[/quote]
,
be the set of integers. Determine all functions 
such that, for all integers 
and 
,
Proposed by Liam Baker, South Africa
such that 
be a convex pentagon such that![\[ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.
\]](http://latex.artofproblemsolving.com/d/3/d/d3d9a82f4318190298a8f008d417a364e03f1fca.png)
The diagonals 
and 
meet at 
.
bisects the side 
.
be an acute-angled triangle with 
.
be the orthocenter of triangle 
,
be the midpoint of the side 
.
be a point on the side 
and 
a point on the side 
such that 
and the points 
is perpendicular to the common chord of the circumscribed circles of triangle 
.
,
,
are fixed on a line in this order. Let 
be a circle passing through 
at 
.
and the line 
and 
be two points inside triangle 
such that![\[ \angle MAB = \angle NAC\quad \mbox{and}\quad \angle MBA = \angle NBC.
\]](http://latex.artofproblemsolving.com/2/0/4/20410178c70693881f348c67454f01057a3fc40c.png)
![\[ \frac {AM \cdot AN}{AB \cdot AC} + \frac {BM \cdot BN}{BA \cdot BC} + \frac {CM \cdot CN}{CA \cdot CB} = 1.
\]](http://latex.artofproblemsolving.com/b/b/6/bb64f80eccb4634b9c9e15cade2f7c0b87b60923.png)
the set of all positive real numbers.
which have the property:![\[f(x)f(y)=2f(x+yf(x))\]](http://latex.artofproblemsolving.com/4/f/f/4ffbde35ff082585a188c1a4368d39338d1b07a4.png)
and 
.
When does equality hold?
and then factor the terms with x and then take log base 2 for both sides? I am not sure how to do it, so could you please help?
,
to the sides 
are 
,
and 
intersect at 
.
from 
.
are colinear
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this is my 100th post! i cant believe it. 
P.S. can everyone who reads this give it an upvote? thx!
P.S. can everyone who reads this give it an upvote? thx!
This post has been edited 1 time. Last edited by JohannIsBach, Mar 10, 2025, 4:08 PM
more posts to get to 1000!
more 2 go
