Found problems: 10
Let $ a,b,c $ be three positive numbers. Show that the equation
$$ a^x+b^x=c^x $$
has, at most, one real solution.
Show that a continuous function $f : \mathbb{R} \to \mathbb{R}$ is increasing if and only if
\[(c - b)\int_a^b f(x)\, \text{d}x \le (b - a) \int_b^c f(x) \, \text{d}x,\]
for any real numbers $a < b < c$.
Let $\mathbb{Z}_{>0} = \{1, 2, 3, \ldots \}$ be the set of all positive integers. Find all strictly increasing functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $f(f(n)) = 3n$.
Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions:
(a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$;
(b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.
Let $f$ be a strictly increasing function defined in the set of natural numbers satisfying the conditions $f(2) = a >
2$ and $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. Determine the smallest possible value of $a$.
We say that a function $f : N \to N$ is increasing if $f(n) < f(m)$ whenever $n < m$, multiplicative if $f(nm) = f(n)f(m)$ whenever $n$ and $m$ are coprime, and completely multiplicative if $f(nm) = f(n)f(m)$ for all $n,m$.
(a) Prove that if $f$ is increasing then $f(n) \ge n$ for each $n$.
(b) Prove that if $f$ is increasing and completely multiplicative and $f(2) = 2$, then $f(n) = n$ for all $n$.
(c) Does (b) remain true if the word ”completely” is omitted?
Let $f : (0,1) \to R$ be a function such that $f(1/n) = (-1)^n$ for all n ∈ N.
Prove that there are no increasing functions $g,h : (0,1) \to R$ such that $f = g - h$.
Let $F$ be an increasing real function defined for all $x, 0 \le x \le 1$, satisfying the conditions
(i) $F (\frac{x}{3}) = \frac{F(x)}{2}$.
(ii) $F(1- x) = 1 - F(x)$.
Determine $F(\frac{173}{1993})$ and $F(\frac{1}{13})$ .
Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions:
(a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$;
(b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.
Assume that $f : [0,1)\to R$ is a function such that $f(x)-x^3$ and $f(x)-3x$ are both increasing functions. Determine if $f(x)-x^2-x$ is also an increasing function.