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.
such that, for each positive divisor 
of , there are integers 
and 
such that 
.
. For the triangle that has the minimum area while satisfying the condition, find the lengths of the other two sides.
be a triangle with circumcenter 
and let the angle bisector of 
intersect 
. The point 
is such that 
and 
. Lines 
and 
intersect at
. Show that the circle centered at and passing through 
is tangent to segment.
be reals such that 
.Prove that
Where 
is divisible by 
.(where 
)Prove that the number 
is divisible by 
for which the inequality ![\[ (a^2+3)(b^2+3)(c^2+3)(d^2+3) + k(a-1)(b-1)(c-1)(d-1) \ge 0 \]](http://latex.artofproblemsolving.com/4/d/b/4db42ebabc7309bba9629a9a7125be43abf113f0.png)
holds for all real numbers 
, 
, 
, and . be an acute triangle (
) with circumcircle 
. Point 
is chosen arbitrarily on arc 
, not containing 
. Points and 
lie on lines 
and respectively, such that 
- parallelogram. Point is chosen on arc , not containing , such that 
. Line 
intersects 
at 
, and line 
intersects 
at 
. Line 
intersects , and 
at points , 
and 
respectively.
.
: 
such :
be a general quadrangle which does not form an orthocentric system. Let , 
, , be the Euler-Poncelet point (
), isogonal center (
), midray homothetic center (
), inscribed square axes crosspoint (
) of , respectively.
be the orthocenter of 
, and define 
, 
, 
cyclically.
(
).
(
).
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
lie on the same conic, here denoted by 
.
be the nine-point conic of . and have 4 distinct real intersection points, and let 
, 
, 
be the intersections, other than , of and . with respect to 
passes through and , and show that the center of and the orthocenter of coincide with .
girls and boys are to be seated in a line. Find the total number of ways such that boys and girls are alternate and a particular boy and girl are never adjacent to each other in any arrangement.
and be complex numbers. For a complex number 
where 
, define the norm 
to be the distance of 
from the origin, or 
. Let 
be the minimum value and be the maximum value of 
for all complex numbers 
where 
. Find 
.
and we have to min/max the quantity 
.
with equality when 
. We also have
and with the two other cases we find 
. Equality when 
.
and 
Prove that
be real numbers . Prove that
be real numbers . Prove that
