2025 SSMO Relay Round 4 Problems
Problem 1
Call a positive integer chuzzed if the sum of the digits in its binary representation is equal to the units digit of its base-
representation. Similarly, call a positive integer chopped if its binary representation does not contain two consecutive ones. Find the number of positive integers less than 
that are chuzzed and chopped.
Problem 2
Let 
Jonathan and Kate are playing a game with 
sticks. On each turn, a player may remove 
or 
sticks. The player who picks up the last stick loses. Kate is first to remove sticks, and both players play optimally. For how many values of in the range ![$\left[T^3,2T^3\right]$](http://latex.artofproblemsolving.com/3/1/1/31124d3f1d2fe3a278328400cfd00e978855ca95.png)
does Kate have a winning strategy?
Problem 3
Let A particle moves in the coordinate plane such that at any time 
its position is ![\[\left(\sum_{a=1}^{T-1} \cos(at),\sum_{a=1}^{T-1} \sin(at)\right).\]](http://latex.artofproblemsolving.com/2/a/f/2aff036f84c6f9ae0cd1f2684885b7c4fc9f0a5d.png)
Over the time interval ![$t\in(0,k],$](http://latex.artofproblemsolving.com/1/d/6/1d6e4f5955235095c51ab884b0552eb11476aea9.png)
the particle lies on at least one coordinate axes 
times. If the minimal value of 
can be written as 
for relatively prime positive integers 
and 
find 
.
