[/quote]
,
Prove that
be integers, let 
be a set with 
elements, and let 
be pairwise distinct non-empty, not necessary disjoint subset of 
is called nice if there exists an index 
such that ![\[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\]](http://latex.artofproblemsolving.com/3/d/a/3da5ad1bcade98ee08caa35763de226027bcb2c6.png)
Prove that the number of nice functions is at least 
.
.![\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]](http://latex.artofproblemsolving.com/b/3/2/b32e0bcba967ca847e878ac186c3168cd6c5fc66.png)
be positive real numbers such that 
Find the minimum value of 
Prove that
Prove that
,
,
are fixed on a line in this order. Let 
be a circle passing through 
.
the intersection of the tangents to 
at 
.
and the line 
Prove that
Where 
be a triangle with circumcentre 
.
and 
are interior points of the sides 
and 
respectively. Let 
and 
be the midpoints of the segments 
and 
.
be the circle passing through 
satisfying
Prove
.
has at most 
and 
Prove that
such that:
for all 
be an acute-angled scalene triangle with circumcenter 
.
,
,
the midpoints of sides 
,
,
,
be the circle passing through 
and tangent to 
at 
at points 
and 
,
be the midpoint of segment 
,
be the intersection of lines 
.
and that triangle 
is isosceles.
Y by Funcshun840
Maria wants to solve a challenge with a deck of cards, each with a different figure. Initially, the cards are distributed randomly into two piles, not necessarily in equal parts. Maria's goal is to get all the cards into the same pile.
On each turn, Maria takes the top card from each pile and compares them. In the rule book, there's a table that indicates, for each card match, which of the two wins. Both cards are then placed on the bottom of the winning card in the order Maria chooses. The challenge ends when all the cards are in one pile.
Show that it is always possible for Maria to solve the challenge. Regardless of the initial distribution of the cards and the table in the rule book.
On each turn, Maria takes the top card from each pile and compares them. In the rule book, there's a table that indicates, for each card match, which of the two wins. Both cards are then placed on the bottom of the winning card in the order Maria chooses. The challenge ends when all the cards are in one pile.
Show that it is always possible for Maria to solve the challenge. Regardless of the initial distribution of the cards and the table in the rule book.
