Weekly Contest #63
(Ended)
Standings
Q1.
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.
Q2.
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)
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.
Two geometric sequences 
and 
have the same common ratio, with 
, 
, and 
. Find 
.