4

Part of 2007 Korea National Olympiad

Problems(2)

two real sequence

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

Two real sequence

\{x_{n}\}

\{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

{x_{n}}^{2}\leq 2007

inductionstrong inductionalgebra unsolvedalgebra

(2007 KMO #8) Product of primes less than n

Source:

For all positive integer

n\geq 2

, prove that product of all prime numbers less or equal than

n

is smaller than

4^{n}

inductioninequalitiesnumber theoryprime numbersrelatively primenumber theory proposed