such that![\[ x f(x + xy) = x f(x) + f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.\]](http://latex.artofproblemsolving.com/b/0/f/b0f1a1d3b9de5d23f1b5a2e9be3b88d8bd575883.png)
be points on the sides 
respectively of a triangle 
.
and 
meet at 
and 
,
and 
,
,
.
to the area of 
.
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
be a metric space and 
be a function such that 
.
Prove that for all 
,
exists, where 
is 
.
Prove that the amount of the limit does not depend on choosing 
.
![\[\iint_{R} \sin(xy) \,dx\,dy, \quad R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]\]](http://latex.artofproblemsolving.com/a/2/0/a205868dd99f4794de53ad1db50f34b17c70a923.png)
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
,
and 
could be chosen to satisfy 
and 
.
be external direct sum of vector spaces 
and 
over a field 
.
and 
and 
is subspaces.
.
be the external direct sum of 
Then 
is
and whose roots are all real.
.
with a finitely generated k-algebra with one generator such that make 
Y by
A competition consists of 
sports, each awarding one gold medal to a winner. athletes participate, each in all sports. There are also experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to . An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number 
such that the experts can make their predictions so that at least of them are guaranteed to be competent regardless of the outcome?
sports, each awarding one gold medal to a winner. athletes participate, each in all sports. There are also experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to . An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number 
such that the experts can make their predictions so that at least of them are guaranteed to be competent regardless of the outcome?This post has been edited 1 time. Last edited by egxa, Apr 18, 2025, 5:21 PM
is equal to 
.
to 
-
where 
is the amount of medals won by athlete 
.
because then if the podium is equal to 
then the most possible amount of judges predicting correctly is equal to 
.
.
.
:
:
of the predictions have these 2 numbers in the same spot so worst case possible 
:
,
