be real numbers such that![\[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \]](http://latex.artofproblemsolving.com/7/e/b/7ebfb2f663d390714310d3c43b9c68e1b101d632.png)
.
is a perfect square.
for which there exist real numbers 
satisfying 
,
and
for 
.
and 
Prove that
Y by Adventure10, Mango247
Let 
be an integer. What is the largest possible number of interior angles greater than 
in an 
-gon in the plane, given that the -gon does not intersect itself and all its sides have the same length?
be an integer. What is the largest possible number of interior angles greater than 
in an 
-gon in the plane, given that the -gon does not intersect itself and all its sides have the same length?
the answer is clearly 
.
a quadrilateral with four equal sides is a rhombus which is convex, so the answer is 
.
,
of these concave angles.
,
.
segments so that they become a bit longer and form an arc inside the original triangle. Then we also make the short side of the triangle slightly larger.
concave angles.
with 
we do the same with a triangle of sides 
(which exists since 
)
.
