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 90-degree triangle with hypotenuse 
. Let 
and 
distinct points on segment and 
be the foot of the perpendicular from to 
and to 
, respectively. 
and 
intersect at 
.
and intersect at 
and lines 
and intersect at 
.
if and only if 
.
be real numbers less than 2.Then prove that 
be an odd integer, 
. Consider 
integers 
such that for any 
, 
and 
. Prove that the number of 
for which satisfy 
for some 
, 
is greater than 
.
of positive integers such that ![\[t^{2}+1=s(s+1).\]](http://latex.artofproblemsolving.com/f/6/6/f665bfe4fd6cec17dbc5b90d42d201274ff7c564.png)
such that 
for all reals 
.
be the sequence of all natural numbers which are sum of squares of two natural numbers.
which 
.
intersect at 
, and there exist a circle 
tangent to the extensions of 
at 
respectively. Circle 
passes through points 
, and is externally tangent to circle at 
. Prove that 
.
such that 
for reals 
.
be the set of all non-empty sub-strings of "Socrates".
but "teS", "sorat", "setarcoS" 
.
and 
as follows: take the string with the first term that appears first in "Socrates" (if both have the same first term, take the shorter one) and WLOG let it be . Then the sum of these two strings will be the string formed by all the letters between the first term of and the last term of . For instance, "Socr" + "te" 
"Socrate" and "cra" + "ates" "crates".
such that
.
is still the same.
that can be written in the form
where 
are pairwise coprime positive integers such that 
, 
, and 
.
such that 
. Prove that
be an odd integer greater than 1 and let 
be integers. For each permutation 
of 
, define 
. Prove that there exist permutations 
of such that 
is a divisor of 
.
.Equality conditions gives us the result.
and again equality conditions.
and again some boring equality cases.
and 
: Prove that :
and 
Prove that
be reals such that 
and 
Prove that
Let 
Prove that
Prove that