AoPS Academy
+1 w
be a strictly increasing sequence of positive integers such that 
What is the remainder when 
is divided by 
?
. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
![\[9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}\]](http://latex.artofproblemsolving.com/5/a/c/5ac98581227bfe75321e66ff87c8e1e881c8abab.png)
is divided by 
.
and 
, respectively? can't u have 2 other roots like 2 and 3, for example where they are not complex or irrational?
with side length 
as its base and apex 
Segments 
and 
are perpendicular. What is the square of the height of the pyramid?
with side length as its base and apex Segments 
and 
are perpendicular. What is the square of the height of the pyramid?
and 
and 
(here 
is obivously the radius of a sidelength- regular octagon). Then we have![\[ AV\perp DV\implies (A-V)\cdot (D-V) = 0\implies (r,0,-h)\cdot (-r/\sqrt{2}, r/\sqrt{2}, -h) = 0\implies h^2 = \frac{r^2}{\sqrt{2}}.\]](http://latex.artofproblemsolving.com/9/f/0/9f0f04906c1d5c03679314ff48ef3a756897c5e6.png)
We can easily derive 
with the law of cosines, so solving for 
and plugging it into the above, we get![\[h^2 = \frac{1}{\sqrt{2}}\left(\frac{1}{2(1 - \frac{1}{\sqrt{2}})}\right) = \frac{1}{2(\sqrt{2} - 1)} = \boxed{\frac{\sqrt{2} + 1}{2}}.\]](http://latex.artofproblemsolving.com/e/0/b/e0babc096dffa982afc24b660bb900503d021c1b.png)
and 
,
,
and 
to meet at 
.
and 
are both right isosceles, so 
.
