Found problems: 4776
Let $f : \left[ a, b\right]\rightarrow\mathbb{R}$ be continuous on $\left[ a, b\right]$ and differentiable on $\left( a, b\right)$. Suppose that $f$ has infinitely many zeros, but there is no $x\in \left( a, b\right)$ with $f(x)=f'(x)=0$.
(a) Prove that $f(a)f(b)=0$.
(b) Give an example of such a function on $\left[ 0, 1\right]$.
(Proposed by Alexandr Bolbot, Novosibirsk State University)
To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied:
(1) $ f(rs) \equal{} f(r)\plus{}f(s)$
(2) $ f(n) \equal{} 0$, if the first digit (from right to left) of $ n$ is 3.
(3) $ f(10) \equal{} 0$.
Find $f(1985)$. Justify your answer.
Given a function $f(x)=a\sin x-\frac{1}{2}\cos 2x+a-\frac{3}{a}+\frac{1}{2}$, where $a\in\mathbb{R}, a\ne 0$.
[b](1)[/b] If for any $x\in\mathbb{R}$, inequality $f(x)\le 0$ holds, find all possible value of $a$.
[b](2)[/b] If $a\ge 2$, and there exists $x\in\mathbb{R}$, such that $f(x)\le 0$. Find all possible value of $a$.
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. \]
A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$.
Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.
Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation
\[ f(f(f(n))) = f(n+1 ) +1 \]
for all $ n\in \mathbb{Z}_{\ge 0}$.
Which of the following functions are differentiable at $x=0$?
$\textbf{(A)}~f(x)=\begin{cases}\tan^{-1}\left(\frac1{|x|}\right)&\text{if }x\ne0\\\frac\pi2&\text{if }x=0\end{cases}$
$\textbf{(B)}~f(x)=|x|^{1/2}x$
$\textbf{(C)}~f(x)=\begin{cases}x^2\left|\cos\frac{\pi}x\right|&\text{if }x\ne0\\0&\text{if }x=0\end{cases}$
$\textbf{(D)}~\text{None of the above}$
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]
$\textbf{(A) }278\qquad
\textbf{(B) }279\qquad
\textbf{(C) }280\qquad
\textbf{(D) }281\qquad
\textbf{(E) }282\qquad$
Suppose $f(x)$ is a rational function such that $3f\left(\dfrac{1}{x}\right)+\dfrac{2f(x)}{x}=x^2$ for $x\neq 0$. Find $f(-2)$.
For every integer $n > 1$, the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $
where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$. Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$.
.
Let $f$ be the function defined by $f(x)=ax^2-\sqrt2$ for some positive $a$. If $f(f(\sqrt2 ))=-\sqrt 2$, then $a=$
$\text{(A)} \ \frac{2-\sqrt2}{2} \qquad \text{(B)} \ \frac12 \qquad \text{(C)} \ 2-\sqrt2 \qquad \text{(D)} \ \frac{\sqrt{2}}{2} \qquad \text{(E)} \ \frac{2+\sqrt2}{2}$
Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.
Find all positive integers $n$ that have precisely $\sqrt{n+1}$ natural divisors.
Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that:
[b]•[/b] For all positive integers $a> 2023^{2023}$ it holds that $f(a) \leq a$.
[b]•[/b] $\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}$ is a positive integer for all $a,b \in \mathbb{N}$.
[i]Proposed by Nikola Velov[/i]
Let $a,b,c$ be non-zero real numbers.Prove that if function $f,g:\mathbb{R}\to\mathbb{R}$ satisfy $af(x+y)+bf(x-y)=cf(x)+g(y)$ for all real number $x,y$ that $y>2018$ then there exists a function $h:\mathbb{R}\to\mathbb{R}$ such that $f(x+y)+f(x-y)=2f(x)+h(y)$ for all real number $x,y$.
In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And
(i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$.
(ii) $a \circ b \neq b \circ a$ when $a \neq b$.
Prove that:
a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$.
b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.
Consider a charged capacitor made with two square plates of side length $L$, uniformly charged, and separated by a very small distance $d$. The EMF across the capacitor is $\xi$. One of the plates is now rotated by a very small angle $\theta$ to the original axis of the capacitor. Find an expression for the difference in charge between the two plates of the capacitor, in terms of (if necessary) $d$, $\theta$, $\xi$, and $L$.
Also, approximate your expression by transforming it to algebraic form: i.e. without any non-algebraic functions. For example, logarithms and trigonometric functions are considered non-algebraic. Assume $ d << L $ and $ \theta \approx 0 $.
$\emph{Hint}$: You may assume that $ \frac {\theta L}{d} $ is also very small.
[i]Problem proposed by Trung Phan[/i]
[hide="Clarification"]
There are two possible ways to rotate the capacitor. Both were equally scored but this is what was meant: [asy]size(6cm);
real h = 7;
real w = 2;
draw((-w,0)--(-w,h));
draw((0,0)--(0,h), dashed);
draw((0,0)--h*dir(64));
draw(arc((0,0),2,64,90));
label("$\theta$", 2*dir(77), dir(77));
[/asy]
[/hide]
Find the function $f(x)$ that satisfies the equation $$f'(x) + x^2f(x) = 0$$ knowing that $f(1) = e$. Graph this function and calculate the tangent of the curve at the point of abscissa $1$.
Let $f$ and $g$ be two continuous, distinct functions from $[0,1] \rightarrow (0,+\infty)$ such that
$\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$
Let
$y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq 0$, natural.
Prove that $(y_n)$ is an increasing and divergent sequence.
Prove that there exists a function $f\colon [\omega_1]^2 \rightarrow \omega _1$ such that
(i) $f(\alpha, \beta)< \mathrm{min}(\alpha, \beta)$ whenever $\mathrm{min}(\alpha,\beta)>0$; and
(ii) if $\alpha_0<\alpha_1<\ldots<\alpha_i<\ldots<\omega_1$ then $\sup\left\{ a_i \colon i<\omega \right\} =\sup \left\{ f(\alpha_i, \alpha_j)\colon i,j<\omega\right\}$.
Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that
\[ a_1 + \dots + a_i \le b_1 + \dots + b_i \text{ for every } i = 1, \dots, n \]
and
\[ a_1 + \dots + a_n = b_1 + \dots + b_n. \]
Suppose that for every real number $m$, the number of pairs $(i,j)$ with $a_i-a_j=m$ equals the numbers of pairs $(k,\ell)$ with $b_k-b_\ell = m$. Prove that $a_i = b_i$ for $i=1,\dots,n$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$:
$(i)$ $f(x)=-f(-x);$
$(ii)$ $f(x+1)=f(x)+1;$
$(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$
Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$, where $0<t<1$. Prove that the area of any triangle inside the region $A$, with two vertices $P(t,t^2)$ and $Q(1,0)$, does not exceed $\frac{1}{4}.$
The function $f(x)$, defined on the entire real line, is known but that for any $a > 1 $ the function $f(x)+f(ax)$ is continuous on the entire line. Prove that $f(x)$ is also continuous along the entire line.
Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the remainder when $S$ is divided by $1000$.