[/quote]
,
be an acute triangle with orthocenter 
.
which is parallel to 
and tangent to the incircles of 
and 
.
are three positive real numbers, prove that 
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
be a positive integer. A positive integer 
digits in decimal representation (it cannot have leading zeros).
is a 
-![\[(20 + 25)^2 = 2025.\]](http://latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
-
be the set of all positive integers greater than or equal to 
and let 
be a function such that 
for all 
if 
,
players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least 
next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
be a fixed positive real number. Find the minimum possible value 
such that for any positive reals 
,
,
,
,
and 
divides 
.
and 
.
such that 
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
![\[
\frac{5^{2000} - 1}{4}
\]](http://latex.artofproblemsolving.com/8/b/a/8ba5d1ed8bafd6653f9e6f351558a9976c43a57a.png)
is divided by 
.
Y by Davi-8191, mathematicsy, Adventure10
Let 
be the area of the triangle 
. A non-regular convex polygon 
is called guayaco if exists a point 
in its interior such that ![\[A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).\]](//latex.artofproblemsolving.com/a/5/e/a5e3a48dd716a4c643d446af8fd8e903bbff8eb8.png)
Show that, for every integer 
, a guayaco polygon of 
sides exists.
be the area of the triangle 
. A non-regular convex polygon 
is called guayaco if exists a point 
in its interior such that ![\[A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).\]](http://latex.artofproblemsolving.com/a/5/e/a5e3a48dd716a4c643d446af8fd8e903bbff8eb8.png)
Show that, for every integer 
, a guayaco polygon of 
sides exists.This post has been edited 1 time. Last edited by adrian97, Aug 18, 2017, 4:48 AM
Reason: Fix typo.
Reason: Fix typo.
,
,
,
congruent isosceles triangles with an angle of 
in such a way that they add up to a flat angle. Then add 2 rectangle triangles with matching areas at the bottom to finish the polygon:
