[/quote]
,
be the incentre of a non-equilateral triangle 
,
be the 
-
be the reflection of 
,
be the reflection of line 
in 
.
,
and line 
analogously. Let 
be the intersection point of 
where 
is the circumcentre of triangle 
and 
.
.
such that 
is a perfect square and 
is a prime number.
,
,
,![\[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \]](http://latex.artofproblemsolving.com/9/f/a/9fa6cca20b3620fbb32ee782196e96f6fffa0063.png)
such that for any real value of 
we have:
,
,![\[ f(x)f(yf(x))=f(x+y). \]](http://latex.artofproblemsolving.com/5/6/0/560049dd70944cafa0423e5f1229d215f78215a0.png)
ones are written on a blackboard. Grisha and Gleb are playing game, by taking turns; Grisha goes first. On one move Grisha squares some 
numbers on the board. On his move, Gleb picks a few (perhaps none) numbers on the board and increases each of them by 
.
moves on the board a number divisible by 
appears, Gleb wins, otherwise Grisha wins. Which of the players has a winning strategy?
be a nonempty subset of the points in the Cartesian plane such that for each 
exactly one of 
or 
also belongs to 
there is a line in the plane (possibly different lines for different 
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)
such that 
is a divisor of 
.
players of a hockey team gather to select their team captain. Initially, they stand in a circle, and each person votes for the person on their left.
and 
then 
(Note that 
and 
then 
be a positive integer that is not divisible by 
.
be a positive integer. Prove that none of the numbers of the form 
is a square number.
,
,
lie on a semicircle 
in this order with diameter 
,
and 
respectively such that 
and 
.
passes through 
.
,
are concurrent.
if the following conditions are satisfied: 
,
,
.
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
.
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 
:
,
