Found problems: 4776
The function $ f$ satisfies \[f(x) \plus{} f(2x \plus{} y) \plus{} 5xy \equal{} f(3x \minus{} y) \plus{} 2x^2 \plus{} 1\] for all real numbers $ x$, $ y$. Determine the value of $ f(10)$.
The image that maps $x$ to $1 - x$ is called [i]complement[/i], the image that maps $x$ to $\frac{1}{x}$ is called [i]invert[/i]. Two numbers $x$ and $y$ are called related if they can be transferred into each other by means of [i]complementation [/i]and/or [i]inversion[/i]. A [i]family [/i] is a collection of numbers where every two elements are related. Determine the maximum size $n$ of such a family.
Show that the number line can be divided into $n$ parts, such that each of those $n$ parts contains exactly one number from each $n$-number family.
Determine all functions $f:\mathbb{N}\to \mathbb{N}$ such that for all $a, b \in \mathbb{N}$ the following conditions hold:
$(i)$ $f(f(a)+b) \mid b^a-1$;
$(ii)$ $f(f(a))\geq f(a)-1$.
Let $\mathbb{R}$ be the set of real numbers. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following condition. Here, $f^{100}(x)$ is the function obtained by composing $f(x)$ $100$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{100 \ \text{times}})(x).$
[b](Condition)[/b] For all $x, y \in \mathbb{R}$, $$f(x + f^{100}(y)) = x + y \ \ \ \text{or} \ \ \ f(f^{100}(x) + y) = x + y$$
assume that k,n are two positive integer $k\leq n$count the number of permutation $\{\ 1,\dots ,n\}\ $ st for any $1\leq i,j\leq k$and any positive integer m we have $f^m(i)\neq j$ ($f^m$ meas iterarte function,)
Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that
\[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \]
where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.
Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$, and $\tan C$, where $A, B$, and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$, and $s.$
An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition.
[quote]For example, 4 can be partitioned in five distinct ways:
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1[/quote]
The number of partitions of n is given by the partition function $p\left ( n \right )$. So $p\left ( 4 \right ) = 5$ .
Determine all the positive integers so that $p\left ( n \right )+p\left ( n+4 \right )=p\left ( n+2 \right )+p\left ( n+3 \right )$.
$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.
Let $n$ and $k$ be two natural numbers such that $k$ is even and for each prime $p$ if $p|n$ then $p-1|k$. let $\{a_1,....,a_{\phi(n)}\}$ be all the numbers coprime to $n$. What's the remainder of the number $a_1^k+.....+a_{\phi(n)}^k$ when it's divided by $n$?
[i]proposed by Yahya Motevassel[/i]
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200$.
Let $ \sigma(\cdot): \mathbb{N}_0 \to \mathbb{N}_0$ be the function from every positive integer $ n$ to the sum of divisors $ \sum_{d \mid n}{d}$ (i.e. $ \sigma(6) \equal{} 6 \plus{} 3 \plus{} 2 \plus{} 1$ and $ \sigma(8) \equal{} 8 \plus{} 4 \plus{} 2 \plus{} 1$).
Find all primes $ p$ such that $ p \mid \sigma(p \minus{} 1)$.
[i](Salvatore Tringali)[/i]
Find all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that
\[ 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n, \quad \forall n\in \mathbb{N}.\]
Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define
\[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\]
Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$
(Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)
Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2\minus{}1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3\minus{}1)$ are distinct modulo $ p^3.$
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$.
[i]Proposed by Japan[/i]
Define for positive integer $n$, a function $f_n(x)=\frac{\ln x}{x^n}\ (x>0).$ In the coordinate plane, denote by $S_n$ the area of the figure enclosed by $y=f_n(x)\ (x\leq t)$, the $x$-axis and the line $x=t$ and denote by $T_n$ the area of the rectagle with four vertices $(1,\ 0),\ (t,\ 0),\ (t,\ f_n(t))$ and $(1,\ f_n(t))$.
(1) Find the local maximum $f_n(x)$.
(2) When $t$ moves in the range of $t>1$, find the value of $t$ for which $T_n(t)-S_n(t)$ is maximized.
(3) Find $S_1(t)$ and $S_n(t)\ (n\geq 2)$.
(4) For each $n\geq 2$, prove that there exists the only $t>1$ such that $T_n(t)=S_n(t)$.
Note that you may use $\lim_{x\to\infty} \frac{\ln x}{x}=0.$
Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]
If $x$, $y$, $k$ are positive reals such that \[3=k^2\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+k\left(\dfrac{x}{y}+\dfrac{y}{x}\right),\] find the maximum possible value of $k$.
Determine all non negative integers $k$ such that there is a function $f : \mathbb{N} \to \mathbb{N}$ that satisfies
\[ f^n(n) = n + k \]
for all $n \in \mathbb{N}$
Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that
\[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C}
\]
Let $ A $ be a subset of $ \mathbb{R} $ and let be a function $ f:A\longrightarrow\mathbb{R} $ satisfying
$$ f(x)-f(y)=(y-x)f(x)f(y),\quad\forall x,y\in A. $$
[b]a)[/b] Show that if $ A=\mathbb{R}, $ then $ f=0. $
[b]b)[/b] Find $ f, $ provided that $ A=\mathbb{R}\setminus\{1\} . $
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$\displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,}$$ for all $x \in \mathbb{R}.$ Find all possible values of $f(2).$
Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$ a monotonic bijective function.
a)Prove that there exist a unique continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ such that $F(x)=f(x),\ (\forall)x\in \mathbb{Q}$.
b)Give an example of a non-injective polynomial function $G:\mathbb{R}\rightarrow \mathbb{R}$ such that $G(\mathbb{Q})\subset \mathbb{Q}$ and it's restriction defined on $\mathbb{Q}$ is injective.