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be non-negative real numbers such that 
Prove that
be the orthocenter of an acute-angled triangle 
. The circle 
centered at the midpoint of 
and passing through intersects the sideline at points 
and 
. Similarly, define the points 
, 
, 
and 
., , , , and are concyclic.
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 
.
, define 
and define 
similarly.
be a positive integer. Then there exists an integer 
, with 
such that 
.
for all , with 
. Then 
and 
are equal to the sam positive integer, say 
. Then we can write 
for 
and 
both smaller then 
. Let 
. Since 
, let 
. Then 
for some real number 
with 
. The key property is that 
. Thus 
Since 
we can easily bound the trailing part by 
. Since 
it must be that the trailing part is equal to 
and 
but this is absurd.
, which all must be distinct.
with 
.
satisfying 
where 
Prove that 
