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 an isosceles triangle with 
Let 
be a point on the circumcircle of on minor arc 
Let 
intersect the extension of 
at 
Let 
be the midpoint of segment 
and let 
be the intersection of 
and 
Let the extension of 
intersect 
and the circumcircle of at 
and 
respectively. Given that 
and 
compute the length of 
that verify the following equation for all 
:
be positive real integer such that 
Prove that
such that 
and 
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.
. Let be a subset of 
such that no sum of three distinct elements of is divisible by 
. What is the maximum value of 
.
, 
, 
and 
. is a point on the bisector of 
such that 
is perpendicular to this bisector, and is a point on the bisector of 
such that is perpendicular to this bisector. Determine the length of 
, , and . is a point on the bisector of such that is perpendicular to this bisector, and is a point on the bisector of such that is perpendicular to this bisector. Determine the length of 
and 
. We have ![$[ABC]=\sqrt{(3+\sqrt{2})(1+\sqrt{2})(-1+\sqrt{2})(3-\sqrt{2})}=\sqrt{7}$](http://latex.artofproblemsolving.com/3/3/b/33b97db2589dfd8149afa6f90e73e5fc1f7ae04d.png)
. This means that the altitude of 
, namely the y-coordinate of , is 
. Let the x-coordinate of be 
. Then, by the Pythagorean Theorem, 
. Next, 
, so 
. Similarly, 
. From here, we get 
and 
. Solving the systems of equations to find the intersections, we get 
and 
, so 
.
