a composite positive integer} \} $?
[b](A)[/b] the set of positive integers
[b](B)[/b] the set of composite positive integers
[b](C)[/b] the set of even positive integers
[b](D)[/b] the set of integers greater than 3
[b](E)[/b] the set of integers greater than 4
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f(xy)=\max\{f(x+y),f(x) f(y)\} \]
for all real numbers $x$ and $y$.
Prove that there is no function $f: \mathbb{R} \rightarrow \mathbb{R}$ with $f(0) >0$, and such that
\[f(x+y) \geq f(x) +yf(f(x)) \text{ for all } x,y \in \mathbb{R}. \]
Let $a_1, a_2... $ be an infinite sequence of real numbers such that $a_{n+1}=\sqrt{{a_n}^2+a_n-1}$. Prove that $a_1 \notin (-2,1)$
[i]Proposed by Oleg Mushkarov and Nikolai Nikolov
[/i]
Determine all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that
\[ f(n)\plus{}f(n\plus{}1)\plus{}f(f(n))\equal{}3n\plus{}1, \quad \forall n\in \mathbb{N}.\]
Let $ a$, $ b$, $ c$ be positive reals such that $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$. Prove that $ \sum\frac1{4 \minus{} ab}\leq1$, where the $ \sum$ sign stands for cyclic summation.
[i]Alternative formulation:[/i] For any positive reals $ a$, $ b$, $ c$ satisfying $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$, prove the inequality
$ \frac{1}{4\minus{}bc}\plus{}\frac{1}{4\minus{}ca}\plus{}\frac{1}{4\minus{}ab}\leq 1$.
For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \]
Find all natural numbers $ n $ such that $ s(n) = 2010 $
Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$
\[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]
Define a sequence $\{a_n\}$ as: $\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.$
Prove that this sequence has a finite limit as $n\to+\infty$ . Also determine the limit.
Show that the system of equations
\begin{align*}
\lfloor x\rfloor^2+\lfloor y\rfloor &=0, \\
3x+y &=2,
\end{align*}
has infinitely many solutions and all these solutions satisfy bounds
\begin{align*}
0<\ &x <4, \\
-9\le\ &y\le 1.
\end{align*}
Prove that there exists a surjective function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ having the property that for all natural numbers $ n\ge 2, $ there exists an infinite set $ A_n $ such that $ f(x)=n, $ for all $ x\in A_n. $
Assume that $a,b$ are integers and $n$ is a natural number. $2^na+b$ is a perfect square for every $n$.Prove that $a=0$.
Prove that$$p (n)= 2+ \left (p (1) + \cdots + p\left ( \left [\frac {n}{2} \right ] + \chi_1 (n)\right ) + \left (p'_2(n) + \cdots + p' _{ \left [\frac {n}{2} \right ] - 1}(n)\right )\right )$$for every $n \in \mathbb {N}$ with $n>2$ where $\chi $ denotes the principal character Dirichlet modulo 2, i.e.$$ \chi _1 (n) = \begin{cases} 1 & \text{if } (n,2)=1 \\ 0 &\text{if } (n,2)>1 \end{cases} $$with $p (n) $ we denote number of possible partitions of $n $ and $p' _m(n) $ we denote the number of partitions of $n$ in exactly $m$ sumands.
4. Let $\mathbb{N}$ denote the strictly positive integers. A function $f$ : $\mathbb{N}$ $\to$ $\mathbb{N}$ has the following properties which hold for all $n \in$ $\mathbb{N}$:
a) $f(n)$ < $f(n+1)$;
b) $f(f(f(n)))$ = 4$n$
Find $f(2022)$.