Weekly Contest #90
(Ended)
Standings
Q1.
Regular polygons with 
and 
sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
Q2.
Eight circles of radius 
are sequentially tangent, and two of the circles are tangent to 
and 
of triangle 
, respectively. 
circles of radius 
can be arranged in the same manner. The inradius of triangle can be expressed as 
, where 
and 
are relatively prime positive integers. Find 
.
![[asy] pair A = (2,1); pair B = (0,0); pair C = (3,0); dot(A^^B^^C); label("$A$", A, N); label("$B$", B, S); label("$C$", C, S); draw(A--B--C--cycle); for(real i=0.62; i<2.7; i+=0.29){ draw(circle((i,0.145), 0.145)); } [/asy]](https://latex.artofproblemsolving.com/e/c/5/ec5d6f1f995c862b1609e69add5733cb092f1f3f.png)
Q3.
Consider the paths of length 
that follow the lines from the lower left corner to the upper right corner on an 
grid. Find the number of such paths that change direction exactly four times, as in the examples shown below.
![[asy] size(10cm); usepackage("tikz");label("\begin{tikzpicture}[scale=.5]\draw(0,0)grid(8,8);\draw[line width=2,red](0,0)--(2,0)--(2,3)--(5,3)--(5,8)--(8,8);\end{tikzpicture}",origin); label("\begin{tikzpicture}[scale=.5]\draw(0,0)grid(8,8);\draw[line width=2,red](0,0)--(0,3)--(3,3)--(3,5)--(8,5)--(8,8);\end{tikzpicture}",E); [/asy]](https://latex.artofproblemsolving.com/2/d/c/2dcb0a2c52c662dbc003bf1192e85a32fb7d35d2.png)
and 
, both greater than 1, such that 
. Find 
.