to satisfy that the equation 
has at least 1 roots 
satisfying the following condition: for every monic polynomial 
of degree at most 
and 
distinct integers 
such that ![\[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]](http://latex.artofproblemsolving.com/e/e/8/ee8adc79dc258b021529245fec16e081da2d1b0a.png)
.
and for 
,
Prove that every natural number occurs exactly once in this sequence.
inscribed in 
,
is the midpoint of arc 
of 
intersects 
at 
.
intersects 
at 
different from 
is the orthocenter of triangle 
= 
and 
Prove that
Let 
Prove that
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
cannot completely cover a square of side length 
,
,
to meet the circumcircle of 
at 
,
.
![$\color[rgb]{0.35,0.35,0.35}v_1=2$](http://latex.artofproblemsolving.com/2/0/4/204b13717fb1dec258e4917232ca9c96344031d2.png)
,![$\color[rgb]{0.35,0.35,0.35}v_2=4$](http://latex.artofproblemsolving.com/b/c/2/bc24e9589aaf75cb848e9fddeef43fb7a6fea2c3.png)
and ![$\color[rgb]{0.35,0.35,0.35} v_{n+1}=3v_n-v_{n-1}$](http://latex.artofproblemsolving.com/8/7/3/8734d09c683e906771abf049d89b273196175585.png)
,![$\color[rgb]{0.35,0.35,0.35}v_n=2F_{2n-1}$](http://latex.artofproblemsolving.com/b/1/e/b1efbac9eef93c5649dab70be54ae2286ca5b2f0.png)
,![$\color[rgb]{0.35,0.35,0.35}F_n$](http://latex.artofproblemsolving.com/0/2/5/025c95f57f5317da3d9739c1620448d8f7cf4e47.png)
are the Fibonacci numbers.
be a point inside equilateral triangle 
,
.
in degrees.
be real numbers so that 
and 
Show that
Y by Adventure10, Mango247
A rectangular piece of paper has top edge 
. A line 
from 
to the bottom edge makes an angle 
with the line . We want to trisect . We take 
and 
on the vertical ege through such that 
. We then fold the paper so that goes to a point 
on the line and goes to a point 
on the horizontal line through . The fold takes to 
. Show that 
and 
are the required trisectors.
. A line 
from 
to the bottom edge makes an angle 
with the line . We want to trisect . We take 
and 
on the vertical ege through such that 
. We then fold the paper so that goes to a point 
on the line and goes to a point 
on the horizontal line through . The fold takes to 
. Show that 
and 
are the required trisectors.
,
,
is horizontal, so 
,
,
is congruent to 
hence 
,
and 
hence 
,
,
hence 
so 
