[/quote]
,
satisfy 
.
and the incircle 
of 
at 
,
meets 
at 
and intersects 
at 
,
meets 
at 
.
(different from 
)
and 
.
.
are being replaced with 
.
will appear on the board.
,
,
,
lie on a semicircle 
in this order with diameter 
,
and 
lie on segments 
respectively such that 
and 
.
passes through 
.
,
are concurrent.
such that 
is a divisor of 
.
such that for all positive real numbers 
,![\[
f^2(x) \geq f(x+y)\left(f(x) + y\right).
\]](http://latex.artofproblemsolving.com/7/1/f/71fb0d51492ee4c3b2bf4049e8224882fec16020.png)
.
the equation: 
,
.
lie on segments 
respectively. Points 
are such that 
and 
are equlateral, 
lie on opposite sides of 
,
lie on the opposite siedes of 
is a parallelogram. Take 
on the line 
so that 
and 
.
.
and 
be two points lying on the circle 
such that 
and 
are perpendicular and parallel to bisector of 
,
and 
bisect two arcs 
.
be given positive integer. Suppose that there is only one positive integer 
which satisfies the equation
for some positive integers 
.
be positive real numbers such that 
.
knowing that the denominators are positive real numbers.
such that 
,
.
with integer coefficients such that for any positive integer 
,
is positive and 
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
.
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 
:
,
