Bounding number of solutions for floor function equation

by Ciobi_, Apr 2, 2025, 12:39 PM

Let $n \geq 2$ be a positive integer. Consider the following equation: $\{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor$ a) For $n=2$ , solve the given equation in $\mathbb{R}$ .
b) Prove that, for any $n \geq 2$ , the equation has at most $2$ real solutions.

floor function

algebra

Bounding

Normal but good inequality

by giangtruong13, Mar 31, 2025, 4:04 PM

Let $a,b,c> 0$ satisfy that $a+b+c=3abc$ . Prove that: $\sum_{cyc} \frac{ab}{3c+ab+abc} \geq \frac{3}{5}$

inequalities

Function on positive integers with two inputs

by Assassino9931, Jan 27, 2025, 10:03 AM

The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$ . Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$ . Determine all possible values of $f(2025, 2025)$ .

This post has been edited 1 time. Last edited by Assassino9931, Jan 27, 2025, 10:06 AM

function

number theory

functional equation

Inequalities

by idomybest, Oct 15, 2021, 9:16 PM

The problem is in the attachment below.

Attachments:

Inequality

inequalities

inequalities unsolved

algebra unsolved

algebra

Concurrency

by Dadgarnia, Mar 12, 2020, 10:54 AM

Let $ABC$ be an isosceles triangle ( $AB=AC$ ) with incenter $I$ . Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$ . $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$ , respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$ . Prove that $AD$ , $MN$ and $BC$ are concurrent.

Proposed by Alireza Dadgarnia

Nice inequality

by sqing, Apr 24, 2019, 1:01 PM

Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be real numbers . Prove that : There exist positive integer $k\in \{1,2,\cdots,n\}$ such that $\sum_{i=1}^{n}\{kx_i\}(1-\{kx_i\})<\frac{n-1}{6}.$ Where $\{x\}=x-\left \lfloor x \right \rfloor.$

inequalities

China

Tiling rectangle with smaller rectangles.

by MarkBcc168, Jul 10, 2018, 11:25 AM

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.

Proposed by Jeck Lim, Singapore

This post has been edited 2 times. Last edited by MarkBcc168, Jul 15, 2018, 12:57 PM

nice system of equations

by outback, Oct 8, 2008, 2:41 PM

Solve in positive numbers the system

$x_1+\frac{1}{x_2}=4, x_2+\frac{1}{x_3}=1, x_3+\frac{1}{x_4}=4, ..., x_{99}+\frac{1}{x_{100}}=4, x_{100}+\frac{1}{x_1}=1$

inequalities

A magician has one hundred cards numbered 1 to 100

by Valentin Vornicu, Oct 24, 2005, 10:21 AM

A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.

How many ways are there to put the cards in the three boxes so that the trick works?

Two circles, a tangent line and a parallel

by Valentin Vornicu, Oct 24, 2005, 10:15 AM

Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$ . Let $AB$ be the line tangent to these circles at $A$ and $B$ , respectively, so that $M$ lies closer to $AB$ than $N$ . Let $CD$ be the line parallel to $AB$ and passing through the point $M$ , with $C$ on $G_1$ and $D$ on $G_2$ . Lines $AC$ and $BD$ meet at $E$ ; lines $AN$ and $CD$ meet at $P$ ; lines $BN$ and $CD$ meet at $Q$ . Show that $EP = EQ$ .