<prompt> Acha x 2 sqrt31000, acha n floor x sakafu, na f x - n. Tafutax1 - f.  <response>: Acha alpha 2 sqrt3 na beta 2 - sqrt3. Kisha zingatia nambarianzaalignN alpha1000 beta1000 2 sqrt31000 2 - sqrt31000 21000 binom10001 2999 sqrt3 binom10002 2998 sqrt3 2 binom10003 sqrt33 dotsb quad 21000 - binom10001 2999 sqrt3 binom10002 2998 sqrt3 2 - binom10003 sqrt33 dotsb.endalignKuongeza 2 sqrt31000 na 2 - sqrt31000, tunaona kwamba masharti yote yenye sqrt 3 itaghairi, kumaanisha kuwa tumebakiwa na nambari kamili.Zaidi ya hayo,beta 2 - sqrt3 frac2 - sqrt32 sqrt32 sqrt3 frac1 2 sqrt3 1,hivyo 0 beta1000 1.Kwa hiyo,N - 1 alpha1000 N,ambayo ina maana n floor alpha1000 floor N - 1.Kishaf x - n alpha1000 - N - 1 1 - beta1000,hivyo 1 - f beta1000. Kwa hivyo,anzaalignx1 - f alpha1000 beta1000 alpha beta1000 2 sqrt32 - sqrt31000 11000 1.mwishopanga 