<prompt> Acha a,b, na c ziwe nambari changamano kama vile a b c 1 nafraca2bc fracb2ac fracc2ab -1.Tafuta thamani zote zinazowezekana za a b c.Ingiza thamani zote zinazowezekana, zikitenganishwa na koma.  <response>: Tangu a 1, a overlinea a2, hivyo overlinea frac1a. Vile vile, overlineb frac1 b na overlinec frac1c.Pia, hebu z a b c. Kishaanzaalignz2 a b c2 a b c overlinea b c a b c overlinea overlineb overlinec a b c kushoto frac1a frac1b frac1c kulia a b c cdot fracab ac bcabc fraca2 b ab2 a2 c ac2 b2 c bc2 3abcabc.endalignTuna hiyoz3 a b c3 a3 b3 c3 3a2 b ab2 a2 c ac2 b2 c bc2 6abc,hivyoanzaalign3z2 frac3a2 b ab2 a2 c ac2 b2 c bc2 3abcabc fracz3 - a3 b3 c3 3abcabc.endalignKutoka kwa mlinganyo fraca2bc fracb2ac fracc2ab -1, a 3 b3 c3 -abc, hivyo3z2 fracz3 4abcabc fracz3abc 4.Kisha3z2 - 4 fracz3abc,hivyokushoto 3z2 - 4 kulia kushoto fracz3abc kulia z3.Acha r z, iwe 3r2 - 4 r3. Ikiwa 3r2 - 4 0, basi4 - 3r2 r3.Hii inakuwa r3 3r2 - 4 0, ambayo huzingatia kama r - 1r 22 0. Kwa kuwa r lazima iwe isiyo hasi, r 1.Ikiwa 3r2 - 4 ge 0, basi3r2 - 4 r3.Hii inakuwa r3 - 3r2 4 0, ambayo huzingatia kama r 1r - 22 0. Kwa kuwa r lazima iwe nonnegtaive, r 2.Hatimaye, lazima tuonyeshe kwamba kwa kila moja ya thamani hizi zinazowezekana za r, kuna nambari changamano zinazolingana a,b, na c.Ikiwa a b 1 na c -1, basi fraca2bc fracb2ac fracc2ab -1, naa b c 1 1.Kama a 1, b frac1 i sqrt32, na c frac1 - i sqrt32 , kisha fraca2bc fracb2ac fracc2ab -1, naa b c 2 2.Kwa hivyo, thamani zinazowezekana za a b c ni 1,2. 