FE with devisibility

by fadhool, May 10, 2025, 4:25 PM

if when i solve an fe that is defined in the set of positive integer i found m|f(m) can i set f(m) =km such that k is not constant and of course it depends on m but after some work i find k=c st c is constant is this correct

number theory

question

algebra

functional equation

Divisibility

Non-homogenous Inequality

by Adywastaken, May 10, 2025, 3:42 PM

$a, b, c\in \mathbb{R_{+}}$ such that $ab+bc+ca=3abc$ . Show that $a^2b+b^2c+c^2a \ge 2(a+b+c)-3$ . When will equality hold?

This post has been edited 1 time. Last edited by Adywastaken, 4 hours ago

Inequality

algebra

inequalities

Classic Diophantine

by Adywastaken, May 10, 2025, 3:39 PM

Find all natural number solutions to $3^x-5^y=z^2$ .

number theory

Diophantine equation

Exponential equation

Cyclic ine

by m4thbl3nd3r, May 10, 2025, 3:34 PM

Let $a,b,c>0$ such that $a^2+b^2+c^2=3$ . Prove that $\sum \frac{a^2}{b}+abc \ge 4$

inequalities

Iranian geometry configuration

by Assassino9931, May 10, 2025, 9:39 AM

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ , such that $CD$ is not a diameter of its circumcircle. The lines $AD$ and $BC$ intersect at point $P$ , so that $A$ lies between $D$ and $P$ , and $B$ lies between $C$ and $P$ . Suppose triangle $PCD$ is acute and let $H$ be its orthocenter. The points $E$ and $F$ on the lines $BC$ and $AD$ , respectively, are such that $BD \parallel HE$ and $AC\parallel HF$ . The line through $E$ , perpendicular to $BC$ , intersects $AD$ at $L$ , and the line through $F$ , perpendicular to $AD$ , intersects $BC$ at $K$ . Prove that the points $K$ , $L$ , $O$ are collinear.

Amir Parsa Hosseini Nayeri, Iran

This post has been edited 1 time. Last edited by Assassino9931, Today at 9:39 AM

Japan MO Finals 2023

by parkjungmin, May 9, 2025, 2:35 PM

It's hard. Help me

Attachments:

Add d or Divide by a

by MarkBcc168, Jul 9, 2023, 4:40 AM

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$ . Let $x_1=1$ , and for $k\geq 1$ , define
$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$ Find, in terms of $a$ and $d$ , the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$ .

GEOMETRY GEOMETRY GEOMETRY

by Kagebaka, Jul 20, 2021, 8:47 PM

Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX.$ Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD,$ respectively. Prove that the lines $BC, EF,$ and $O_1O_2$ are concurrent.

Alice and Bob play, 8x8 table, white red black, minimum n for victory

by parmenides51, Jul 22, 2019, 3:19 AM

The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice can win the game no matter how Bob plays.

f(m + n) >= f(m) + f(f(n)) - 1

by orl, Jul 13, 2008, 12:47 PM

Consider those functions $f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
$f(m + n) \geq f(m) + f(f(n)) - 1$
for all $m,n \in \mathbb{N}.$ Find all possible values of $f(2007).$

Author: Nikolai Nikolov, Bulgaria

This post has been edited 2 times. Last edited by orl, Feb 12, 2009, 5:18 PM

algebra

Functional inequality

IMO Shortlist

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