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.
, let 
and 
be the circumcenter and orthocenter, respectively. Let 
and 
be the midpoints of 
and 
, respectively, and let 
and 
be the feet of the perpendiculars from 
and 
to the opposite sides, respectively. Show that if 
is the intersection of 
and 
, then 
is perpendicular to 
.
. Prove that:
. Prove that:![$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$](http://latex.artofproblemsolving.com/d/3/c/d3cf439c79cb84ca23083e51b6190ab5118f34e8.png)
denote the sum of all integers 
such that 
and exactly one of 
and 
is divisible by 
. Compute the remainder when is divided by 
.
.
of positive integers that satisfy
.
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](http://latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](http://latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](http://latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](http://latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
with two altitudes 
and 
.Let be the point on the line such that 
.The lines 
and 
intersect at point , and 
is the point on segment 
such that 
.Suppose that bisects .Prove that triangle 
is isosceles.
![\[\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}\]](http://latex.artofproblemsolving.com/5/d/6/5d6881be29327bcdc3e7bd76c221968578c7f50b.png)
is isosceles.
, 
, and 
are the roots of 
, compute 
.
Prove that one can find pairwise coprime integers 
such that the set of prime divisors of the numbers 
and 
coincides.
be an acute triangle, let 
be a median, and let 
and 
be the altitudes. Let 
be the line perpendicular to passing through 
. Let be the intersection point of with , and let 
be the intersection point of with . is the midpoint of .
be the circumscribed circle of the triangle 
, and let 
and 
be any two circles that have two points 
and 
in common, and are tangent to the segment and the arc of not containing the point . Prove that points 
are collinear.
for an odd natural number 
Prove that 
divides 
for all 
.
cannot divide . On the other hand, divides for 
. It is a case of "generalized Wilson theorem", proved very much like the original Wilson theorem: all residues 
modulo such that 
are split into pairs with product 1, and 8 square roots of 1 modulo have product 1.
is easy.
by induction hypothesis, as desired.
for an odd natural number Prove that divides for all . is true because 
, so now assume that 
for some 
. The key claim is that:
which is true because:
Then from we have that 
(since 
more than an even number is even), so multiplying these gives 
. Then, because of the above result, we have 
, completing the proof.
.
and 
,
and 
. Prove that
,such that 
P
