Found problems: 313
A function $f: R \to R$ satisfies the conditions:
$f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$.
Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.
Prove that there is no continuous function $ f: \mathbb{R} \to \mathbb{R} $ satisfying the condition $ f(f(x)) = - x $ for every $ x $.
Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$. Also $f(10) = 1$. How many three digit numbers $n$ satisfy $f(n) = f(N)$, where $N$ is the "tower" $2, 3, 4, 5$, in other words, it is $2^a$, where $a = 3^b$, where $b = 4^5$?
Determine all functions $f : R \to R$ such that $f(xy + f(x)) = xf(y) + f(x)$ holds for any $x, y \in R$.
Find all functions $f : R \to R$ such that
$$2f(x)f(x + y) -f(x^2) =\frac{x}{2}(f(2x) + 4f(f(y)))$$
for all $x, y \in R$.
Determine all strictly increasing functions $f: R \to R$ satisfying $f (f(x) + y) = f(x + y) + f (0)$ for all $x,y \in R$.
Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.
Let $N =\{1, 2, 3, ...\}$ be the set of all natural numbers and $f : N\to N$ be a function.
Suppose $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(2n) + 1$ for all natural numbers $n$.
(i) Calculate the maximum value $M$ of $f(n)$ for $n \in N$ with $1 \le n \le 1994$.
(ii) Find all $n \in N$, with 1 \le n \le 1994, such that $f(n) = M$.
A function $f : Z \to Z$ is given so that $f(m + n) = f(m) + f(n) + 2mn - 2548$ for all positive integers $m, n$. Given that $f(2548) = -2548$, find the value of $f(2)$.
Determine all polynomials $P$ such that for every real number $x$, $P(x)^2 +P(-x) = P(x^2)+P(x)$
Find all functions $f : R \to R$ such that for all real $x,y$, $f(f(x)+y) = f(x^2 -y)+4y f(x)$
Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.
Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.
a) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \in N$?
b) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \in N$?
I. Voronovich
Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions:
a) $f (a b) = f (a) + f (b)$,
b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$,
c) $f (10) = 0$.
Find $f (2016).$
Show that there exists no function $f : Z \to Z$ such that for all $m, n \in Z$
$$f(m + f(n)) = f(m) - n.$$
The function $g : [0, \infty) \to [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$, for all $x \ge 0$.
You are also told that for $2 \le x \le 3$: $g(x) = \frac{x + 1}{2}$.
(a) Find $g(2021)$.
(b) Find $g(1/2021)$.
Find all functions satisfying the equality $$(x-1)f \left(\dfrac{x+1}{x-1}\right)- f(x) = x$$ for all $x \ne 1$.
Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.
For which integers $k$ does there exist a function $f : N \to Z$ such that
$f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$?
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.
Indicate (justifying your answer) if there exists a function $f: R \to R$ such that for all $x \in R$ fulfills that
i) $\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)$
ii) $f (f(x)) = f(x)$
where $\{m\}$ denotes the fractional part of $m$. That is, $\{2.657\} = 0.657$, and $\{-1.75\} = 0.25$.
Let $a > 0$ be a real number and let $f : R \to R$ be a function satisfying $$f(x_1) + f(x_2) \ge a f(x_1 + x_2), \forall x_1 ,x_2 \in R.$$ Prove that $$f(x_1) + f(x_2) +(x_3) \ge \frac{3a^2}{a+2} f(x_1+ x_2 + x_3), \forall x_1 ,x_2,x_3 \in R$$.