[/quote]
,
be an acute-angled triangle with 
.
be the incentre of triangle 
be the points where the incircle touches the sides 
respectively. Let 
meet the line 
at 
respectively. Further assume that both 
and 
are outside the triangle 
are concyclic.
.
such that 
is a perfect square but none of the integers 
are perfect squares.
such that![\[
f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1,
\quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/7/a/5/7a51a7be2306d18f4e80db1f5aac6e9bbf25c25c.png)
and 
,
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
and 
Prove that
,
,
be non-negative real numbers such that![\[2a+2b+2c+2d+ab+bc+cd+da+3=abcd.\]](http://latex.artofproblemsolving.com/d/e/8/de8ccd816cfe032a97d940982192b1abea21dc18.png)
prove that : ![\[\sqrt[4]{abc}+\sqrt[4]{bcd}+\sqrt[4]{cda}+\sqrt[4]{dab}\le\sqrt[4]{27(1+a)(1+b)(1+c)(1+d)}.\]](http://latex.artofproblemsolving.com/a/f/c/afc556cd8f5c7f4dedf153ffb92fb1f978c56823.png)
such that 
.
May everyone not to upload solutions on this problem anymore until May 15/2025, this is an active problem on Mathematical Reflection! (Thank you Victoria_Discalceata1)
and 
Where 
Where 
Note: consider 
and 
Y by
Consider the numbers from 
to 
. A game is made by placing all the numbers in pairs and replacing each pair with the largest prime divisor of the sum of the numbers of that couple. For example, if we match the numbers as: 
, we get the following list of 
numbers: 
. where there are repetitions. The game continues in a similar way until in the end only one number remains. Determine the highest possible value from the number that remains at the end.
to 
. A game is made by placing all the numbers in pairs and replacing each pair with the largest prime divisor of the sum of the numbers of that couple. For example, if we match the numbers as: 
, we get the following list of 
numbers: 
. where there are repetitions. The game continues in a similar way until in the end only one number remains. Determine the highest possible value from the number that remains at the end.
can be achieved given a game of 
is impossible as a final answer because it is 
.
