Found problems: 4776
Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.
Find all function $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that
forall positive integers $x$ and $y$, $\frac{f(x+y)-f(x)}{f(y)}$ is again a positive integer not exceeding $2022^{2022}$.
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
Given $X=\{0,a,b,c\}$, let $M(X)=\{f|f: X\to X\}$ denote the set of all functions from $X$ into itself. An addition table on $X$ is given us follows:
$+$ $0$ $a$ $b$ $c$
$0$ $0$ $a$ $b$ $c$
$a$ $a$ $0$ $c$ $b$
$b$ $b$ $c$ $0$ $a$
$c$ $c$ $b$ $a$ $0$
a)If $S=\{f\in M(X)|f(x+y+x)=f(x)+f(y)+f(x)\forall x,y\in X\}$, find $|S|$.
b)If $I=\{f\in M(X)|f(x+x)=f(x)+f(x)\forall x\in X\}$, find $|I|$.
Let be an open interval $ I $ and a convex function $ f:I\longrightarrow\mathbb{R} . $
Prove that the lateral derivatives of $ f $ are left-continuous on $ \mathbb{R} $ and also right-continuous on $ \mathbb{R} . $
[i]Marin Tolosi[/i]
Let $D$ be the interior of the circle $C$ and let $A \in C$. Show that the function $f : D \to \mathbb R, f(M)=\frac{|MA|}{|MM'|}$ where $M' = AM \cap C$, is strictly convex; i.e., $f(P) <\frac{f(M_1)+f(M_2)}{2}, \forall M_1,M_2 \in D, M_1 \neq M_2$ where $P$ is the midpoint of the segment $M_1M_2.$
Let $a_1,a_2,\ldots$ be a sequence defined by $a_1=a_2=1$ and $a_{n+2}=a_{n+1}+a_n$ for $n\geq 1$. Find \[\sum_{n=1}^\infty \dfrac{a_n}{4^{n+1}}.\]
Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$.
[i]David Yang.[/i]
Let $\, a,b \,$ be odd positive integers. Define the sequence $\, (f_n ) \,$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for $\, n \,$ sufficiently large and determine the eventual value as a function of $\, a \,$ and $\, b$.
[b]3.[/b] Let the density function $f(x)$ of the random variable $\xi$ bean even function; let further $f(x)$ be monotonically non-increasing for $x>0$. Suppose that
$D^{2}= \int_{-\infty }^{\infty }x^{2}f(x) dx$
exists. Prove that for every non negative $\lambda $
$P(\left |\xi \right |\geq \lambda D)\leq \frac{1}{1+\lambda ^{2}}$. [b](P. 7)[/b]
Let $f_1,f_2$ be nonnegative and concave functions. Then prove that
$$(f_1f_2)^{\frac{2^n-1}{n\cdot2^n}}\left(\frac{\displaystyle\prod_{k=1}^n\left(\sqrt[2^k]{f_1}+\sqrt[2^k]{f_2}\right)}{f_1+f_2}\right)^{\frac1n}$$
is concave.
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
Let $(f_i)$ be a sequence of functions defined by: $f_1(x)=x, f_n(x) = \sqrt{f_{n-1}(x)}-\dfrac14$. ($n\in \mathbb{N}, n\ge2$)
(a) Prove that $f_n(x) \le f_{n-1}(x)$ for all x where both functions are defined.
(b) Find for each $n$ the points of $x$ inside the domain for which $f_n(x)=x$.
Find all injective functions$f:\mathbb{Z}\to \mathbb{Z}$ that satisfy:
$\left| f\left( x \right)-f\left( y \right) \right|\le \left| x-y \right|$ ,for any $x,y\in \mathbb{Z}$.
Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.
Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.
For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides \[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\] where $N=6^{1999}$.
[i]Proposed by David Altizio[/i]
Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a map such that the image of every compact set is compact, and the image of every connected set is connected. Prove that $f$ is continuous.
Given $a\in (0,1)$ and $C$ the set of increasing functions
$f:[0,1]\to [0,\infty )$ such that $\int\limits_{0}^{1}{f(x)}dx=1$ . Determine:
$(a)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{f(x)dx}$
$(b)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{{{f}^{2}}(x)dx}$
Let $\displaystyle M$ be a set composed of $\displaystyle n$ elements and let $\displaystyle \mathcal P (M)$ be its power set. Find all functions $\displaystyle f : \mathcal P (M) \to \{ 0,1,2,\ldots,n \}$ that have the properties
(a) $\displaystyle f(A) \neq 0$, for $\displaystyle A \neq \phi$;
(b) $\displaystyle f \left( A \cup B \right) = f \left( A \cap B \right) + f \left( A \Delta B \right)$, for all $\displaystyle A,B \in \mathcal P (M)$, where $\displaystyle A \Delta B = \left( A \cup B \right) \backslash \left( A \cap B \right)$.
Determine functions $f : \mathbb{R} \rightarrow \mathbb{R},$ with property that
\[
f(f(x)) + y \cdot f(x) \le x + x \cdot f(f(y)),
\]
for every $x$ and $y$ are real numbers.
Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is itself an arithmetic progression.
[i]Proposed by Gabriel Carroll, USA[/i]
Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have
\[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.
Suppose that $|x+y|+|x-y|=2$. What is the maximum possible value of $x^2-6x+y^2$?
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9
$
Let $a,b$ be two constants within the open interval $(0,\frac{1}{2})$. Find all continous functions $f(x)$ such that \[f(f(x))=af(x)+bx\] holds for all real $x$.
This is much harder than the problems we had in the 1st TST...
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.