◆❍Ú◆● ❍➀◆● ✣➃◆● ❚❍Ù❈ ✣⑩◆● ◆❍❰ ●❱✿ ❚r➛♥ ❱➠♥ ❚✉➙♥ õ tt ỵ tt tự ✤→♥❣ ♥❤ỵ (A + B)2 = A2 + 2AB + B (1) (A − B)2 = A2 − 2AB + B (2) A2 − B = (A + B)(A − B) (3) (A + B)3 = A3 + 3A2 B + 3AB + B (4) (A − B)3 = A3 − 3A2 B + 3AB − B (5) A3 + B = (A + B) A2 − AB + B (6) A3 + B = (A + B)3 − 3AB (A + B) (6 ) A3 − B = (A − B) A2 + AB + B (7) A3 − B = (A − B)3 + 3AB (A − B) (7 ) ✷✳ ▼ët sè ❤➡♥❣ ✤➥♥❣ t❤ù❝ ♠ð rë♥❣ (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca (8) (a − b + c)2 = a2 + b2 + c2 − 2ab − 2bc + 2ca (9) (a + b + c)3 = a3 + b3 + c3 + 3(a + b)(b + c)(c + a) (10) a3 + b3 + c3 − 3abc = (a + b + c) a2 + b2 + c2 − ab − bc ca (11) ú ỵ a + b + c = t❤➻ a3 + b3 + c3 = 3abc✳ ❇✳ ❇➔✐ t➟♣ ❇➔✐ ✶✳ ❚➼♥❤ 5b ❛✮ 3a + ❞✮ (3a + 2) ❜✮ (2x − 3y)2 b ❡✮ a − 2 ❝✮ 4x 7y + 10 7y 4x − 10 ❢✮ (2a + b − 5) (2a − b + 5) ❇➔✐ ✷✳ ❱✐➳t ❝→❝ ❜✐➸✉ t❤ù❝ s❛✉ t❤➔♥❤ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ ♠ët ✤❛ t❤ù❝ ❛✮ 25x2 + 10 x+ ❜✮ x2 + 2xy + 5y ❇➔✐ ✸✳ ❘ót ❣å♥ ❝→❝ ❜✐➸✉ t❤ù❝ s❛✉ ❝✮ − y + y8 ❛✮ A = (3x + 4y)2 + (4x − 3y)2 − (5x + y)(5x − y)✳ ❜✮ B = 3(x − y)2 − 2(x + y)2 − (x − y)(x + y)✳ ❝✮ C = (2x + 3)2 + (2x + 3)(2x − 6) + (x − 3)2 ✳ ❞✮ D = (x − 1)2 − 2(x − 1)(x − 3) + (x − 3)2 ✳ ❡✮ E = (2x + 7y)2 − 2(2x + 7y)(3x + 8y) + (3x + 8y)2 ✳ ❢✮ F = (x + y)3 + 3(x − y)(x + y)2 + 3(x − y)2 (x + y) + (x − y)3 ✳ ❣✮ G = (x − 2)3 − x(x2 − x + 1) + 5x2 + 8✳ ❤✮ H = (x + 3)3 − 9(x + 1)2 − 9x + 1✳ ✐✮ I = (x + y − z)2 + (z − y)2 + 2(x + y − z)(z − y)✳ ❥✮ J = (x2 + x + 1)(x2 − x + 1)(x4 − x2 + 1)(x8 − x4 + 1)✳ ❦✮ K = (x − 2)(x + 2)(x2 + 4) − (x2 − 3)(x2 + 3)✳ ❧✮ L = (x − 2)3 − (x + 1)(x2 − x + 1) + 6(x − 1)2 ✳ ♠✮ M = (x + 1)4 − 6(x + 1)2 − (x2 − 2)(x2 + 2)✳ ♥✮ N = x2 − − x4 + x2 + x2 − ✳ ♦✮ O = (x4 − 3x2 + 9)(x2 + 3) − + x2 ✳ ♣✮ P = (x − 3)3 − (x − 3)(x2 + 3x + 9) + 6(x + 1)2 ✳ q✮ Q = 2x2 + 2x + 2x2 − 2x + − 2x2 + r✮ R = (x − y − 1)3 − (x − y + 1)3 + (x − y)2 3 s✮ S = a2 + b2 + − a2 + b2 − − 12 a2 + b2 2 t✮ T = a2 + b2 − c2 − a2 − b2 + c2 ✉✮ U = (a + b + c)2 − (a − b − c)2 − (a + b)2 ✈✮ V = (a + b + c)2 + (a − b + c)2 + (a + b − c)2 + (b + c − a)2 ①✮ X = (a + b + c)3 − (a + b − c)3 − (b + c − a)3 − (c + a − b)3 ❇➔✐ ✹✳ ❚➻♠ x✱ ❜✐➳t ❛✮ 25x2 − = 0✳ ❜✮ (x + 4)2 − (x + 1)(x − 1) = 16✳ ❝✮ (2x − 1)2 + (x + 3)2 − 5(x + 7)(x − 7) = 0✳ ❞✮ x2 − = 8(x − 2)✳ ❡✮ (x − 1)3 + 3(x + 1)2 = (x2 − 2x + 4)(x + 2)✳ ❢✮ x2 − 4x + = 9(x − 2)✳ ❣✮ 4x2 − 12x + = (5 − x)2 ✳ ❤✮ x3 − 3x2 + 3x − = 0✳ ✐✮ 7x3 − 3x2 − 3x − = 0✳ ❇➔✐ ✺✳ ❈❤ù♥❣ ♠✐♥❤ ❣✐→ trà ❝õ❛ ♠é✐ ❜✐➸✉ t❤ù❝ s❛✉ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ❣✐→ trà ❝õ❛ ❜✐➳♥ ❛✮ A = (x + 2)3 + (x − 2)3 − 2x(x2 + 12)✳ ❜✮ B = (x − 1)3 − (x + 1)3 + 6(x + 1)(x − 1)✳ ❝✮ C = y(x2 − y )(x2 + y ) − y(x4 − y )✳ 1 + 2x 4x2 − x + − 8x3 − ✳ 3 27 ❡✮ E = (x − 1)3 − (x − 1)(x2 + x + 1) − 3(1 − x)x✳ ❞✮ ❇➔✐ ✻✳ ❈❤ù♥❣ ♠✐♥❤ tự s ổ tr ữỡ ợ ♠å✐ x✳ ❛✮ A = x2 − 2x + 3✳ ❜✮ B = 4x2 − 12x + 13✳ ❝✮ C = x2 + 3x + 3✳ ❞✮ D = x2 − 5x + 7✳ ❡✮ E = 3x2 − 4x + 5✳ ❢✮ F = 2x2 − 5x + 4✳ ❇➔✐ ✼✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❝→❝ ❜✐➸✉ t❤ù❝ s❛✉ ❛✮ A = x2 + 4x + 1✳ ❜✮ B = x2 − x + 1✳ ❝✮ C = x2 + 5x + 7✳ ❞✮ D = 3x2 − 8x + 15✳ ❡✮ E = x2 + 5y + 2xy − 2y + 2021✳ ❇➔✐ ✽✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❝→❝ ❜✐➸✉ t❤ù❝ s❛✉ ❛✮ A = −x2 − 6x + 4✳ ❜✮ B = −x2 + 3x + 2✳ ❝✮ C = 6x − x2 − 5✳ ❞✮ D = −2x2 + 7x − 1✳ ❡✮ E = −x2 − 2y + 2xy − y + 1✳ ❇➔✐ ✾✳ ❈❤♦ x − y = −3✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ M = (x − y)3 − x2 + 2xy − y ❇➔✐ ✶✵✳ ❈❤♦ x + y = 1✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ N = 2(x3 + y ) − 3(x2 + y ) ❇➔✐ ✶✶✳ ❈❤♦ x3 + y3 = 1✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ P = 2x6 + 3x3 y + y + y ❇➔✐ ✶✷✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❝→❝ ❜✐➸✉ t❤ù❝ s❛✉ ❛✮ A = x2 − 2x + ❜✮ B = x2 − x + ❝✮ C = (x − 2) (x − 5) x2 − 7x − 10 ❞✮ x (x + 1) x2 + x − ❡✮ E = x2 + 2y + 2xy − 4y + ❢✮ F = 5x2 + y + 4xy − 6x + 8y + ❣✮ G = 2x2 + y − 2xy + 6x − 2y + ❤✮ H = x2 + 5y + 4xy − 2x − 2y + ❇➔✐ ✶✸✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❝→❝ ❜✐➸✉ t❤ù❝ s❛✉ ❛✮ A = −x2 + 2x + ❜✮ B = −x2 + 3x + ❝✮ C = −2x2 + 5x + ❞✮ D = −2x2 − 4y + 4xy + 4x − ❡✮ E = −x2 + 2xy − 4y + 2x + 10y − ❇➔✐ ✶✹✳ ❚➻♠ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ♠➣♥ ❛✮ x2 + y + 2x − 6y + 10 = ❜✮ 4x2 + y − 4x + 2y + = ❝✮ 2x2 + y − 4x + 4y + = ❞✮ 3x2 + 2y + 6x − 8y + 11 = ❇➔✐ ✶✺✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c, d t❤ä❛ ♠➣♥ a + b + c + d = 0✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a3 + b3 + c3 + d3 = (b + c) (ad − bc) ❇➔✐ ✶✻✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c = t❤ä❛ ♠➣♥ a + c = ✈➔ a = b + c✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❇➔✐ ✶✼✳ a3 + b3 a+b = a3 + c a+c ❈❤♦ ❝→❝ sè t❤ü❝ a, y, a, b, c t❤ä❛ ♠➣♥ x + y = a, x2 + y = b, x3 + y = c ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a3 − 3ab + 2c = 0✳ ❇➔✐ ✶✽✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ♠➣♥ x + y = ✈➔ x.y = 3✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❝→❝ ❜✐➸✉ t❤ù❝ s❛✉ ❛✮ A = x2 + y ❜✮ B = x3 + y ❝✮ C = x4 + y ❞✮ D = x5 + y ❇➔✐ ✶✾✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ a − b = ✈➔ a.b = 3✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❝→❝ ❜✐➸✉ t❤ù❝ s❛✉ ❛✮ M = a2 + b2 ❜✮ N = a4 + b4 ❝✮ P = a6 + b6 ❇➔✐ ✷✵✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ♠➣♥ x + y = −5 ✈➔ x2 + y2 = 11✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ P = x4 + y ✳ ❇➔✐ ✷✶ ✭❑❍❚◆ ✷✵✶✺✱ ✈á♥❣ ✶✮✳ ●✐↔ sû a, b ❧➔ ❤❛✐ sè t❤ü❝ ♣❤➙♥ ❜✐➺t t❤ä❛ ♠➣♥ a2 + 3a = b2 + 3b = 2✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❛✮ a + b = −3 ❜✮ a3 + b3 = −45 ❇➔✐ ✷✷✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y, z t❤ä❛ ♠➣♥ x2 + 2y + = y + 2z + = z + 2x + = ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ Q = x2019 + y 2020 + z 2021 ❇➔✐ ✷✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ sè n ✈✐➳t ✤÷đ❝ t❤➔♥❤ tê♥❣ ❝õ❛ ❤❛✐ sè ❝❤➼♥❤ ♣❤÷ì♥❣ t❤➻ sè 2n ❝ơ♥❣ ✈✐➳t ✤÷đ❝ t❤➔♥❤ tê♥❣ ❝õ❛ ❤❛✐ sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ❇➔✐ ✷✹ ✭❝❤✉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 ✳ ❇➔✐ ✷✺✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ a + b + c = ✈➔ a2 + b2 + c2 = 1✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ M = a4 + b4 + c4 ✳ ❇➔✐ ✷✻ ✭❚❤✐ ❝❤✉②➯♥ ❚P ❍❈▼✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ a + b + c = ✈➔ a2 = (a + c + 1) (a + b − 1) ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ M = a2 + b2 + c2 ✳ ❇➔✐ ✷✼✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c = t❤ä❛ ♠➣♥ a + b + c = 0✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❝→❝ ❜✐➸✉ t❤ù❝ ab bc ca ❛✮ P = + + ✳ 2 2 2 2 a +b −c b +c −a c +a −b 2 b c2 a + + ✳ ❜✮ Q = a − b2 − c2 b2 − c2 − a2 c2 − a2 − b2 số a ỗ 2n ỳ số số b ỗ n ỳ số ự a + b + số ữỡ số a ỗ 2n ỳ số số b ỗ n + ỳ số số c ỗ n ỳ số ✭n ∈ N∗✮✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a + b + c + ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ a+b+c = 0✳ ❇➔✐ ✷✽✳ ❇➔✐ ✷✾✳ ❇➔✐ ✸✵ ✭❙P ✷✵✶✵✱ ✈á♥❣ ✶✮✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a2 + b2 + c2 = a4 + b4 + c4 ❇➔✐ ✸✶ ✭❚❤✐ ❝❤✉②➯♥ ❚P ❍❈▼✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ a + b + c = 1✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ Q = a3 + b3 + c3 − (ab + c) (c − 1) ❇➔✐ ✸✷ ✭❑❍❚◆ ✷✵✶✺✱ ✈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✳ ❇➔✐ ✸✸ ✭❆♠s ✷✵✶✻✱ ❝❤✉②➯♥ ❚✐♥✮ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ a + b + c = ✈➔ a3 + b3 + c3 = 3abc✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ a2 b2 c2 P = + + b + c2 c2 + a2 a2 + b2 ❇➔✐ ✸✹ ✭❆♠s ✷✵✶✻✱ ❝❤✉②➯♥ ❚♦→♥✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ t❤ä❛ ♠➣♥✿ a3 + b3 + c3 = 3abc ✈➔ a, b, c = 0✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ 2 P = ab bc ca + + a2 + b2 − c2 b2 + c2 − a2 c2 + a2 − b2 ❇➔✐ ✸✺ ✭❙P ✷✵✶✹✱ ✈á♥❣ ✷✮✳ ●✐↔ sû a, b, c, x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❦❤→❝ t❤ä❛ ♠➣♥ a b c + + = ✈➔ x y z x y z + + =1 a b c x2 y z ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ + + = 1✳ a b c ❇➔✐ ✸✻✳ a + b + c = ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥✿ a3 + b3 + c3 = ✳ ❈❤ù♥❣ ♠✐♥❤ a2019 + b2019 + c2019 = ❇➔✐ ✸✼ ✭❍❙● ◆❛♠ ✣à♥❤ ✷✵✶✷✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ ✈➔ t❤ä❛ ♠➣♥✿ a2 − b = b2 − c = c2 − a✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ (a + b + 1) (b + c + 1) (c + a + 1) = −1 ❇➔✐ ✸✽ ✭❍❙● ❍➔ ❚➽♥❤ ✷✵✶✹✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ a + b + c = a2 + b2 + c2 = a3 + b3 + c3 = ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ M = a2020 + b2021 + c2022 ❇➔✐ ✸✾ ✭❝❤✉②➯♥ ▲❛♠ ❙ì♥ ✷✵✷✵✮✳ ❈❤♦ x, y, z số tỹ tọ ỗ tớ ❝→❝ ✤✐➲✉ ❦✐➺♥ x + y + z = 2045 ✈➔ (x − 18)3 + (y − 7)3 + (z − 2020)3 = ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ P = (x − 18)2021 + (y − 7)2021 + (z − 2020)2021