$N$
nonnegative integers whose sum is $100,$
the integer $3$
occurs more often than any other. What is the largest possible value of $N?$

Why not find a valid list and try to increase the number of integers in the list?

What are the sensible integers in the largest list?

… remember that you can use
$0$
as well. 
… for example, having
$5$
is not sensible because we can have$3,$
$2$
and$0$
instead and have more integers. 
How about expressing the number of other sensible integers in terms of the number of
$3$
s to find$N?$

Have you considered all of the constraints given?

Having numbers greater than
$3$
in the list is not useful, since we can have more integers for the same value by splitting that number into some$3$
s,$1$
s and$0$
s. A$2$
can be converted into two$1$
s and two$2$
s can be converted into$3,1$
and$0.$
Since any two
$2$
s can be converted to$3,1$
and$0,$
which doesn’t affect our quota of$1$
s and$0$
s (bounded by number of$3$
s), the optimal solution will have at most one$2.$
Although there is an optimal solution that uses$2$
(see below), it is possible to reach optimum without$2$
s. So for now, the only numbers we will consider are$0, 1,$
and$3.$
We also notice that we can exchange a
$3$
and a$0$
for 3$1$
s. Hence our strategy is to maximize the number of$3$
s such that we can fit as many$1$
s without making the$1$
s more common. We then add$0$
s up to the number of$3$
s. Let$t$
be the number of$3$
s which means we’ll have$1003t$
of$1$
s and$t1$
of$0$
s. In total, we’ll have$N=t+1003t+t1=100t1,$
conditioned by$t>1003t,$
which gives$t=26$
and$N=73.$
The two solutions are 26
$3$
s, 22$1$
s, 25$0$
s or 25$3$
s, 1$2$
, 23$1$
s, 24$0$
s.