[/quote]
,
\end{table}
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
such that 
and 
for any real numbers 
and 
.
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
,
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
and 
as ![\[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\]](http://latex.artofproblemsolving.com/4/d/5/4d5049d5c3b26757a25c5dd34462c5462228b83d.png)
where each 
or 
is a digit between 0 and 9. Prove that there exist at least 1000 values of 
between 
and 
such that 
.
of positive integers with 
is called good if we can colour each integer red, blue, green or purple, in such a way that
of each 
of each 
consecutive integers at least one is coloured green;
of each 
consecutive integers at least one is coloured purple.
.
,
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
such that :
.
such that 
,
and 
to obtain 
and 
,
be the intersection of the altitude through 
and the median through 
.
be the intersection of the altitude through 
and line 
.
be the intersection of the median through 
.
either the sum of the first 
are positive real numbers satisfied 
.
Y by PikaPika999
9Poll:
How was the difficulty of the problems compared to last year?
165 Votes
59%
(97)
25%
(41)
4%
(7)
5%
(9)
2%
(4)
4%
(7)
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I personally thought the problems were much easier than last year, but I didn't really improve as much as I would of liked to improve.
