[/quote]
,
with 
and 
.
?
of positive integers such that
be a circle inscribed inside a rhombus 
.
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to 
is the same.
,
such that any graph with 
vertices and at least 
passing through point 
is an oblique strophoid.
.
and 
externally on the sides 
and 
respectively.
meets 
at 
,
meets 
at 
and 
denotes circumcenter of 
and 
meets again on 
.
be an acute triangle and let 
be the foot of the 
be the midpoint of 
with diameter 
meets 
at 
,
with diameter 
at 
.
,
are concyclic.
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
having 
distinct real roots 
)
: 
.
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
.
such that 
.
![$P\in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/7/4/c/74cb2247069abb8a22666e87fcbd69a55a015004.png)
for which there exists 
such that for all 
we have: ![\[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]](http://latex.artofproblemsolving.com/9/0/9/90954cdeade9d13149e1aaa03df2c87dff157041.png)
Y by
Let's construct a family 
of geometric figures following the pattern shown in pictures:
where each hexagon (like the starting one) is constructed by cutting the two corners tops of a square, in such a way that the two figures removed are identical isosceles triangles, and the three resulting upper sides have the same length.
Continuing like this, a pattern is produced with which we can build the figures
, for integer 
. Then, we denote by 
and 
the perimeter and area of the figure , respectively. If the side of square to build 
measures 
:
(a) Calculate
and 
(in terms of the length ).
(b) Find an explicit formula for, and for , in terms of and of 
. Simplify your answers.
(c) If
, find the measure of the six sides of the figure , in its simplest form.
of geometric figures following the pattern shown in pictures:
Continuing like this, a pattern is produced with which we can build the figures
, for integer 
. Then, we denote by 
and 
the perimeter and area of the figure , respectively. If the side of square to build 
measures 
:(a) Calculate
and 
(in terms of the length ).(b) Find an explicit formula for
, and for , in terms of and of 
. Simplify your answers.(c) If
, find the measure of the six sides of the figure , in its simplest form.This post has been edited 1 time. Last edited by parmenides51, Mar 24, 2024, 2:10 PM
