2020 CIME I Problems/Problem 3

Problem 3

In a math competition, all teams must consist of between $12$ and $15$ members, inclusive. Mr. Beluhov has $n > 0$ students and he realizes that he cannot form teams so that each of his students is on exactly one team. Find the sum of all possible values of $n$.


We see that the integers from 1 - 11 cannot be achieved. We similarly see that the integers from 16-23 cannot be reached. We then see that the integers from 31-35 cannot be reached. We then see that 46 and 47 cannot be reached. We then attempt to show that 61 can be reached. We see that 12x4 and 13 get the trick done. The answer is then 480.

Solution 2 (Methodic Process)

Let $t$ be the total number of students in his class. Then, that means that $t\pmod{12}>3\lfloor{\frac{t}{12}}\rfloor$ This translates to when there are $12$ students on a team, there are more remaining students than slots on those teams, where each team has a maximum of three slots for $\lfloor{\frac{t}{12}}\rfloor$ teams. Those three slots are the $13$th $14$th and $15$th members of the team. We see that $3\lfloor{\frac{t}{12}}\rfloor$ has to be less than $12$, so $\lfloor{\frac{t}{12}}\rfloor$ is between $0$ and $3$ inclusive. Plugging in $x= {0,1,2,3,}$ give the numbers $1-11$, $16-23$, $31-35$, and $46-47$ respectively. Adding these up, we get $\boxed{480}$


See also

2020 CIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All CIME Problems and Solutions

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