In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam.

HIDE: original wording En un examen fueron propuestos 6 problemas. Cada problema fue resuelto por exactamente 1000 estudiantes, pero en ningun caso ha ocurrido que dos estudiantes en conjunto, hayan resuelto los 6 problemas. Determinar el menor numero de participantes que pudo haber en dicho exame