Japan Junior Mathematical Olympiad

Since the smallest and the next to smallest possible numbers of the bulbs to be planted are $52$ and $64$, and since each participating student is asked to plant at least $1$ bulb, the total number of participating students should be $52$, and the number of students in the grade with the smallest number of participating student is $12$. Let us denote by $<a,b,c>$ the total number of the bulbs to be planted if each 7th grader plants $a$ bulbs, 8th grader $b$ bulbs and 9th grader $c$ bulbs. Then we can check that the total number of bulbs to be planted will be less than $100$ only in the cases resulting in $<1,1,1>$, $<1,1,2>$, $<1,2,1>$, $<1,2,2>$, $<2,1,1>$, $<2,1,2>$, $<2,2,1>$. Since there are only $6$ possibilities for the total number of bulbs to be planted, distribution of the number of students in different grades should be such that among these $7$ numbers there be a pair which should be equal and others are all distinct. Taking into account that the total number of students is $52$ and the number of students in the grade with the smallest number of participating students is $12$, we can conclude that the possible student distributions among different grades must be such that either the number of students in one grade is equal to the number of students in one other grade, or the number of students in one grade is equal to the sum of the numbers of students in two other grades, and this forces the possibility for the distribution of students among different grades to be one of the following three cases: $(12,12,28)$, $(12,20,20)$, $(12,14,26)$. Among these three, we can check there is only one case $(12,12,28)$ for which there are exactly $6$ possible numbers of bulbs under $100$ to be planted.

Alternatively, suppose we denote by $A$, $B$, $C$ the number of participating students in different grades. We may assume without loss of generality that $A\le B\le C$. Then, from the fact that $52=A+B+C$, $64=2A+B+C$, we obtain $A=12$, $B+C=40$. Since $A\le B\le C$, we also have $12\le B\le 20$. Let us denote by $<a,b,c>$ the total number of bulbs planted if each student in the grade with $A$, $B$, $C$ students plants $a$, $b$, $c$ bulbs, respectively. Now, if the condition $13\le B\le 20$ is satisfied, there are $7$ distinct numbers $<1,1,1>$, $<2,1,1>$, $<3,1,1>$, $<4,1,1>$, $<1,2,1>$, $<2,1,1>$, $<3,2,1>$, which are all less than $100$. On the other hand, we can check that the case for $B=12$ yields exactly $6$ distinct possibilities for the total number of bulbs to be planted. Therefore, $(12,12,28)$ is the only possible distribution of students among different grades.