Решил задачу, но валидатор не пропускает. Взял чужое решение, валидатор пропустил - но работает не правильно.

public class Solution {
    public static void main(String[] args) {
        System.out.println(isDateOdd("JANUARY 1 2000"));
    }

    public static boolean isDateOdd(String date) {

        Date yearStartTime = new Date();
        Date currentTime = new Date(date);

        yearStartTime.setSeconds(0);
        yearStartTime.setMinutes(0);
        yearStartTime.setHours(0);

        yearStartTime.setDate(1);
        yearStartTime.setMonth(0);
        yearStartTime.setYear(currentTime.getYear());

        long msTimeDistancce = currentTime.getTime() - yearStartTime.getTime();
        long msDay = 24 * 60 * 60 * 1000;

        int countDay = (int) (msTimeDistancce/msDay);

        if(countDay % 2 != 0)
            return true;
        else
            return  false;
    }
}