Found problems: 4776
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?
Let $ m$ a positive integer and $ p$ a prime number, both fixed. Define $ S$ the set of all $ m$-uple of positive integers $ \vec{v} \equal{} (v_1,v_2,\ldots,v_m)$ such that $ 1 \le v_i \le p$ for all $ 1 \le i \le m$. Define also the function $ f(\cdot): \mathbb{N}^m \to \mathbb{N}$, that associates every $ m$-upla of non negative integers $ (a_1,a_2,\ldots,a_m)$ to the integer $ \displaystyle f(a_1,a_2,\ldots,a_m) \equal{} \sum_{\vec{v} \in S} \left(\prod_{1 \le i \le m}{v_i^{a_i}} \right)$.
Find all $ m$-uple of non negative integers $ (a_1,a_2,\ldots,a_m)$ such that $ p \mid f(a_1,a_2,\ldots,a_m)$.
[i](Pierfrancesco Carlucci)[/i]
Find all functions from the reals to the reals satisfying
\[f(xf(y) + x) = xy + f(x)\]
Find the $ \mathcal{C}^1 $ class functions $ f:[0,2]\longrightarrow\mathbb{R} $ having the property that the application $ x\mapsto e^{-x} f(x) $ is nonincreasing on $ [0,1] , $ nondecreasing on $ [1,2] , $ and satisfying
$$ \int_0^2 xf(x)dx=f(0)+f(2) . $$
[i]Cristinel Mortici[/i]
Let $f(x)$ be a continuous function defined on the closed interval $0 \leq x \leq 1$. Let $G(f)$ denote the graph of $f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leq$$ x \leq 1, y = f(x) \}$. Let $G_a(f)$ denote the graph of the translated function $f(x - a)$ (translated over a distance $a$), defined by $G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}$. Is it possible to find for every $a, \ 0 < a < 1$, a continuous function $f(x)$, defined on $0 \leq x \leq 1$, such that $f(0) = f(1) = 0$ and $G(f)$ and $G_a(f)$ are disjoint point sets ?
Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.
How many polynomial functions $f$ of degree $\ge 1$ satisfy
\[ f(x^2)=[f(x)]^2=f(f(x)) \ ? \]
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \text{finitely many but more than 2} \\ \qquad\textbf{(E)}\ \text{infinitely many} $
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that:
$$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$
For a quadratic trinomial $f(x)$ and the different numbers $a$ and $b$ it is known that $f(a)=b$ and $f(b)=a$. We call such $a$ and $b$ [i]conjugate[/i] for $f(x)$. Prove that $f(x)$ has no other [i]conjugate[/i] numbers.
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$.
Prove that
$ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$.
The function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(x^2)f^{\prime\prime}(x)=f^\prime (x)f^\prime (x^2)$ for all real $x$. Given that $f(1)=1$ and $f^{\prime\prime\prime}(1)=8$, determine $f^\prime (1)+f^{\prime\prime}(1)$.
A set $G$ with elements $u,v,w...$ is a Group if the following conditions are fulfilled:
$(\text{i})$ There is a binary operation $\circ$ defined on $G$ such that $\forall \{u,v\}\in G$ there is a $w\in G$ with $u\circ v = w$.
$(\text{ii})$ This operation is associative; i.e. $(u\circ v)\circ w = u\circ (v\circ w)$ $\forall\{u,v,w\}\in G$.
$(\text{iii})$ $\forall \{u,v\}\in G$, there exists an element $x\in G$ such that $u\circ x = v$, and an element $y\in G$ such that $y\circ u = v$.
Let $K$ be a set of all real numbers greater than $1$. On $K$ is defined an operation by $ a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}$. Prove that $K$ is a Group.
Let $g: \mathbb{C} \rightarrow \mathbb{C}$ be a function, $w\in\mathbb{C}$ and $w^3=1$. Show that there exists a function $f:\mathbb{C} \rightarrow \mathbb{C}$ such that $f(z)+f(wz+a)=g(z), \forall z\in\mathbb{C}$. When there is an unique function $f$ with this property? Find it.
Let $\mathbb R^+$ denote the set of positive real numbers.Find all functions $f:\mathbb R^+\rightarrow \mathbb R^+$ satisfying the following equation for all $x,y\in \mathbb R^+$:
$$f(xy+f(y)^2)=f(x)f(y)+yf(y)$$
A natural number $k$ is said $n$-squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$, the smallest natural $k$ that is $n$-squared.
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$.
\[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]
A function $f:[0,\infty)\to[0,\infty)$ is integrable and
$$\int_0^\infty f(x)^2 dx<\infty,\quad \int_0^\infty xf(x) dx <\infty$$
Prove the following inequality.
$$\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)$$
Prove the following inequalities:
(1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$
(2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$
In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
$\textbf{(A) } 9
\qquad\textbf{(B) } 11
\qquad\textbf{(C) } 12
\qquad\textbf{(D) } 13
\qquad\textbf{(E) } 15
$
The function $f(x) = x^2+ax+b$ has two distinct zeros. If $f(x^2+2x-1)=0$ has four distinct zeros $x_1<x_2<x_3<x_4$ that form an arithmetic sequence, compute the range of $a-b$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 11)[/i]
Suppose that $ x_1, x_2, x_3, \ldots$ are positive real numbers for which \[ x^n_n \equal{} \sum^{n\minus{}1}_{j\equal{}0} x^j_n\] for $ n \equal{} 1, 2, 3, \ldots$ Prove that $ \forall n,$ \[ 2 \minus{} \frac{1}{2^{n\minus{}1}} \leq x_n < 2 \minus{} \frac{1}{2^n}.\]
Let $A= \{a_1,a_2, \cdots, a_n \}$ and $B=\{b_1,b_2 \cdots, b_n \}$ be two positive integer sets and $|A \cap B|=1$. $C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}$. Function $f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \}$ is injective. For any $\{x,y\} \in C$, denote $|f(x)-f(y)|$ as the $\textsl{mark}$ of $\{x,y\}$. If $n \geq 6$, prove that at least two elements in $C$ have the same $\textsl{mark}$.
let $f:R\to R$ be a continuous function tf(t)>0 for $t\neq 0$. Prove that there exists a non-zero differentiable function $y:[0,\infty)\to R$ such that $y'(t)=f(y(t-1))\,\forall t>1$ and the roots of y are bounded.