❚❖⑩◆ ❇■➌◆ ✣✃■ ✣❸■ ❙➮ ❇➔✐ ✶ ✭❍❙● ❍➔ ◆ë✐ ✷✵✶✽✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ a + b + c = 2018 ✈➔ 1 2017 + + = a + b b + c c + a 2018 ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ P = b +a c + c +b a + a +c b ✳ √ ❇➔✐ ✷✳ ❈❤♦ a, b, c > t❤ä❛ ♠➣♥✿ a + b + c + abc = 4✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ Q= a (4 − b) (4 − c) + b (4 − c) (4 − a) + ❇➔✐ ✸✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥✿ c (4 − a) (4 − b) a+b+c=0 a3 + b3 + c3 = ✳ ❈❤ù♥❣ ♠✐♥❤ a2019 + b2019 + c2019 = ❇➔✐ ✹ ✭❍❙● ◆❛♠ 2✣à♥❤ ✷✵✶✷✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ 2 ♥❤❛✉ ✈➔ t❤ä❛ ♠➣♥✿ a − b = b − c = c − a✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ (a + b + 1) (b + c + 1) (c + a + 1) = −1 ❇➔✐ ✺ ✭❍❙● ❍➔ ❚➽♥❤ ✷✵✶✹✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ a2 + b2 + c2 = a3 + b3 + c3 = ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ M = a2012 + b2013 + c2014 ❇➔✐ ✻✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y, z t❤ä❛ ♠➣♥ x2 + 2y + = y + 2z + = z + 2x + = ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ A = x2019 + y2019 + z 2019✳ ❇➔✐ ✼ ✭❙P ✷✵✶✷✱ ✈á♥❣ ✷✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ♣❤➙♥ ❜✐➺t ✈➔ t❤ä❛ ♠➣♥✿ 2 a (b + c) = b (c + a) = 2012 ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ A = c2 (a + b)✳ ❇➔✐ ✽✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ ✈➔ ❤❛✐ sè t❤ü❝ x, y t❤ä❛ ♠➣♥✿ 3 a + ax + y = 0, b + bx + y = 0, c + cx + y = ✶ ❈❤ù♥❣ ♠✐♥❤✿ a + b + c = 0✳ ❇➔✐ ✾ ✭❙P ✷✵✶✼✱ ✈á♥❣ ✶✮✳ ●✐↔ sû x, y ❧➔ ❤❛✐ sè t❤ü❝ ♣❤➙♥ ❜✐➺t t❤ä❛ ♠➣♥✿ 1 + = ✳ x2 + y + xy + S= ❍➣② t➼♥❤ x2 1 + + + y + xy + ❇➔✐ ✶✵ ✭❙P ✷✵✶✺✱ ✈á♥❣ ✷✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ♠➣♥ < x < 1, < y < ✈➔ x y + =1 1−x 1−y ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ P = x + y + x2 − xy + y2✳ ❇➔✐ ✶✶ ✭❙P ✷✵✶✽✱ ✈á♥❣ ✶✮✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤✿ x3 − x − = 0✳ ●✐↔ sû x0 ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ t➻♥❤ ✤➣ ❝❤♦✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ x0 > 0✳ x20 − ❜✮ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ M = x3 2x20 + 3x0 + 2✳ ❇➔✐ ✶✷2 ✭❑❍❚◆2 ✷✵✶✺✱ ✈á♥❣ ✶✮✳ ●✐↔ sû a, b ❧➔ ❤❛✐ sè t❤ü❝ ♣❤➙♥ ❜✐➺t t❤ä❛ ♠➣♥ a + 3a = b + 3b = 2✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 3 ❛✮ a + b = −3 ❜✮ a + b = −45 ❇➔✐ ✶✸ ✭❝❤✉②➯♥ ♥❣ú ✷✵✵✺✮✳❈❤♦ ❝→❝ sè t❤ü❝ x, y, z = t❤ä❛ ♠➣♥✿ (x − y)2 +(y − z)2 +(z − x)2 = (x + y − 2z)2 +(y + z − 2x)2 +(z + x − 2y)2 ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ x = y = z ✳ ❇➔✐ ✶✹ ✭❑❍❚◆ ✷✵✶✺✱ ✈á♥❣ ✷✮✳ ❱ỵ✐ a, b, c ❧➔ ❝→❝ sè t❤ü❝ t❤ä❛ ♠➣♥ (3a + 3b + 3c)3 = 24 + (3a + b − c)3 + (3b + c − a)3 + (3c + a − b)3 ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ (a + 2b)(b + 2c)(c + 2a) = 1✳ ❇➔✐ ✶✺ ✭❑❍❚◆ ✷✵✶✼✱ ✈á♥❣ ✶✮✳ ✈ỵ✐ a, b ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥✿ ab + a + b = 1✱ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ a b + = + a2 + b2 + ab (1 + a2 ) (1 + b2 ) ❇➔✐ ✶✻ ✭❆♠s ✷✵✶✻✱ ❝❤✉②➯♥ ❚✐♥✮ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ 3 a+b+c=0 ✈➔ a + b + c = 3abc✳ P = ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ a2 b2 c2 + + b2 + c2 c2 + a2 a2 + b2 ✷ ❇➔✐ ✶✼ ✭❆♠s ✷✵✶✻✱ ❝❤✉②➯♥ ❚♦→♥✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ 3 ♥❤❛✉ t❤ä❛ ♠➣♥✿ a t❤ù❝ + b + c = 3abc ✈➔ a, b, c = 0✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ ab2 bc2 ca2 P = + + a + b2 − c2 b2 + c2 − a2 c2 + a2 − b2 ❚➻♠ t➜t ❝↔ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ❇➔✐ ✶✽ ✭❙P ✷✵✶✻✱ ✈á♥❣ ✷✮✳ √ x2 − y − x−1+ + +8=4 x y ♠➣♥ y−1 ❇➔✐ ✶✾ ✭❆♠s ✷✵✶✽✱ ❝❤✉②➯♥ ❚♦→♥✮✳ ❱ỵ✐ x, y, z ❧➔ ❝→❝ sè t❤ü❝ t❤ä❛ ♠➣♥ xyz = ✈➔ (xy + x + 1)(yz + y + 1)(zx + z + 1) = ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 1 + + =1 xy + x + yz + y + zx + z + ❇➔✐ ✷✵ ✭❑❍❚◆ ✷✵✶✹✱ ✈á♥❣ ✶✮✳ ●✐↔ sû x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x + y + z = xyz ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ x 2y 3z xyz (5x + 4y + 3z) + + = + x2 + y + z (x + y) (y + z) (z + x) ❇➔✐ ✷✶ ✭❑❍❚◆ ✷✵✶✹✱ ✈á♥❣ ✷✮✳ ●✐↔ sû x, y ❧➔ ♥❤ú♥❣ sè t❤ü❝ ❞÷ì♥❣ ♣❤➙♥ ❜✐➺t t❤ä❛ ♠➣♥ 2y 4y 8y y + + + =4 x + y x2 + y x4 + y x8 − y ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ 5y = 4x✳ ❇➔✐ ✷✷✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y = 0✳ ✣➦t a = x + x1 , b = y + y1 , c = z + z1 ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a2 + b2 + c2 − abc = 4✳ ❇➔✐ ✷✸ ✭❙P ✷✵✵✻✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c = ✈➔ ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ t❤ä❛ ♠➣♥✿ a + b + c = 0✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ b c a + + b−c c−a a−b b−c c−a a−b + + a b c =9 ❇➔✐ ✷✸ ✭❙P ✷✵✶✸✱ ✈á♥❣ ✷✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c tọ ỗ tớ tự (a + b)(b + c)(c + a) = abc✳ ✐✐✮ a3 + b3 b3 + c3 c3 + a3 ❈❤ù♥❣ ♠✐♥❤ abc = 0✳ = a3 b c ✳ ❇➔✐ ✷✹ ✭❙P ✷✵✶✹✱ ✈á♥❣ ✷✮✳ ●✐↔ sû a, b, c, x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❦❤→❝ t❤ä❛ ♠➣♥ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a b c + + = ✈➔ x y z x2 y z + + = 1✳ a2 b c x y z + + =1 a b c ❇➔✐ ✷✺ ✭❑❍❚◆ ✷✵✶✸✱ ✈á♥❣ ✶✮✳ ●✐↔ sû a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❦❤→❝ t❤ä❛ ♠➣♥✿ (a + b) (b + c) (c + a) = 8abc✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a b c ab bc ca + + = + + + a + b b + c c + a (a + b) (b + c) (b + c) (c + a) (c + a) (a + b) ❇➔✐ ✷✻ ✭❍❙● ◆❛♠ ✣à♥❤ ✷✵✶✻✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y, z t❤ä❛ ♠➣♥ x + y + z = 2, x2 + y + z = 18, xyz = −1 ❚➼♥❤ S = xy +1z − + yz +1x − + zx + y 1− − ✳ ❇➔✐ ✷✼ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✾✱ ✈á♥❣ ✶✮✳ ❱ỵ✐ a, b, c ❧➔ ♥❤ú♥❣ sè t❤ü❝✱ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ 2b + c b + 2c 2c + a c + 2a 2a + b a + 2b + + a−b b−c b−c c−a c−a a−b = −3 ❇➔✐ ✷✽ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✾✱ ✈á♥❣ ✷✮✳ ❱ỵ✐ a, b ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥ a + b = 1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a b (ab − 2) + = b3 − a3 − a2 b + ❇➔✐ ✷✾ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✾✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 2a + b 2b + c 2c + a (2a + b) (2b + c) (2b + c) (2c + a) (2c + a) (2a + b) + + = 3+ + + a−b b−c c−a (a − b) (b − c) (b − c) (c − a) (c − a) (a − b) ❇➔✐ ✸✵ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✾✱ ✈á♥❣ ✷✮✳ ❱ỵ✐ a, b, c ❧➔ ❝→❝ sè t❤ü❝ t❤ä❛ ♠➣♥ a + b + c = abc✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a2 − b2 − c2 − + + =1+ a2 + b2 + c2 + ✹ (1 + a2 ) (1 + b2 ) (1 + c2 ) ❇➔✐ ✸✶ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✽✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ t❤ä❛ ♠➣♥ (1 − ab) (1 − bc) (1 − ca) = 0✳ ❈❤ù♥❣ ♠✐♥❤ ❜✐➸✉ t❤ù❝ + ab b + c b + c ac + ac + 1 + ab + + − ab b − c b − c ac − ac − 1 − ab trà ❧➔ sè ♥❣✉②➯♥ ✈ỵ✐ ♠å✐ a, b, c✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët M= ❧✉æ♥ ♥❤➟♥ ❣✐→ ❇➔✐ ✸✷ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✽✱ ✈á♥❣ ✷✮✳ ❦❤→❝ ♥❤❛✉✳ ❈❤ù♥❣ ♠✐♥❤ ❜✐➸✉ t❤ù❝ c b a c b a + + a−b c−b b−c a−c c−a b−a ❧➔ sè ♥❣✉②➯♥ ✈ỵ✐ ♠å✐ a, b, c✳ Q= ❧✉æ♥ ♥❤➟♥ ❣✐→ trà ❇➔✐ ✸✸ ✭❍❙● ❚P ❍❈▼ ✷✵✶✻✮✳ ❈❤♦ ❤❛✐ sè t❤ü❝ ♣❤➙♥ ❜✐➺t a, b t❤ä❛ ♠➣♥✿ ab = a − b✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ a b + − ab b a a, b, c, x, y, z t❤ä❛ ♠➣♥ M= ❇➔✐ ✸✹✳ ❈❤♦ ❝→❝ sè t❤ü❝ ❈❤ù♥❣ x = by + cz, y = ax + cz, z = ax + by ♠✐♥❤ +1 a + +1 b + +1 c = 2✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✈➔ x, y, z = t❤ä❛ ♠➣♥ b c a = = a + b + c = a2 + b2 + c2 = ✈➔ x y z ❇➔✐ ✸✺✳ ❈❤ù♥❣ ♠✐♥❤✿ xy + yz + zx = 1✳ ❇➔✐ ✸✻ ✭❍❙● ❍➔ ◆ë✐ ✷✵✶✹✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c = t❤ä❛ ♠➣♥ ✈➔ a + b + c = a1 + 1b + 1c ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ tr♦♥❣ ❜❛ sè a, b, c ❝â ➼t ♥❤➜t ♠ët sè ❜➡♥❣ ✶✳ ❇➔✐ ✸✼✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ t❤ä❛ ♠➣♥ abc = ❚➼♥❤ b−c c−a a−b + + = 2019 (a − b) (a − c) (b − c) (b − a) (ac − a) (c − b) ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ P = a −2 b + b −2 c + c −2 a ✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ abc = 2019✳ ❚➼♥❤ ❣✐→ trà ❇➔✐ ✸✽✳ ❜✐➸✉ t❤ù❝ Q= a b 2019c + + ab + a + 2019 bc + b + ac + 2019c + 2019 ✺ ❝õ❛ ❇➔✐ ✸✾ ✭❍❙● ◆✐♥❤ ❇➻♥❤ ✷✵✶✺✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ a, b t❤ä❛ ♠➣♥ a100 + b100 = a101 + b101 = a102 + b102 ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ M = a2015 + b2015✳ ❇➔✐ ✹✵✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y, z t❤ä❛ ♠➣♥ x + y + z = 1, x2 + y + z = 1, x3 + y + z = ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ Q = x4 + y5 + z 6✳ ❇➔✐ 5✹✶✳ ❈❤♦ ❝→❝ sè 3t❤ü❝3a, b,3c t❤ä❛ ♠➣♥✿ a + b + c = 0✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a + b + c a2 + b2 + c2 a + b + c5 = ❛✮ 7 4 ❜✮ a + b + c = 7abc a + b + c4 ❝✮ a7 + b7 + c7 = 7abc a2b2 + b2c2 + c2a2 ❞✮ 10 a7 + b7 + c7 = a2 + b2 + c2 a5 + b5 + c5 ✻ ... 20 19 (a − b) (a − c) (b − c) (b − a) (ac − a) (c − b) ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ P = a −2 b + b −2 c + c −2 a ✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ abc = 20 19? ?? ❚➼♥❤ ❣✐→ trà ❇➔✐ ✸✽✳ ❜✐➸✉ t❤ù❝ Q= a b 2019c... c t❤ä❛ ♠➣♥ abc = 20 19? ?? ❚➼♥❤ ❣✐→ trà ❇➔✐ ✸✽✳ ❜✐➸✉ t❤ù❝ Q= a b 2019c + + ab + a + 20 19 bc + b + ac + 2019c + 20 19 ✺ ❝õ❛ ❇➔✐ ✸✾ ✭❍❙● ◆✐♥❤ ❇➻♥❤ ✷✵✶✺✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ a, b t❤ä❛ ♠➣♥ a100 + b100... ❦❤→❝ ♥❤❛✉ t❤ä❛ ♠➣♥✿ a + b + c = 0✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ b c a + + b−c c−a a−b b−c c−a a−b + + a b c =9 ❇➔✐ ✷✸ ✭❙P ✷✵✶✸✱ ✈á♥❣ ✷✮✳ ❈❤♦ số tỹ a, b, c tọ ỗ tớ ❤❛✐ ✤➥♥❣ t❤ù❝ ✸ ✐✮ (a + b)(b + c)(c + a)