Hard Functional Equation in the Complex Numbers

by yaybanana, Apr 9, 2025, 3:29 PM

Find all functions $f:\mathbb {C}\rightarrow \mathbb {C}$ , s.t :

$f(xf(y)) + f(x^2+y) = f(x+y)x + f(f(y))$

for all $x,y \in \mathbb{C}$

functional equation

algebra

function

complex numbers

a/b+b/c+c/a=3

by ilovemath0402, Jan 1, 2025, 11:41 AM

Find all integer $a,b,c$ such that
$\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}=3$ some confuse

Primes and sets

by mathisreaI, Jul 13, 2022, 2:53 AM

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$ .

This post has been edited 1 time. Last edited by v_Enhance, Jul 16, 2022, 12:38 AM

center of (XYZ) lies on a fixed circle

by VicKmath7, Apr 19, 2022, 5:35 PM

An acute-angled triangle $ABC$ is fixed on a plane with largest side $BC$ . Let $PQ$ be an arbitrary diameter of its circumscribed circle, and the point $P$ lies on the smaller arc $AB$ , and the point $Q$ is on the smaller arc $AC$ . Points $X, Y, Z$ are feet of perpendiculars dropped from point $P$ to the line $AB$ , from point $Q$ to the line $AC$ and from point $A$ to line $PQ$ . Prove that the center of the circumscribed circle of triangle $XYZ$ lies on a fixed circle.

points on sides of a triangle, intersections, extensions, ratio of areas wanted

by parmenides51, Jul 28, 2018, 10:13 PM

Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$ . Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$ , and $AP$ and $BQ$ meet at $C'$ , such that $AB' = B'C', BC' =C'A'$ , and $CA'= A'B'$ . Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$ .

ratio

geometry

areas

integer functional equation

by ABCDE, Jul 7, 2016, 7:52 PM

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that $f(x-f(y))=f(f(x))-f(y)-1$ holds for all $x,y\in\mathbb{Z}$ .

IMO Shortlist

functional equation

IMO Shortlist 2013, Algebra #2

by lyukhson, Jul 9, 2014, 4:32 PM

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d$ , such that $\left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}.$

Three numbers cannot be squares simultaneously

by WakeUp, May 18, 2011, 4:48 PM

Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.

inequalities

quadratics

number theory proposed

number theory

xf(x + xy) = xf(x) + f(x^2)f(y)

by orl, Sep 10, 2008, 9:45 PM

Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that
$x f(x + xy) = x f(x) + f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.$

function

algebra

Problem G5 - IMO Shortlist 2007

by April, Jul 13, 2008, 1:53 AM

Let $ABC$ be a fixed triangle, and let $A_1$ , $B_1$ , $C_1$ be the midpoints of sides $BC$ , $CA$ , $AB$ , respectively. Let $P$ be a variable point on the circumcircle. Let lines $PA_1$ , $PB_1$ , $PC_1$ meet the circumcircle again at $A'$ , $B'$ , $C'$ , respectively. Assume that the points $A$ , $B$ , $C$ , $A'$ , $B'$ , $C'$ are distinct, and lines $AA'$ , $BB'$ , $CC'$ form a triangle. Prove that the area of this triangle does not depend on $P$ .

Author: Christopher Bradley, United Kingdom

geometry

circumcircle

IMO Shortlist