Found problems: 4776
Let $ A = \{ 1, 2, 3, \cdots , 12 \} $. Find the number of one-to-one function $ f :A \to A $ satisfying following condition: for all $ i \in A $, $ f(i)-i $ is not a multiple of $ 3 $.
Find all functions $f: \mathbb N \to \mathbb N$ Such that:
1.for all $x,y\in N$:$x+y|f(x)+f(y)$
2.for all $x\geq 1395$:$x^3\geq 2f(x)$
Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.
[i]Proposed by Evan Chen[/i]
$P,Q,R$ are non-zero polynomials that for each $z\in\mathbb C$, $P(z)Q(\bar z)=R(z)$.
a) If $P,Q,R\in\mathbb R[x]$, prove that $Q$ is constant polynomial.
b) Is the above statement correct for $P,Q,R\in\mathbb C[x]$?
Prove that any continuos function $ f: \mathbb{R}\rightarrow \mathbb{R}$ with
\[ f(x)\equal{}\left\{ \begin{aligned} a_1x\plus{}b_1\ ,\ \text{for } x\le 1 \\
a_2x\plus{}b_2\ ,\ \text{for } x>1 \end{aligned} \right.\]
where $ a_1,a_2,b_1,b_2\in \mathbb{R}$, can be written as:
\[ f(x)\equal{}m_1x\plus{}n_1\plus{}\epsilon|m_2x\plus{}n_2|\ ,\ \text{for } x\in \mathbb{R}\]
where $ m_1,m_2,n_1,n_2\in \mathbb{R}$ and $ \epsilon\in \{\minus{}1,\plus{}1\}$.
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties:
(i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$;
(ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite.
Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic.
[i]Proposed by Ang Jie Jun, Singapore[/i]
In order to draw a graph of $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, a table of values was constructed. These values of the function for a set of equally spaced increasing values of $ x$ were $ 3844$, $ 3969$, $ 4096$, $ 4227$, $ 4356$, $ 4489$, $ 4624$, and $ 4761$. The one which is incorrect is:
$ \textbf{(A)}\ 4096 \qquad\textbf{(B)}\ 4356 \qquad\textbf{(C)}\ 4489 \qquad\textbf{(D)}\ 4761 \qquad\textbf{(E)}\ \text{none of these}$
The set of all real numbers $ x$ for which
\[ \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))
\]is defined is $ \{x|x > c\}$. What is the value of $ c$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2001^{2002} \qquad \textbf{(C)}\ 2002^{2003} \qquad \textbf{(D)}\ 2003^{2004} \qquad \textbf{(E)}\ 2001^{2002^{2003}}$
Find all continuous real functions $ f,g$ and $ h$ defined on the set of positive real numbers and satisfying the relation \[ f(x\plus{}y)\plus{}g(xy)\equal{}h(x)\plus{}h(y)\] for all $ x>0$ and $ y>0$.
[i]Z. Daroczy[/i]
[b]a)[/b] Show that there is no function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that
$$ f(f(x))=\left\{ \begin{matrix} \sqrt{2007} ,& \quad x\in\mathbb{Q} \\ 2007, & \quad x\not\in \mathbb{Q} \end{matrix} \right. , $$
for any real number $ x. $
[b]b)[/b] Prove that there is an infinite number of functions $ g:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that
$$ g(g(x))=\left\{ \begin{matrix} 2007 ,& \quad x\in\mathbb{Q} \\ \sqrt{2007}, & \quad x\not\in \mathbb{Q} \end{matrix} \right. , $$
for any real number $ x. $
Let $a_n$ be a sequence recursively de fined by $a_0 = 0, a_1 = 1$ and $a_{n+2} = a_{n+1} + a_n$. Calculate the sum of $a_n\left( \frac 25\right)^n$ for all positive integers $n$. For what value of the base $b$ we get the sum $1$?
For any positive integer $n$, let $f(n)$ denote the $n$- th positive nonsquare integer, i.e., $f(1) = 2, f(2) = 3, f(3) = 5, f(4) = 6$, etc. Prove that $f(n)=n +\{\sqrt{n}\}$ where $\{x\}$ denotes the integer closest to $x$.
(For example, $\{\sqrt{1}\} = 1, \{\sqrt{2}\} = 1, \{\sqrt{3}\} = 2, \{\sqrt{4}\} = 2$.)
On each face of two dice some positive integer is written. The two dice are thrown and the numbers on the top face are added. Determine whether one can select the integers on the faces so that the possible sums are $2,3,4,5,6,7,8,9,10,11,12,13$, all equally likely?
Let $n$ be a positive integer and let $f_1(x), f_2(x) \dots f_n(x)$ be affine functions from $\mathbb{R}$ to $\mathbb{R}$ such that, amongst the graph of these functions, no two are parallel and no three are concurrent. Let $S$ be the set of all convex functions $g(x)$ from $\mathbb{R}$ to $\mathbb{R}$ such that for each $x \in \mathbb{R}$, there exists $i$ such that $g(x) = f_i(x)$.
Determine the largest and smallest possible values of $|S|$ in terms of $n$.
(A function $f(x)$ is affine if it is of form $f(x) = ax + b$ for some $a, b \in \mathbb{R}$. A function $g(x)$ is convex if $g(\lambda x + (1 - \lambda) y) \leq \lambda g(x) + (1-\lambda)g(y)$ for all $x, y \in \mathbb{R}$ and $0 \leq \lambda \leq 1$)
Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.
Prove that for positive number $t$, the function $F(t)=\int_0^t \frac{\sin x}{1+x^2}dx$ always takes positive number.
1972 Tokyo University of Education entrance exam
Let $x,y,z$ is positive. Solve:
$\begin{cases}{x\left( {6 - y} \right) = 9}\\
{y\left( {6 - z} \right) = 9}\\
{z\left( {6 - x} \right) = 9}\end{cases}$
Prove that if $ a_i(i=1,2,3,4)$ are positive constants, $ a_2-a_4 > 2$, and $ a_1a_3-a_2 > 2$, then the solution $ (x(t),y(t))$ of the system of differential equations \[ \.{x}=a_1-a_2x+a_3xy,\] \[ \.{y}=a_4x-y-a_3xy \;\;\;(x,y \in \mathbb{R}) \] with the initial conditions $ x(0)=0, y(0) \geq a_1$ is such that the function $ x(t)$ has exactly one strict local maximum on the interval $ [0, \infty)$.
[i]L. Pinter, L. Hatvani[/i]
A function \[\text{f:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] is called contract if, for every numbers $x,y\in \text{(0,}\infty \text{)}$ we have, $\underset{n\to \infty }{\mathop{\lim }}\,\left( {{f}^{n}}\left( x \right)-{{f}^{n}}\left( y \right) \right)=0$ where ${{f}^{n}}=\underbrace{f\circ f\circ ...\circ f}_{n\ f\text{'s}}$
a) Consider \[f:\text{(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] a function contract, continue with the property that has a fixed point, that existing ${{x}_{0}}\in \text{(0,}\infty \text{) }$ there so that $f\left( {{x}_{0}} \right)={{x}_{0}}.$ Show that $f\left( x \right)>x,$ for every $x\in \text{(0,}{{x}_{0}}\text{)}\,$ and $f\left( x \right)<x$, for every $x\in \text{(}{{x}_{0}}\text{,}\infty \text{)}\,$.
b) Show that the given function \[f\text{:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] given by $f\left( x \right)=x+\frac{1}{x}$ is contracted but has no fix number.
Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that
(a) $f(1)=1$
(b) $f(n+2)+(n^2+4n+3)f(n)=(2n+5)f(n+1)$ for all $n \in \mathbb{N}$.
(c) $f(n)$ divides $f(m)$ if $m>n$.
There is an equation $\sum_{i=1}^{n}{\frac{b_i}{x-a_i}}=c$ in $x$, where all $b_i >0$ and $\{a_i\}$ is a strictly increasing sequence. Prove that it has $n-1$ roots such that $x_{n-1}\le a_n$, and $a_i \le x_i$ for each $i\in\mathbb{N}, 1\le i\le n-1$.
Given two fractions $a/b$ and $c/d$ we define their [i]pirate sum[/i] as:
$\frac{a}{b} \star \frac{c}{d} = \frac{a+c}{b+d}$ where the two initial fractions are simplified the most possible, like the result.
For example, the pirate sum of $2/7$ and $4/5$ is $1/2$.
Given an integer $n \ge 3$, initially on a blackboard there are the fractions:
$\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}$.
At each step we choose two fractions written on the blackboard, we delete them and write at their place their pirate sum. Continue doing the same thing until on the blackboard there is only one fraction.
Determine, in function of $n$, the maximum and the minimum possible value for the last fraction.
Prove that there is exactly a function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{\ge 0} $ satisfying the following two properties:
$ \text{(i)} x\in\mathbb{R}_{> 0}\implies \left( f(x)+f(f(x)) =4018020x \wedge f(x)>0 \right) $
$ \text{(ii)} 0=f(0)+f(f(0)) $
We have a closed path on a vertices of a $ n$×$ n$ square which pass from each vertice exactly once . prove that we have two adjacent vertices such that if we cut the path from these points then length of each pieces is not less than quarter of total path .
Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relation $u_{n+1}-u_n=3+4(n-1)$, $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:
$\textbf{(A) }3\qquad
\textbf{(B) }4\qquad
\textbf{(C) }5\qquad
\textbf{(D) }6\qquad
\textbf{(E) }11$