Weekly Contest #81
(Ended)
Standings
Q1.
The repeating decimals 
and 
satisfy
![\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]](https://latex.artofproblemsolving.com/1/a/b/1ab5008009f306f1d32c9fd9123a272d593dfcbd.png)
where 
, 
, and 
are (not necessarily distinct) digits. Find the three digit number 
.
Q2.
Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer 
, no collection of 
pairs made by the child contains the shoes from exactly of the adults is 
, where m and n are relatively prime positive integers. Find 
Q3.
Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs
. Find the sum of all positive integers 
for which ![\[\left |\sum_{k=1}^n\log_{10}f(k)\right|=1.\]](https://latex.artofproblemsolving.com/3/8/9/389a540c1dec657ab2e68535a6098d77f575cdcb.png)
Q5.
Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability 
and each of the other five sides has probability 
. Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is 
, where 
and 
are relatively prime positive integers. Find 
.