Found problems: 4776
Define $ x \heartsuit y$ to be $ |x\minus{}y|$ for all real numbers $ x$ and $ y$. Which of the following statements is [b]not[/b] true?
$\textbf{(A)}\ x \heartsuit y \equal{} y \heartsuit x \text{ for all } x \text{ and } y$
$\textbf{(B)}\ 2(x \heartsuit y) \equal{} (2x) \heartsuit (2y) \text{ for all } x \text{ and } y$
$\textbf{(C)}\ x \heartsuit 0 \equal{} x \text{ for all } x$
$\textbf{(D)}\ x \heartsuit x \equal{} 0 \text{ for all } x$
$\textbf{(E)}\ x \heartsuit y > 0 \text{ if } x \ne y$
Let $a$ and $b$ be real numbers in the interval $\left(0,\tfrac{1}{2}\right)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(0) \in \mathbb Q$ and
\[f(x+f(y)^2 ) = {f(x+y)}^2.\]
(25 points)
Let be the set $ G=\{ (u,v)\in \mathbb{C}^2| u\neq 0 \} $ and a function $ \varphi :\mathbb{C}\setminus\{ 0\}\longrightarrow\mathbb{C}\setminus\{ 0\} $ having the property that the operation $ *:G^2\longrightarrow G $ defined as
$$ (a,b)*(c,d)=(ac,bc+d\varphi (a)) $$
is associative.
[b]a)[/b] Show that $ (G,*) $ is a group.
[b]b)[/b] Describe $ \varphi , $ knowing that $(G,*) $ is a commutative group.
[i]Marius Perianu[/i]
Find all functions $f: (0, +\infty)\cap\mathbb{Q}\to (0, +\infty)\cap\mathbb{Q}$ satisfying thefollowing conditions:
[list=1]
[*] $f(ax) \leq (f(x))^a$, for every $x\in (0, +\infty)\cap\mathbb{Q}$ and $a \in (0, 1)\cap\mathbb{Q}$
[*] $f(x+y) \leq f(x)f(y)$, for every $x,y\in (0, +\infty)\cap\mathbb{Q}$
[/list]
Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.
Do there exist integers $m, n$ and a function $f\colon \mathbb R \to \mathbb R$ satisfying simultaneously the following two conditions?
$\bullet$ i) $f(f(x))=2f(x)-x-2$ for any $x \in \mathbb R$;
$\bullet$ ii) $m \leq n$ and $f(m)=n$.
Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute
\[
\frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}.
\]
i) Prove that if a function $f$ is continuous on the closed interval $[0, \pi]$ and
$$ \int_{0}^{\pi} f(t) \cos t \; dt= \int_{0}^{\pi} f(t) \sin t \; dt=0,$$
then there exist points $0 < \alpha < \beta < \pi$ such that $f(\alpha) =f(\beta) =0.$
ii) Let $R$ be a bounded, convex, and open region in the Euclidean plane. Prove with the help of i) that the centroid of $R$ bisects at least three different chords of the boundary of $ R.$
a) Show that for each function $f:\mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{R}$, there exists a function $g:\mathbb{Q}\rightarrow \mathbb{R}$ with $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{Q}$.
b) Find a function $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, for which there is no function $g:\mathbb{Q}\rightarrow \mathbb{R}$ such that $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{R}$.
Let $f: [0,1]\rightarrow\mathbb{R}$ be a continuous function satisfying $xf(y)+yf(x)\le 1$ for every $x,y\in[0,1]$.
(a) Show that $\int^1_0 f(x)dx \le \frac{\pi}4$.
(b) Find such a funtion for which equality occurs.
$a$ is an integer and $p$ is a prime number and we have $p\ge 17$. Suppose that $S=\{1,2,....,p-1\}$ and $T=\{y|1\le y\le p-1,ord_p(y)<p-1\}$. Prove that there are at least $4(p-3)(p-1)^{p-4}$ functions $f:S\longrightarrow S$ satisfying
$\sum_{x\in T} x^{f(x)}\equiv a$ $(mod$ $p)$.
[i]proposed by Mahyar Sefidgaran[/i]
Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.
Find all functions $ f: \mathbb{R}\longrightarrow \mathbb{R}$ such that
\[f(x\plus{}y)\plus{}f(y\plus{}z)\plus{}f(z\plus{}x)\ge 3f(x\plus{}2y\plus{}3z)\]
for all $x, y, z \in \mathbb R$.
Find the smallest positive integer solution to $\tan 19x^\circ=\frac{\cos 96^\circ+\sin 96^\circ}{\cos 96^\circ-\sin 96^\circ}.$
Find all tuples $ (x_1,x_2,x_3,x_4,x_5)$ of positive integers with $ x_1>x_2>x_3>x_4>x_5>0$ and
$ {\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.$
Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions:
1) $f$ is surjective
2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$)
3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$).
Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.
Find all functions $f:Q\rightarrow Q$ such that \[ f(x+y)+f(y+z)+f(z+t)+f(t+x)+f(x+z)+f(y+t)\ge 6f(x-3y+5z+7t) \] for all $x,y,z,t\in Q.$
Let $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying:
[list]
[*] for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$;
[*] for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$.
[/list]
[i]Proposed by Sutanay Bhattacharya[/i]
Suppose that $ V$ is a locally compact topological space that admits no countable covering with compact sets. Let $ \textbf{C}$
denote the set of all compact subsets of the space $ V$ and $ \textbf{U}$ the set of open subsets that are not contained in any compact set. Let $ f$ be a function from $ \textbf{U}$ to $ \textbf{C}$ such that $ f(U)\subseteq U$ for all $ U \in \textbf{U}$. Prove that either
(i) there exists a nonempty compact set $ C$ such that $ f(U)$ is not a proper subset of $ C$ whenever $ C \subseteq U \in \textbf{U}$,
(ii) or for some compact set $ C$, the set \[ f^{-1}(C)= \bigcup \{U \in \textbf{U}\;: \ \;f(U)\subseteq C\ \}\] is an element of $ \textbf{U}$, that is, $ f^{-1}(C)$ is not contained in any compact set.
[i]A. Mate[/i]
On the $x$-$y$ plane, 3 half-lines $y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)$ intersect with the circle with the center the origin $O$, radius $r\geq 1$ at $A,\ B,\ C$ respectively. Note that $\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}$.
If the area of quadrilateral $OABC$ is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then
evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .$
[i]2011 Waseda University of Education entrance exam/Science[/i]
Let $\mathbb R^\ast$ denote the set of nonzero reals. Find all functions $f: \mathbb R^\ast \to \mathbb R^\ast$ satisfying \[ f(x^2+y)+1=f(x^2+1)+\frac{f(xy)}{f(x)} \] for all $x,y \in \mathbb R^\ast$ with $x^2+y\neq 0$.
[i]Proposed by Ryan Alweiss[/i]
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that:
$xf(y)+yf(x)=(x+y)f(x^2+y^2), \forall x,y \in \mathbb{N}$
Prove that for any three positive real numbers $ a,b,c, \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \ge \frac{9}{2} \cdot \frac{1}{a\plus{}b\plus{}c}$.
Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$.
[i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]