Here is a problem that one of cousins asked me. He is currently studying mathematics at IIT. I am not really sure how to solve it, so please help. I am comfortable with algebraic concepts and number theory at AIME level but don't know too much beyond that. How would I go about solving this problem.
For integers q $\geq4$ find all positive reals $x_{1},x_{2},x_{3}...x_{q}$ such that $x_{n}^2=9x_{n+1}+10x_{n+2}+11x_{n+3}$ holds for all $n=1,2,3....q$ with indices taken modulo q.
For the largest number in the sequence, say $A$, you have: $$A^2 \leqslant 9A+10A+11A \implies A\leqslant 30$$ Similarly, for the smallest number in the sequence, say $a$, you have: $$a^2 \geqslant 9a+10a+11a \implies a\geqslant 30$$ Since $A \geqslant x_i \geqslant a$, this forces all the numbers in the sequence to be $30$, which will thus be your only solution.