How can I prove boundness?

by davichu, Apr 27, 2025, 5:55 PM

Solve $f(t^2+u)=tf(t)+f(u)$ over $\mathbb{R}$

Is easy to show that f satisfies Cauchy's functional equation, but I can't find any other property to show that $f$ is linear

Functional Equations

algebra

functional equation

An equation from the past with different coefficients

by Assassino9931, Apr 27, 2025, 1:00 PM

Let $n$ be an integer. Prove that $n^4 - 12n^2 + 144$ is not a perfect cube of an integer.

all functions satisfying f(x+yf(x))+y = xy + f(x+y)

by falantrng, Apr 27, 2025, 11:52 AM

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ ,
$f(x+yf(x))+y = xy + f(x+y).$
Proposed by Giannis Galamatis, Greece

This post has been edited 1 time. Last edited by falantrng, Today at 12:02 PM
Reason: added author

functional equation

function

algebra

external bisector in 2 angle

by crocodilepradita, Aug 22, 2024, 5:51 AM

Given a $\triangle ABC$ with incenter $I$ . Line $BI$ and $CI$ intersects $CA$ and $AB$ at $E$ and $F$ , respectively. Let $M$ and $N$ be the midpoints of $BI$ and $CI$ . Line $FM$ meets the external bisector of angle $B$ at $K$ and line $EN$ meets the external bisector of angle $C$ at $L$ . Prove that $K, B, L, C$ are concylic.

This post has been edited 1 time. Last edited by crocodilepradita, Aug 22, 2024, 5:51 AM

geometry

GJMO 2022/1: Cyclic Isosceles Pentagon

by CyclicISLscelesTrapezoid, May 15, 2022, 6:34 AM

Let $ABCDE$ be a cyclic pentagon with $AB=CD$ and $BC=DE$ . Let $P$ and $Q$ be points on $\overline{CB}$ and $\overline{CD}$ , respectively, such that $BPQD$ is cyclic. Let $M$ be the midpoint of $\overline{BD}$ . Prove that lines $CM$ , $AP$ , and $\text{[math]}$ concur.

Proposed by Tiger Zhang, USA

This post has been edited 5 times. Last edited by CyclicISLscelesTrapezoid, Feb 3, 2025, 3:54 AM
Reason: edited in real name

geometry

GJMO

GJMO 2022

FE over R

by jasperE3, Apr 27, 2021, 2:21 AM

Find all functions $f:\mathbb R\to\mathbb R$ such that $f(x+f(y))+f(xy)=f(x+1)f(y+1)-1\forall x,y\in\mathbb R$ .

function

algebra

functional equation

FE over R+

by jasperE3, Apr 5, 2021, 2:35 AM

Find all functions $f:\mathbb R^+\to\mathbb R^+$ such that for any $x,y>0$ ,
$x^2\left(f(x)+f(y)\right)=(x+y)f\left(f(x)y\right).$

This post has been edited 1 time. Last edited by jasperE3, Apr 5, 2021, 2:36 AM

functional equation

function

algebra

Inequality with condition a+b+c = ab+bc+ca (and special equality case)

by DoThinh2001, May 2, 2019, 11:41 AM

Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$
Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases.

(Edit: Proposed by sir Leonard Giugiuc, Romania)

This post has been edited 1 time. Last edited by DoThinh2001, May 3, 2019, 10:09 AM

inequalities

Problem 5

by codyj, Jul 11, 2015, 6:29 AM

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation $f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$ for all real numbers $x$ and $y$ .

Proposed by Dorlir Ahmeti, Albania

This post has been edited 5 times. Last edited by djmathman, Jun 14, 2018, 4:22 PM
Reason: italicized author

IMO ShortList 2002, geometry problem 7

by orl, Sep 28, 2004, 1:00 PM

The incircle $\Omega$ of the acute-angled triangle $ABC$ is tangent to its side $BC$ at a point $K$ . Let $AD$ be an altitude of triangle $ABC$ , and let $M$ be the midpoint of the segment $AD$ . If $N$ is the common point of the circle $\Omega$ and the line $KM$ (distinct from $K$ ), then prove that the incircle $\Omega$ and the circumcircle of triangle $BCN$ are tangent to each other at the point $N$ .

Attachments:

This post has been edited 1 time. Last edited by orl, Oct 25, 2004, 12:16 AM

Submit

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it started when I was an infant
wow and like 1.5 million views
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tbh it scares me too I still dont know where those views came from
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this blog started when i was like 2 months old
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1579363 visits rn, curious how many views (if any) this blog gets now
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EpicSkills32 reaped on Jul 27, 18:44:19 and gained 10 minutes, 57 seconds

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1.5M views pro
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BTW, lets talk about the NBA playoffs 2012
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MY BLOG IS MORE BORING
ALONG WITH OCCASIONAL MISUSE OF CAPS
by Grindar, May 13, 2012, 5:38 PM
contrib???
by El_Ectric, May 10, 2012, 7:13 PM
by EpicSkills32, May 2, 2012, 5:22 AM
by EpicSkills32, Apr 26, 2012, 5:11 PM

536 shouts

Seems

by davichu, Apr 27, 2025, 5:55 PM

by Assassino9931, Apr 27, 2025, 1:00 PM

by falantrng, Apr 27, 2025, 11:52 AM

by crocodilepradita, Aug 22, 2024, 5:51 AM

by CyclicISLscelesTrapezoid, May 15, 2022, 6:34 AM

by jasperE3, Apr 27, 2021, 2:21 AM

by jasperE3, Apr 5, 2021, 2:35 AM

by DoThinh2001, May 2, 2019, 11:41 AM

by codyj, Jul 11, 2015, 6:29 AM

by orl, Sep 28, 2004, 1:00 PM