AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a triangle such that 
,
and 
.Let 
be a point on side 
and 
one on side 
so that the line 
is tangent to the incircle of 
.
.
satisfying the equation![\[ 4x^2 - 3y^2 = 1 \]](http://latex.artofproblemsolving.com/d/2/8/d284c2c3a7d7fbe12a7988a4534df16e7adc8947.png)
such that 
.
such that if the positive integer 
is some divisor then 
is a prime power.
satisfying the following equations: 
. 
amount of people stand on coordinates 
where 
. Every person got a water cup and two people are considered to be neighbour if the distance between them is 
. At the first minute, the person standing on coordinates 
got litres of water, and the other 
people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups.
will not have more than 
litres of water.
. Find .
terms of this sequence, she writes the permutation of 
. To define the following terms, she does the following: She takes the last terms of the sequence, and takes its median. How many values could this sequence's 
'th term could get? numbers, you write them in an increasing order,and take the number in the middle)
be positive real numbers such that 
. Prove that
.
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
such that 
.
.
and 
. How many other points with integer coordinates are on the line and strictly between 
and 
?
the points
cake sliced into square units, and you put one diagonal strip of red icing on the cake from the lower left to the lower right, how many square slices of cake get red icing?
to 
, with 
and 
relatively prime, you will NEVER hit a lattice point on the interior. You only get the two exterior lattice points. to 
, with 
relatively prime, then 
is the GCF of 
and 
, and you hit an interior lattice point 
times - you repeat 
rectangles, with the lattice points only coming there.
slices that get hit, and you subtract off the endpoints.
such that: 
.
.
Let 
.
Let 
.
Let 
.
.
,
should be in point 
where 
,
,
value, so the values are: 
,
.
exclusive that 
is 
.
.
has to divide 
.
.
,
.