Found problems: 4776
Consider the set
$\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}$.
Prove that there is no function in this function space that gives us the minimum of $S=\int_{-1}^1x^2f'(x)^2dx$. What is the infimum of $S$ for the functions of this space?
Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$
Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. What is the least possible value of $f (999)$ ?
Define the function $f_k(x)$ (where $k$ is a positive integer) as follows: \[f_k(x)=(\cos kx)(\cos x)^k+(\sin kx)(\sin x)^k-(\cos 2x)^k.\] Find the sum of all distinct value(s) of $k$ such that $f_k(x)$ is a constant function.
Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n) \ge n+(-1)^{n}.\]
For every positive integer $n$, let $s(n)$ be the sum of the exponents of $71$ and $97$ in the prime factorization of $n$; for example, $s(2021) = s(43 \cdot 47) = 0$ and $s(488977) = s(71^2 \cdot 97) = 3$. If we define $f(n)=(-1)^{s(n)}$, prove that the limit
\[ \lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n} \]
exists and determine its value.
Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute
$\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.
A function $f : \Bbb{N}_0 \to \Bbb{R}$ satisfies $f(1) = 3$ and \[f(m + n) + f(m - n) - m + n - 1 =\frac{f(2m) + f(2n)}{2},\]
for any non-negative integers $m$ and $n$ with $m \geq n.$ Find all such functions $f$.
Let $f, g:\mathbb{N}\to\mathbb{N}$ be functions that satisfy the following equation:
\[f(f(n))+g(f(n)) = n,\ \forall\ n\in\mathbb{N}\ .\]
Prove that $g$ is the zero function on $\mathbb{N}$.
Show that for all $n \in \mathbb{N}$, \[n = \sum^{}_{d \vert n}\phi(d).\]
Let $ f : \mathbf{R} \to \mathbf{R} $ be a continuous function with $ \displaystyle\int_{0}^{1} f(x) f'(x) \, \mathrm{d}x = 0 $ and $ \displaystyle\int_{0}^{1} f(x)^2 f'(x) \, \mathrm{d}x = 18 $. What is $ \displaystyle\int_{0}^{1} f(x)^4 f'(x) \, \mathrm{d} x $?
Suppose $f : \mathbb{N} \longrightarrow \mathbb{N}$ is a function that satisfies $f(1) = 1$ and
$f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array}$
$(a)$ Prove that $f(f(n)-n+1)$ is either $n$ or $n+1$.
$(b)$ Determine$f$.
Let $k$ be a natural number. For each function $f : \mathbb{N}\to \mathbb{N}$ define the sequence of functions $(f_{m})_{m\geq 1}$ by $f_{1}= f$ and $f_{m+1}= f \circ f_{m}$ for $m \geq 1$ . Function $f$ is called $k$-[i]nice[/i] if for each $n \in\mathbb{N}: f_{k}(n) = f (n)^{k}$.
(a) For which $k$ does there exist an injective $k$-nice function $f$ ?
(b) For which $k$ does there exist a surjective $k$-nice function $f$ ?
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.
Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity.
Let be given natural numbers $a_1,a_2,...,a_n$ and a function $f : Z \to R$ such that $f(x) = 1$ for all integers $x < 0$ and $f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)$ for all integers $x \ge 0$. Prove that there exist natural numbers $s$ and $t$ such that for all integers $x > s$ it holds that $f(x+t) = f(x)$.
For which of the following values of real number $t$, the equation $x^4-tx+\dfrac 1t = 0$ has no root on the interval $[1,2]$?
$
\textbf{(A)}\ 6
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 9
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Is it true that if a function $f: R \to R$ is continuous and takes rational values at rational points,
then at least at one point it is differentiable?
Let $\{a_n\}_{n\geq 0}$ be an arithmetic sequence with difference $d$ and $1\leq a_0\leq d$. Denote the sequence as $S_0$, and define $S_n$ recursively by two operations below:
Step $1$: Denote the first number of $S_n$ as $b_n$, and remove $b_n$.
Step $2$: Add $1$ to the first $b_n$ numbers to get $S_{n+1}$.
Prove that there exists a constant $c$ such that $b_n=[ca_n]$ for all $n\geq 0$, where $[]$ is the floor function.
Given non-constant linear functions $p(x), q(x), r(x)$. Prove that at least one of three trinomials $pq+r, pr+q, qr+p$ has a real root.
[b]proposed by S. Berlov[/b]
Find the volume of the solid of the domain expressed by the inequality $x^2-x\leq y\leq x$, generated by a rotation about the line $y=x.$
In four functions $y=\sin|x|,y=\cos|x|,y=|\cot x|,y=\lg|\sin x|$, which one is even function, and increases on $\left(0,\frac{\pi}{2}\right)$, with period of $\pi$?
$\text{(A)}y=\sin|x|\qquad\text{(B)}y=\cos|x|\qquad\text{(C)}y=|\cot x|\qquad\text{(D)}y=\lg|\sin x|$
Let $n \geq 0$ be an integer and $f: [0, 1] \rightarrow \mathbb{R}$ an integrable function such that: $$\int^1_0f(x)dx = \int^1_0xf(x)dx = \int^1_0x^2f(x)dx = \ldots = \int^1_0x^nf(x)dx = 1$$ Prove that: $$\int_0^1f(x)^2dx \geq (n+1)^2$$
Let $N$ be a natural number. Find (with prove) the number of solutions in the segment $[1,N]$ of the equation $x^2-[x^2]=(x-[x])^2$, where $[x]$ means the floor function of $x$.
Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$.
[i]Sammy Luo and Alex Zhu.[/i]