Found problems: 4776
$f(x)=a\sin x+b\sqrt[3]{x}+4$. If $f(\lg\log_{3}10)=5$, then the value of $f(\lg\lg 3)$ is
$\text{(A)}-5\qquad\text{(B)}-3\qquad\text{(C)}3\qquad\text{(D)}$ not sure
Let $N$ be the number of functions $f$ from $\{1,2,\ldots, 8\}$ to $\{1,2,3,\ldots, 255\}$ with the property that:
[list]
[*] $f(k)=1$ for some $k \in \{1,2,3,4,5,6,7,8\}$
[*] If $f(a) =f(b)$, then $a=b$.
[*] For all $n \in \{1,2,3,4,5,6,7,8\}$, if $f(n) \neq 1$, then $f(k)+1>\frac{f(n)}{2} \geq f(k)$ for some $k \in \{1,2,\ldots, 7,8\}$.
[*] For all $k,n \in \{1,2,3,4,5,6,7,8\}$, if $f(n)=2f(k)+1$, then $k<n$.
[/list]
Compute the number of positive integer divisors of $N$.
[i]2021 CCA Math Bonanza Individual Round #15[/i]
$f$ is a function from the set of positive integers to the set of all integers that satisfies the following.
[b]$\cdot$[/b] $f(1)=1, f(2)=-1$
[b]$\cdot$[/b] $f(n)+f(n+1)+f(n+2)=f(\left\lfloor\frac{n+2}{3}\right\rfloor)$
Find the number of positive integers $k$ not exceeding $1000$ such that $f(3)+f(6)+\cdots+f(3k-3)+f(3k)=5$.
How many times changes the sign of the function \[ f(x)\equal{}\cos x\cos\frac{x}{2}\cos\frac{x}{3}\cdots\cos\frac{x}{2009}\] at the interval $ \left[0, \frac{2009\pi}{2}\right]$?
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?
The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to:
${{ \textbf{(A)}\ t^3 \qquad\textbf{(B)}\ t^2+t \qquad\textbf{(C)}\ |t^2+t| \qquad\textbf{(D)}\ t\sqrt{t^2+1} }\qquad\textbf{(E)}\ |t|\sqrt{1+t^2} } $
Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that
\[f(x+t) - f(x) = P(x)\]
for all $x \in \mathbb{R}$.
$g(x):\mathbb{Z}\rightarrow\mathbb{Z}$ is a function that satisfies $$g(x)+g(y)=g(x+y)-xy.$$ If $g(23)=0$, what is the sum of all possible values of $g(35)$?
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in [i]almost-order[/i] with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(b) How many different ways are there to line up $20$ people in [i]almost-order[/i] if their heights are $120, 125, 130,$ $135,$ $140,$ $145,$ $150,$ $155,$ $160,$ $164, 165, 170, 175, 180, 185, 190, 195, 200, 205$, and $210$ centimeters? (Note that there is someone of height $164$ centimeters.)
Find all non empty subset $ S$ of $ \mathbb{N}: \equal{} \{0,1,2,\ldots\}$ such that $ 0 \in S$ and exist two function $ h(\cdot): S \times S \to S$ and $ k(\cdot): S \to S$ which respect the following rules:
i) $ k(x) \equal{} h(0,x)$ for all $ x \in S$
ii) $ k(0) \equal{} 0$
iii) $ h(k(x_1),x_2) \equal{} x_1$ for all $ x_1,x_2 \in S$.
[i](Pierfrancesco Carlucci)[/i]
Let $f: \mathbb{R} \to (0, \infty)$ be a continuously differentiable function. Prove that there exists $\xi \in (0,1)$ such that $$e^{f'(\xi)} \cdot f(0)^{f(\xi)} = f(1)^{f(\xi)}$$
Define the regions $M, N$ in the Cartesian Plane as follows:
\begin{align*}
M &= \{(x, y) \in \mathbb R^2 \mid 0 \leq y \leq \text{min}(2x, 3-x)\} \\
N &= \{(x, y) \in \mathbb R^2 \mid t \leq x \leq t+2 \}
\end{align*}
for some real number $t$. Denote the common area of $M$ and $N$ for some $t$ be $f(t)$. Compute the algebraic form of the function $f(t)$ for $0 \leq t \leq 1$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 5)[/i]
Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If
\[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\]
then
\[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]
The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$, $|f''(x)| \le B$ for some positive $A, B$ (and all $x$). Show that $|f'(x)| \le C$, for some fixed$ C$, which depends only on $A$ and $B$. What is the smallest possible value of $C$?
Let $a_1\ge a_2\ge \ldots \ge a_n>0$ be $n$ reals. Prove the inequality
\[a_1a_2\ldots a_{n-1}+(2a_2-a_1)(2a_3-a_2)\ldots (2a_n-a_{n-1})\ge 2a_2a_3\ldots a_n\]
let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending.
and for all $x\in \mathbb{R}$
$p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ .
prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$
$f(q(p(x))+h(x))=f(x)^{2}+1$
Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.
Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous and nondecreasing function.
[b]a)[/b] Show that the sequence $ \left( \frac{1}{2^n}\sum_{i=1}^{2^n} f\left(\frac{i}{2^n}\right) \right)_{n\ge 1} $ is nonincreasing.
[b]b)[/b] Prove that, if there exists some natural index at which the sequence above is equal to $ \int_0^1 f(x)dx, $ then $ f $ is constant.
Consider a finite set $ A, $ and two functions $ f,g: A\longrightarrow A. $ Prove that:
$$ |\{ x\in A| g(f(x))\neq x \} | =|\{ x\in A| f(g(x))\neq x \} | $$
[i]Cristinel Mortici[/i]
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by
\[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\]
Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
A calculating ruler is a ruler for doing algebric calculations. This ruler has three arms, two of them are sationary and one can move freely right and left. Each of arms is gradient. Gradation of each arm depends on the algebric operation ruler does. For eaxample the ruler below is designed for multiplying two numbers. Gradations are logarithmic.
[img]http://aycu05.webshots.com/image/5604/2000468517162383885_rs.jpg[/img]
For working with ruler, (e.g for calculating $x.y$) we must move the middle arm that the arrow at the beginning of its gradation locate above the $x$ in the lower arm. We find $y$ in the middle arm, and we will read the number on the upper arm. The number written on the ruler is the answer.
1) Design a ruler for calculating $x^{y}$. Grade first arm ($x$) and ($y$) from 1 to 10.
2) Find all rulers that do the multiplication in the interval $[1,10]$.
3) Prove that there is not a ruler for calculating $x^{2}+xy+y^{2}$, that its first and second arm are grade from 0 to 10.
Let $x, y, z$ be real numbers satisfying $x+y+z=0$ and $x^2+y^2+z^2=6$. Find the maximum value of
\[ |(x-y)(y-z)(z-x) | \]
Define $f(x)=\int^x_0\int^x_0e^{u^2v^2}dudv$. Calculate $2f''(2)+f'(2)$.
Let $g(x)$ be a function that has a continuous first derivative $g'(x)$. Suppose that $g(0)=0$ and $|g'(x)| \leq |g(x)|$ for all values of $x.$ Prove that $g(x)$ vanishes identically.
i) Let $a,b$ be real numbers such that $b\leq 0$ and $1+ax+bx^{2} \geq 0$ for every $x\in [0,1]$.
Prove that
$$\lim_{n\to \infty} n \int_{0}^{1}(1+ax+bx^{2})^{n}dx= \begin{cases}
-\frac{1}{a} &\text{if}\; a<0,\\
\infty & \text{if}\; a \geq 0.
\end{cases}$$
ii) Let $f:[0,1]\rightarrow[0,\infty)$ be a function with a continuous second derivative and let $f''(x)\leq0$ for every $x\in [0,1]$. Suppose that $L=\lim_{n\to \infty} n \int_{0}^{1}(f(x))^{n}dx$ exists and $0<L<\infty$. Prove that $f'$ has a constant sign and $\min_{x\in [0,1]}|f'(x)|=L^{-1}$.