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
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
such that 
.
be a quadrilateral inscribed in a circle 
, and let 
be a point on the extension of 
such that 
and 
are tangent to . The tangent at 
intersects at 
and the line 
at 
. Let 
be the second point of intersection between 
and . Prove that 
, , are collinear.
be a triangle with incenter 
, and suppose the incircle is tangent to 
and 
at and 
. Denote by 
and 
the reflections of and over . Let be the intersection of 
with 
, and let 
be the midpoint of . Prove that 
and 
are perpendicular.
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 function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter , centroid and orthocenter . Prove that 
is constant.
opposite the vertex 
be tangent to the side at the point 
. Define the points 
on and 
on analogously, using the excircles opposite and , respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled. and , Jaqueline has a ruler that allows tracing the line 
. Jaqueline also has a special object that allows the construction of a circle of diameter . Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle . Show that Jaqueline can construct the incenter of .
be triangle with point 
and on 
and 
are on the same side of be midpoint of segment and 
be midpoint of segment 
be intersection of 
with 
and 
be positive real integer such that 
Prove that
such that 
and 
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 
.
and 
be the lines tangent to a circle 
drawn from a point 
and belong to the circle). A line through cuts at and with between and , and 
. If is the intersection point of and 
, calculate 
.
and be the lines tangent to a circle drawn from a point and belong to the circle). A line through cuts at and with between and , and . If is the intersection point of and , calculate .
is a harmonic bundle, and so 
. WLOG, let 
and 
. Then:![\[\frac{2}{x} = \frac{5}{3 - x}\]](http://latex.artofproblemsolving.com/9/7/6/9762f85e73aa5a7da618192ca5894357bb6d7b9d.png)
and so 
. It follows that 
.
