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.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
be a triangle inscribed in circle 
such that points 
are fixed, while 
moves on major arc 
of . The tangents through 
and 
to intersect at 
. The circle with diameter 
intersects 
and 
at 
and 
, respectively. Prove that 
is tangent to a fixed circle whose radius is half the radius of .
and 
be the circumcenter, centroid and symmedian point of 
respectively. is an arbitrary point on Euler line 
Show that the orthocorrespondent of WRT 
falls on 
be a subset of 
such that if 
, then the residue obtained when we divide 
by 
also belongs to .
.
, such that for any convex polygon with area 
, there exist a point 
in the plane, such that the area of convex hull of 
is at most , where 
denotes the image of under central symmetry with respect to .
be positive integers such that 
is a square. Prove that 
is not prime.
with parallel sides 
and 
, assume that there exist points 
on line 
outside segment , and 
inside segment 
such that 
. Denote by 
the point of intersection of and 
, and by 
the point of intersection of and . Let 
be the midpoint of segment , assume it does not lie on line . Prove that belongs to the circumcircle of 
if and only if belongs to the circumcircle of 
.
such that 
is the square of an integer.
prove that
satisfy 
. Prove that:
is the amount of bananas Kyle currently has in his garage (where 
is in hours), then what does 
mean in this context with units or something.
hours, the rate at which the amount of Bananas Kyle has is decreasing by 
bananas per hour per hour)
,
.
