[/quote]
,
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
be an acute triangle. Let 
be a point on side 
be a point on side 
are parallel. Let 
be an interior point of 
.
and 
meet side 
,
and 
.
and 
intersect at a point 
.
,
are collinear.
.
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
is given. Let 
to 
be the intersection point of 
lies on the arc 
.
.
with diameter 
meets 
and 
(where 
.
is perpendicular to 
.
![\[ x + y^2 = p^m \]](http://latex.artofproblemsolving.com/e/c/f/ecf553036f22327dd457dc06611b6bc1b0ef072c.png)
![\[ x^2 + y = p^n \]](http://latex.artofproblemsolving.com/5/b/d/5bd672934a395aabd6865ca4b50d8da75c0fe07e.png)
For 
positive integers and 
being a prime.
and 
.
Equality holds when 
or 
.
intersect again at 
.
Y by
Let 
be a tetrahedron that is not degenerate and not necessarily regular, where sides 
and 
have the same length 
, sides 
and 
have the same length 
, side 
has length 
and the side 
has length 
. There is a point 
for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities 
and .
be a tetrahedron that is not degenerate and not necessarily regular, where sides 
and 
have the same length 
, sides 
and 
have the same length 
, side 
has length 
and the side 
has length 
. There is a point 
for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities 
and .This post has been edited 2 times. Last edited by parmenides51, Mar 25, 2021, 7:37 AM
