Weekly Contest #67
(Ended)
Standings
Q1.
Rectangles 
and 
are drawn such that 
are collinear. Also, 
all lie on a circle. If 
,
,
, and 
, what is the length of 
?
![[asy] import graph; unitsize(0.1cm); pair A = (0,0);pair B = (70,0);pair C = (70,16);pair D = (0,16);pair E = (3,16);pair F = (90,16);pair G = (90,33);pair H = (3,33); dot(A^^B^^C^^D^^E^^F^^G^^H); label("$A$", A, S);label("$B$", B, S);label("$C$", C, N);label("$D$", D, N);label("$E$", E, S);label("$F$", F, S);label("$G$", G, N);label("$H$", H, N); draw(E--D--A--B--C--E--H--G--F--C); [/asy]](https://latex.artofproblemsolving.com/1/2/f/12f0a4224f27ffa406ded2c01cfd349ecd0c04de.png)
Q2.
Jen enters a lottery by picking 
distinct numbers from 
numbers are randomly chosen from 
She wins a prize if at least two of her numbers were 
of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of her winning the grand prize given that she won a prize is 
where 
and 
are relatively prime positive integers. Find 
.
Q3.
Alice and Bob play the following game. A stack of tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either 
token or tokens from the stack. Whoever removes the last token wins. Find the number of positive integers less than or equal to 
for which there exists a strategy for Bob that guarantees that Bob will win the game regardless of Alice's play.
Q4.
Every morning Aya goes for a 
-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of 
kilometers per hour, the walk takes her 4 hours, including 
minutes spent in the coffee shop. When she walks 
kilometers per hour, the walk takes her 2 hours and 24 minutes, including minutes spent in the coffee shop. Suppose Aya walks at
kilometers per hour. Find the number of minutes the walk takes her, including the minutes spent in the coffee shop.