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two real sequence

Source: 2007 Korean MO, 2nd Round, A.M.

August 18, 2007
inductionstrong inductionalgebra unsolvedalgebra

Problem Statement

Two real sequence {xn} \{x_{n}\} and {yn} \{y_{n}\} satisfies following recurrence formula; x_{0}\equal{} 1, y_{0}\equal{} 2007 x_{n\plus{}1}\equal{} x_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(y_{n}\plus{}y_{n\plus{}1}), y_{n\plus{}1}\equal{} y_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(x_{n}\plus{}x_{n\plus{}1}) Then show that for all nonnegative integer n n, xn2≤2007 {x_{n}}^{2}\leq 2007.
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