Prove that
and 
Prove that
Prove that
Where 
Y by
find all function F: IR->IR such that 
G
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Hard Functional Equation in the Complex Numbers
yaybanana 6
N
by jasperE3
Source: Own
Find all functions 
, s.t :
for all
, s.t :
for all
6 replies
FE with conditions on $x,y$
Adywastaken 3
N
by jasperE3
Source: OAO
Find all functions 
such that 
,
![\[
f(x^2+f(y))=f(xf(x))+y
\]](//latex.artofproblemsolving.com/d/e/d/dedd2338779fcc6723077ce586713fb4a68b9931.png)
such that 
,![\[
f(x^2+f(y))=f(xf(x))+y
\]](http://latex.artofproblemsolving.com/d/e/d/dedd2338779fcc6723077ce586713fb4a68b9931.png)
3 replies
Point satisfies triple property
62861 36
N
by cursed_tangent1434
Source: USA Winter Team Selection Test #2 for IMO 2018, Problem 2
Let 
be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at 
. Denote by 
and 
the midpoints of 
and 
. Rays 
and 
meet 
and 
at 
and 
, respectively. Prove that there exists a point 
, lying outside quadrilateral , such that
[list]
[*] ray
bisects both angles 
, 
, and
[*]
.
[/list]
Proposed by Evan Chen
be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at 
. Denote by 
and 
the midpoints of 
and 
. Rays 
and 
meet 
and 
at 
and 
, respectively. Prove that there exists a point 
, lying outside quadrilateral , such that[list]
[*] ray
bisects both angles 
, 
, and[*]
.[/list]
Proposed by Evan Chen
36 replies
Find the minimum
sqing 27
N
by sqing
Source: Zhangyanzong
Let 
be positive real numbers such that 
Find the minimum value of 
be positive real numbers such that 
Find the minimum value of 
27 replies
Sharygin 2025 CR P15
Gengar_in_Galar 7
N
by Giant_PT
Source: Sharygin 2025
A point 
lies on the bisector of an acute angle with vertex . Let 
, 
be the projections of to the sidelines of the angle. The circle centered at with radius 
meets the sidelines at points 
and 
such that 
. Prove that the circle with center touching 
and the circle with center touching 
are tangent.
Proposed by: A.Zaslavsky
lies on the bisector of an acute angle with vertex . Let 
, 
be the projections of to the sidelines of the angle. The circle centered at with radius 
meets the sidelines at points 
and 
such that 
. Prove that the circle with center touching 
and the circle with center touching 
are tangent.Proposed by: A.Zaslavsky
7 replies
Inequality olympiad algebra
Foxellar 1
N
by sqing
Given that 
are nonzero real numbers such that
![\[
\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b},
\]](//latex.artofproblemsolving.com/6/d/9/6d9aabd8130b99d7128de9a3f063eb4c7124febf.png)
let 
be the maximum value of the expression
![\[
\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}.
\]](//latex.artofproblemsolving.com/a/b/2/ab28769126365574d4c0108500c370826082de6b.png)
Determine the sum of the numerator and denominator of the simplified fraction representing .
are nonzero real numbers such that![\[
\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b},
\]](http://latex.artofproblemsolving.com/6/d/9/6d9aabd8130b99d7128de9a3f063eb4c7124febf.png)
let 
be the maximum value of the expression![\[
\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}.
\]](http://latex.artofproblemsolving.com/a/b/2/ab28769126365574d4c0108500c370826082de6b.png)
Determine the sum of the numerator and denominator of the simplified fraction representing .
1 reply
interesting function equation (fe) in IR
G
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G
H
Source: mine
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or 
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Y by
skellyrah wrote:
find all function F: IR->IR such that
If IR means irrational numbers, and the problem is to find all
such that 
for all 
then setting 
gives a contradictionThis post has been edited 1 time. Last edited by jasperE3, Apr 23, 2025, 9:43 PM
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G
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