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.
that verify the following equation for all 
:
is a prime number
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
. when![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers. be the circumcenter of an acute-angled triangle with 
. The line 
meets the side at . The circumcenters of the triangles 
and 
are 
and 
, respectively. Extend the sides 
and 
beyond 
, and choose on the respective extensions points 
and such that 
and 
. Prove that the quadrilateral 
is a rectangle if and only if 
.
satisfying
Prove
is a positive integer. Call all positive divisors of which are different from 
and beautiful divisors.We call a special number when it has at least 
beautiful divisors and difference of any beautiful divisors divides as well. Find all special numbers.
that verify the following equation for all 
:
be a triangle with incenter and circumcircle 
. Circles 
passing through 
and 
passing through 
are tangent at . Let meet minor arc 
of at and at 
, and let meet minor arc 
of at and at 
. Rays 
and 
meet at 
. Let 
be a point such that 
is tangent to and 
is tangent to .
are collinear.
be an acute triangle and let be an interior point for it such that 
. Prove that
When does equality hold?
denote the set of all positive integers less than 
that are relatively prime to . Let 
. Is the polynomial 
reducible over the rational numbers, given that it has integer coefficients?
blocks, each weighing at least ; their total weight is 
. Prove that for every real number 
with 
you can choose a subset of the blocks whose total weight is at least but at most 
.
be an acute-angled triangle. lies on with between . Let 
be the angle bisector of 
with on . Prove that 
on the segment 
(
) such that 
is the angle bisector of 
.
which is a contradiction since 
. lies on the segment 
with between 
and obtain 
,
it's bigger than 45 degrees.
.
,
by LOS. Also, 
.![\[\frac{CD}{AD} < \frac{BC}{AB}\]](http://latex.artofproblemsolving.com/d/3/6/d36d6a4b21ca2f5e714f180472886fb9c067b90d.png)
.![\[\frac{CD}{AD} < \frac{CE}{AE}\]](http://latex.artofproblemsolving.com/a/7/a/a7a2ec336c4bccf0cf627c7f090ce665c3787d3f.png)
.
.![\[\frac{CF}{AF} < \frac{CE}{AE}\]](http://latex.artofproblemsolving.com/6/e/e/6ee7e3685ca8eb4385835540e5e123ce3150ec84.png)
It follows that 
,
.
is acute,we get 
,
,
,
,
,
.
