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.
= 2∠AMP.
, and an odd prime 
not dividing any of them, and an integer 
such that if one considers the infinite sequence 
and looks at the highest power of that divides each of them, these powers are not all zero, and are all at most . Prove that there exists some 
(which may depend on 
) such that whenever divides an element of this sequence, the maximum power of that divides that element is exactly 
.
be a positive integer. Consider 
points on the circumference of a circle. pairs, and line segments are drawn connecting the points in each pair.
} such that for every 2 different number 
and 
in this set at last in one of the permutation comes exactly after
: 
such that:
convex polygons, determine the maximum possible number of edges in the obtained figure. polygons with 
vertices and 
edges in the resulting figure, then 
.)
write all its divisors in increasing order: 
. such that 
.
be an inscribed quadrilateral. Circles of diameters 
and 
intersect at points 
and 
, and circles of diameters 
and 
intersect at points 
and 
. The circles of diameters 
and 
intersect in two points 
and 
. Prove that the lines 
and 
are concurrent.
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
for which there exists a sequence 
of positive integers satisfying the following conditions:[list]
for all integers 
for all integers 
the sequence 
is a permutation of
[/list]
.
be a cyclic quadrilateral. Let 
be the midpoint of the arc 
of its circumcircle which does not contain 
or 
. Let the lines 
and 
meet at 
and the lines 
and 
meet at 
. Prove that the lines 
and are parallel.
to bring it in plane 
, then, if 
is the new position of , 
is an isosceles trapezoid, hence 
, but 
.
and the sum of measures of the angles 
and 
equals 
degrees. Prove that the measure of the angle 
.