CEMC Fermat, 2026

The smallest positive integer less than $100$ that can be written as the sum of
three consecutive integers is $1+2+3=6$.

The second smallest positive integer less than $100$ that can be written as the sum of
three consecutive integers is $2+3+4=9$, which is an increase of $3$ from the previous sum.

In general, for positive integers $a$, the sum of three consecutive positive
integers is

$a+(a+1)+(a+2)=3a+3$.

The next largest sum of three consecutive positive integers is $(a+1)+(a+2)+(a+3)=3a+6$ which is $(3a+6)-(3a+3)=3$ more than the previous
sum.

That is, successive integers in the list of such sums will continue to
increase by $3$.

The largest positive integer less than $100$ that can be written as the sum of
three consecutive integers is $32+33+34=99$. (We note that $\text{33+34+35=102 gt;100}$.)

Thus the positive integers less than $100$ that can be written as the sum of
three consecutive positive integers are of the form $3a+3$ for positive integers $1\leq a\leq32$, and so there are $32$ such numbers.