Found problems: 32
Given a function $ f:\mathbb{N}\longrightarrow\mathbb{N} , $ find the necessary and sufficient condition that makes the sequence
$$ \left(\left( 1+\frac{(-1)^{f(n)}}{n+1} \right)^{(-1)^{-f(n+1)}\cdot(n+2)}\right)_{n\ge 1} $$
to be monotone.
Find all monotonic functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that
$$ (f(\sin x))^2-3f(x)=-2, $$
for any real numbers $ x. $
[i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]
Solve in the real numbers the system:
$$ \left\{ \begin{matrix} x^2+7^x=y^3\\x^2+3=2^y \end{matrix} \right. $$
[i]Eduard Buzdugan[/i]
Consider a sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ and a primitivable function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $
[b]a)[/b] Prove that $ f $ is monotonic and continuous if for any natural numbers $ n $ and real numbers $ x, $ the inequality
$$ f\left( x+x_n \right)\geqslant f(x) $$
is true.
[b]b)[/b] Show that $ f $ is convex if for any natural numbers $ n $ and real numbers $ x, $ the inequality
$$ f\left( x+2x_n \right) +f(x)\geqslant 2f\left( x+x_n \right) $$
is true.
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
Solve in $ \mathbb{R}^3 $ the system:
$$ \left\{ \begin{matrix} 3^x+4^x=5^y \\8^y+15^y=17^z \\ 20^z+21^z=29^x \end{matrix} \right. $$
[i]Cristinel Mortici[/i]
[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality
$$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$
holds.
[b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?
Solve in the positive real numbers the following system.
$$ \left\{\begin{matrix} x^y=2^3\\y^z=3^4\\z^x=2^4 \end{matrix}\right. $$
[i]Aurel Ene[/i]
Solve in the reals the following system.
$$ \left\{ \begin{matrix} \log_2|x|\cdot\log_2|y| =3/2 \\x^2+y^2=12 \end{matrix} \right. $$
[i]Gheorghe Marchitan[/i]
Solve in the reals the equation $ 2^{\lfloor\sqrt[3]{x}\rfloor } =x. $
[i]Nedelcu Ion[/i]
Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.
Let be a number $ x $ and three positive numbers $ a,b,c $ such that $ a^x+b^x=c^x. $
Prove that $ a^y,b^y,c^y $ are the lenghts of the sides of an obtuse triangle if and only if $ y<x<2y. $
Let $\mathbb{N}_0$ denote the set of non-negative integers. Determine all non-negative integers $k$ for which there exists a function $f: \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and $f(f(n)) \leq f(n+1) - f(n)$ for all non-negative integers $n$.
Let be an even natural number $ n $ and a function $ f:[0,\infty )\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\int_0^x \prod_{k=0}^n (s-k) ds. $$
Show that
[b]a)[/b] $ f(n)=0. $
[b]b)[/b] $ f $ is globally nonnegative.
[i]Gheorghe Grigore[/i]
let P be a finite set with at least 2 elements. P is a partially ordered and connected set. $p:P^3 \to P$ is a 3-variable, monotone function which satisfies p(x,x,y)=y. Prove that there exists a non-empty subset $I \subset P$ such that $\forall x \in P$ $\forall y \in I$, we have $p(x, y, y) \in I$.
[P is connected means that if each element is replaced by vertices and there is an edge between 2 vertices iff the 2 elements can be compared, then the graph is connected.
p is monotone means that if $x_1\leq y_1 , x_2\leq y_2 , x_3\leq y_3$ , then $p(x_1,x_2,x_3)\leq p(y_1,y_2,y_3)$.]
Solve in the real numbers the equation:
$$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$
[i]Gheorghe Mihai[/i]
Construct a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following properties:
$ \text{(i)} f $ is not monotonic on any real interval.
$ \text{(ii)} f $ has Darboux property (intermediate value property) on any real interval.
$ \text{(iii)} f(x)\leqslant f\left( x+1/n \right) ,\quad \forall x\in\mathbb{R} ,\quad \forall n\in\mathbb{N} $
[i]Alexandru Cioba[/i]
A function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has property $ \mathcal{F} , $ if for any real number $ a, $ there exists a $ b<a $ such that $ f(x)\le f(a), $ for all $ x\in (b,a) . $
[b]a)[/b] Give an example of a function with property $ \mathcal{F} $ that is not monotone on $ \mathbb{R} . $
[b]b)[/b] Prove that a continuous function that has property $ \mathcal{F} $ is nondecreasing.
Let be the sequence $ \left( I_n \right)_{n\ge 1} $ defined as $ I_n=\int_0^{\pi } \frac{dx}{x+\sin^n x +\cos^n x} . $
[b]a)[/b] Study the monotony of $ \left( I_n \right)_{n\ge 1} . $
[b]b)[/b] Calculate the limit of $ \left( I_n \right)_{n\ge 1} . $
We know that the function $ f: \left[ 0,\frac{\pi }{2}\right]\longrightarrow [a,b], f(x)=\sqrt[n]{\cos x } +\sqrt[n]{\sin x} , $ is surjective for a given natural number $ n\ge 2. $ Determine the numbers $ a,b, $ and the monotony of $ f. $
Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ Show that:
[b]a)[/b] $ f $ is nondecreasing, if $ f+g $ is nondecreasing for any increasing function $ g:\mathbb{R}\longrightarrow\mathbb{R} . $
[b]b)[/b] $ f $ is nondecreasing, if $ f\cdot g $ is nondecreasing for any increasing function $ g:\mathbb{R}\longrightarrow\mathbb{R} . $
[i]Cristian Mangra[/i]
Solve in the real numbers the equation $ \sqrt[5]{2^x-2^{-1}} -\sqrt[5]{2^x+2^{-1}} =-1. $
[i]Mihai Neagu[/i]
Let be a twice-differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the properties that:
$ \text{(i) supp} f''=f\left(\mathbb{R}\right) $
$ \text{(ii)}\exists g:\mathbb{R}\longrightarrow\mathbb{R}\quad\forall x\in\mathbb{R}\quad f(x+1)=f(x)+f'\left( g(x)\right)\text{ and } f'(x+1)=f'(x)+f''\left( g(x)\right) $
Prove that:
[b]a)[/b] any such $ g $ is injective.
[b]b)[/b] $ f $ is of class $ C^{\infty } , $ and for any natural number $ n, $ any real number $ x $ and any such $ g, $
$$f^{(n)}(x+1)=f^{(n)}(x)+f^{(n+1)}\left( g(x)\right) . $$
[i]Laurențiu Panaitopol[/i]
Let be a function $ f:(0,\infty )\longrightarrow\mathbb{R} $ satisfying the following two properties:
$ \text{(i) } 2\lfloor x \rfloor \le f(x) \le 2 \lfloor x \rfloor +2,\quad\forall x\in (0,\infty ) $
$ \text{(ii) } f\circ f $ is monotone
Can $ f $ be non-monotone? Justify.
Solve in the real numbers the equation $ \arctan\sqrt{3^{1-2x}} +\arctan {3^x} =\frac{7\pi }{12} . $
[i]Ovidiu Țâțan[/i]
Prove that for any real numbers $ a,b,c, $ the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ f(x)=\sqrt{(x-c)^2+b^2} +\sqrt{(x+c)^2+b^2} $$ is decreasing on $ (-\infty ,0] $ and increasing on $ [0,\infty ) . $