An olympiad question

[jagged oracle]

first, subtract f(999) from both sides of the equation f(1)+f(2)+f(3)+...+f(n)=(n^2)(f(n))
f(1)+f(2)+f(3)+...+f(n) - f(999) = (n^2)(f(n)) - f(999)
then factor out f(999) on the right-hand side.
f(1)+f(2)+f(3)+...+f(n) - f(999) = f(999)(n^2 - 1)
we know that f(1) + f(2) + f(3) + ... + f(n) = (n^2)(f(n)),so we can substitute this into the equation above.
f(n^2)(f(n)) - f(999) = f(999)(n^2 - 1)
we can now divide both sides of the equation by f(999).
(n^2)(f(n)) - 1 = n^2 - 1
finally, we can solve this equation for f(n).
f(n)=1 so f(999)=1

if you didn't understand this way, i can write it down. good luck!