On tap thi chuyen su pham

❇⑨■ ❚❾P ➷◆ ❚❍■ ❈❍❯❨➊◆ ❙× P❍❸▼ ❇➔✐ ✶ ✭❙P ✷✵✵✽✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❜✐➸✉ t❤ù❝ √ P =√ a+b √ : a+ b a a+b b √ +√ − a − b b − ab ab + a − a− √ b ✈ỵ✐ a > 0, b > 0, a = b✳ ❛✮ ❘ót ❣å♥ P ✳ ❜✮ ❚➻♠ a, b ❜✐➳t P = −1 ✈➔ b = (a + 1)2✳ ❇➔✐ ✷ ✭❙P ✷✵✵✽✱ ✈á♥❣ ✶✮✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤✿ x2 + m2 + x + m = 2✱ m ❧➔ t❤❛♠ sè✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ ữỡ tr ổ õ t ợ m✳ ❜✮ ●å✐ x1, x2 ❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ ❚➻♠ t➜t ❝→❝ ❝→❝ ❣✐→ trà ❝õ❛ m s❛♦ ❝❤♦ 2x − 2x − 55 x2 + = x1 x2 + x1 x1 x2 ❇➔✐ ✸ ✭❙P ✷✵✵✽✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c, d ✈ỵ✐ b = 1, c = tọ ỗ tớ ac a − c = b − 2b ✐✐✮ bd − b − d = c2 − 2c ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ ad + b + c = bc + a + d✳ ❇➔✐ ✹ ✭❙P ✷✵✵✾✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❝→❝ ❜✐➸✉ t❤ù❝ A= 20a + 92 + √ a4 + 16a2 + 64 B = a4 + 20a3 + 102a2 + 40a + 200 ❛✮ ❘ót ❣å♥ A✳ ❜✮ ❚➻♠ a ✤➸ A + B = 0✳ ❈❤♦ ♣❛r❛❜♦❧ (P ) : y = x2 ✈➔ ✤÷í♥❣ t❤➥♥❣ d : y = mx + 1✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✤÷í♥❣ t❤➥♥❣ ❞ ❧✉æ♥ ❝➢t ♣❛r❛❜♦❧ (P ) t↕✐ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ✈ỵ✐ ♠å✐ ❣✐→ trà ❝õ❛ m✳ ❜✮ ●å✐ A(x1; y1), B(x2; y2) ❧➔ ❝→❝ ❣✐❛♦ ✤✐➸♠ ❝õ❛ ❞ ✈➔ (P )✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✺ ✭❙P ✷✵✵✾✱ ✈á♥❣ ✶✮✳ M = (y1 − 1)(y2 − 1) √ ❇➔✐ ✻ ✭❙P ✷✵✵✾✱ ✈á♥❣ ✷✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ♠➣♥ xy = √ xy = − 2✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❜✐➸✉ t❤ù❝ s❛✉ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ x, y √ √ 2xy xy − 2xy xy √ √ √ √ P = + − x2 y − 2xy + xy + xy − ✶ ✈➔ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ x2 + bx + c = 0✱ tr♦♥❣ ✤â b, c ❧➔ ❝→❝ t❤❛♠ sè t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ b + c = 4✳ ❚➻♠ ❝→❝ ❣✐→ trà ❝õ❛ b ✈➔ c ✤➸ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t x1 , x2 s❛♦ ❝❤♦ x1 = x22 + x2 ✳ ❇➔✐ ✽ ✭❙P ✷✵✶✵✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❜✐➸✉ t❤ù❝ ❇➔✐ ✼ ✭❙P ✷✵✵✾✱ ✈á♥❣ ✷✮✳ A= x4 + x3 − x(4x − 1) − x2 + 29x + 78 − x4 − : x +1 x + 6x6 − x − 3x2 + 12x − 36 ❛✮ ❘ót ❣å♥ A✳ ❜✮ ❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà ♥❣✉②➯♥ ❝õ❛ x s❛♦ ❝❤♦ A ❝â ❣✐→ trà ♥❣✉②➯♥✳ ❇➔✐ ✾ ✭❙P ✷✵✵✾✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ d1 : y = 2m2 + x + 2m − d2 : y = m2 x + m − ✈ỵ✐ m ❧➔ t❤❛♠ sè✳ ❛✮ ❚➻♠ tå❛ ✤ë ❣✐❛♦ ✤✐➸♠ I ❝õ❛ d1 ✈➔ d2 t❤❡♦ m✳ ❜✮ ❑❤✐ m t❤❛② ✤ê✐✱ ❝❤ù♥❣ ♠✐♥❤ ✤✐➸♠ I ❧✉æ♥ t❤✉ë❝ ♠ët ✤÷í♥❣ t❤➥♥❣ ❝è ✤à♥❤✳ ❇➔✐ ✶✵ ✭❙P ✷✵✶✵✱√✈á♥❣ ✷✮✳ √ ❈❤♦ a, b ❧➔ ❤❛✐ sè t❤ü❝ ❞÷ì♥❣ ❦❤→❝ ♥❤❛✉ ✈➔ t❤ä❛ ♠➣♥ a − b = − b2 − − a2✳ ❈❤ù♥❣ ♠✐♥❤ a2 + b2 = ❇➔✐ ✶✶ ✭❝❤✉②➯♥ ♥❣ú ✷✵✶✶✮✳ ❈❤♦ ❜✐➸✉ t❤ù❝ √ A= 1 √ +√ x y 1 √ : √ + + x+ y x y √ √ x3 + y x + x y + xy + y3 x3 y ❛✮ ❘ót ❣å♥ A✳ ❜✮ ❚➻♠ x, y ❜✐➳t xy = 36 ✈➔ A = 5✳ ❇➔✐ ✶✷ ✭❙P ✷✵✶✶✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❜✐➸✉ t❤ù❝ A= x−y x2 + y + y − + 2y − x 2y + xy − x2 4x4 + 4x2 y + y − : x2 + y + xy + x ❱ỵ✐ x > 0, y > 0, x = 2y, y = − 2x2✳ ❛✮ ❘ót ❣å♥ A✳ ❜✮ ❈❤♦ y = 1✱ ❤➣② t➻♠ x s❛♦ ❝❤♦ A = 25 ✳ ❇➔✐ ✶✸ ✭❙P ✷✵✶✶✱ ✈á♥❣ ✷✮✳ ❈❤♦ ♣❛r❛❜♦❧ (P ) : y = x2 ✈➔ ✤÷í♥❣ t❤➥♥❣ ✷ d : y = mx − m2 + 3✱ m ❧➔ t❤❛♠ sè✳ ❛✮ ❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà ❝õ❛ m ✤➸ ✤÷í♥❣ t❤➥♥❣ ❞ ❝➢t ♣❛r❛❜♦❧ (P ) t↕✐ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝â ❤♦➔♥❤ ✤ë x1, x2✳ ❜✮ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ m t❤➻ x1, x2 ❧➔ ✤ë ❞➔✐ ❝→❝ ❝↕♥❤ ❣â❝ ✈✉æ♥❣ ❝õ❛ ♠ët t❛♠ ❣✐→❝ ✈✉æ♥❣ ❝â ✤ë ❞➔✐ ❝↕♥❤ ❤✉②➲♥ ❜➡♥❣ 52 ✳ √ ❈❤♦ a = 12 + 18 − 28 ✳ √ √ ❛✮ ❈❤ù♥❣ ♠✐♥❤ 4a2 + 2a − = 0✳ √ ❜✮ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ S = a2 + a4 + a + 1✳ ❇➔✐ ✶✺ ✭❙P ✷✵✶✷✱ ✈á♥❣ ✷✮✳ ❈❤♦ ❜✐➸✉ t❤ù❝ ❇➔✐ ✶✹ ✭❙P ✷✵✶✶✱ ✈á♥❣ ✷✮✳ √ P = √ a−b a−b √ +√ a+b+ a−b a2 − b2 − a + b a2 + b2 √ a2 − b2 ❱ỵ✐ a > b > 0✳ ❛✮ ❘ót ❣å♥ P ✳ ❜✮ ❇✐➳t a − b = 1✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ P ✳ ❇➔✐ ✶✻ ✭❙P ✷✵✶✷✱ ✈á♥❣ ✶✮✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy ✱ ❝❤♦ ♣❛r❛❜♦❧ (P ) : y = −x2 ✈➔ ✤÷í♥❣ t❤➥♥❣ d : y = mx − m − 2✱ m ❧➔ t❤❛♠ sè✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ♠å✐ ❣✐→ trà ❝õ❛ m✱ ✤÷í♥❣ t❤➥♥❣ ❞ ❧✉ỉ♥ ❝➢t ♣❛r❛❜♦❧ (P ) t↕✐ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t√❝â ❤♦➔♥❤ ✤ë x1 , x2 ✳ ❜✮ ❚➻♠ m ✤➸ |x1 − x2| = 20✳ ❇➔✐ ✶✼ ✭❙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❝ M = c2.(a + b)✳ ❇➔✐ ✶✽ ✭❙P ✷✵✶✸✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❜✐➸✉ t❤ù❝ Q= a−b √ √ a+ b √ √ + 2a a + b b √ 3a2 + 3b ab √ ab − a √ + √ a a−b a ✈ỵ✐ a > 0, b > 0, a = b✳ ❈❤ù♥❣ ♠✐♥❤ ❜✐➸✉ t❤ù❝ Q ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ a ✈➔ b✳ ❇➔✐ ✶✾ ✭❙P ✷✵✶✸✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ a+b+c = 0✳ ❈❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ a2 + b2 + c2 = a4 + b4 + c4 ✸ ❈❤♦ ♣❛r❛❜♦❧ (P ) : y = x2 ✈➔ ✤÷í♥❣ t❤➥♥❣ d : y = −mx + ✱ ✈ỵ✐ m = ❧➔ t❤❛♠ sè✳ 2m2 ự ợ m = ữớ t ❞ ❧✉æ♥ ❝➢t ♣❛r❛❜♦❧ (P ) t↕✐ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t✳ ❜✮ ●å✐ A(x1; y1), B(x2; y2) ❧➔ ❝→❝ ❣✐❛♦ ✤✐➸♠ ❝õ❛ ❞ ✈➔ (P )✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸ t❤ù❝ M = y12 + y22✳ ❇➔✐ ✷✶ ✭❙P ✷✵✶✸✱ ✈á♥❣ ✶✮✳ ●✐↔ sû a, b, c ❧➔ ❝→❝ sè t❤ü❝✱ a = b s❛♦ ❝❤♦ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ x2 + ax + = 0, x2 + bx + c = ❝â ♥❣❤✐➺♠ ❝❤✉♥❣ ✈➔ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ x2 + x + a = 0, x2 + cx + b = ❝â ♥❣❤✐➺♠ ❝❤✉♥❣✳ ❚➼♥❤ P = a + b + c✳ ❇➔✐ ✷✷ ✭❙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 = a3b3c3✳ ❈❤ù♥❣ ♠✐♥❤ abc = 0✳ ❇➔✐ ✷✸ ✭❙P ✷✵✶✹✱ ✈á♥❣ số tỹ ữỡ a, b ợ a = b✳ ❈❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ ❇➔✐ ✷✵ ✭❙P ✷✵✶✸✱ ✈á♥❣ ✶✮✳ √ √ (a − b)3 − b b + 2a a √ √ √ a− b 3a + ab √ + =0 √ b−a a a−b b ❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy✱ ❝❤♦ ♣❛r❛❜♦❧ (P ) : y = x2 ✈➔ ✤÷í♥❣ t❤➥♥❣ d : y = − (m + 1) x + ✱ ✈ỵ✐ m ❧➔ t❤❛♠ sè✳ 3 ❛✮ ❈❤ù♥❣ ợ tr m ữớ t ❧✉æ♥ ❝➢t ♣❛r❛❜♦❧ (P ) t↕✐ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t✳ ❜✮ ●å✐ x1, x2 ❧➔ ❤♦➔♥❤ ✤ë ❝→❝ ❣✐❛♦ ✤✐➸♠ ❝õ❛ ❞ ✈➔ (P )✳ ✣➦t f (x) = x3 + (m + 1) x2 − x✳ ❈❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ ❇➔✐ ✷✹ ✭❙P ✷✵✶✹✱ ✈á♥❣ ✶✮✳ f (x1 ) − f (x2 ) = − (x1 − x2 )3 ●✐↔ sû x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x + y + z = xyz ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❇➔✐ ✷✸ ✭❑❍❚◆ ✷✵✶✹✱ ✈á♥❣ ✶✮✳ x 2y 3z xyz (5x + 4y + 3z) + + = 2 1+x 1+y 1+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✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy✱ ❝❤♦ ♣❛r❛❜♦❧ (P ) : y = x ✈➔ ✤÷í♥❣ t❤➥♥❣ d : y = mx + 2✱ m ❧➔ t❤❛♠ sè✳ ự ợ tr m ữớ t❤➥♥❣ ❞ ❧✉æ♥ ❝➢t ♣❛r❛❜♦❧ (P ) t↕✐ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t✳ ❜✮ ●å✐ A(x1; y1), B(x2; y2) ❧➔ ❝→❝ ❣✐❛♦ ✤✐➸♠ ❝õ❛ ❞ ✈➔ (P )✳ ❚➻♠ ❣✐→ trà ❝õ❛ m ✤➸ A = y12 + y22 ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ ❇➔✐ ✷✼ ✭❙P ✷✵✶✺✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❜✐➸✉ t❤ù❝ ❇➔✐ ✷✻ ✭❝❤✉②➯♥ ♥❣ú ✷✵✶✺✮✳ P = a b + +1 b a a2 b + − b a2 1 − a b a b + b a ✈ỵ✐ a > 0, b > 0, a = b✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ P = ab1 ✳ √ ❜✮ ●✐↔ sû a, b t❤❛② ✤ê✐ s❛♦ ❝❤♦ 4a + b + ab = 1✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ P ✳ ❇➔✐ ✷✽ ✭❙P ✷✵✶✺✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ x − my = − 4m mx + y = 3m + ợ m t số ữỡ tr➻♥❤ ❦❤✐ m = 2✳ ❜✮ ❈❤ù♥❣ ♠✐♥❤ ❤➺ ❧✉æ♥ ❝â ♥❣❤✐➺♠ ✈ỵ✐ ♠å✐ ❣✐→ trà ❝õ❛ m✳ ●✐↔ sû (x0; y0) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❤➺✳ ❈❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ x20 + y02 − (x0 + y0 ) + 10 = ❇➔✐ ✷✾ ✭❙P ✷✵✶✺✱ ✈á♥❣ ✶✮✳ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳ a (x − a)2 + b (x − b)2 = ❈❤ù♥❣ ♠✐♥❤ |a| = |b|✳ ❇➔✐ ✸✵ ✭❙P ✷✵✶✻✱ ✈á♥❣ ✶✮✳ √ P = √ ❈❤♦ a, b ❧➔ ❝→❝ sè t❤ü❝ ❦❤→❝ 0✳ ❇✐➳t r➡♥❣ ❈❤♦ ❜✐➸✉ t❤ù❝ 1+a 1−a √ +√ 1+a− 1−a − a2 − + a ✺ 1 − − a2 a ✈ỵ✐ < a < 1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ P = −1✳ ❈❤♦ ♣❛r❛❜♦❧ (P ) : y = −x2 ✈➔ ✤÷í♥❣ t❤➥♥❣ d : y = 2mx − 1✱ ✈ỵ✐ m ❧➔ t❤❛♠ sè✳ ❛✮ ❚➻♠ tå❛ ✤ë ❣✐❛♦ ✤✐➸♠ ❝õ❛ ❞ ✈➔ (P ) ❦❤✐ m = 1✳ ❜✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ♠å✐ ❣✐→ trà ❝õ❛ m✱ ❞ ❧✉ỉ♥ ❝➢t (P ) t↕✐ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t A, B ✳ ●å✐ y1, y2 ❧➔ t✉♥❣ ✤ë ❝õ❛ A, B ✳ ❚➻♠ m s❛♦ ❝❤♦ ❇➔✐ ✸✶ ✭❙P ✷✵✶✻✱ ✈á♥❣ ✶✮✳ √ y12 − y22 = ❇➔✐ ✸✷ ✭❙P ✷✵✶✼✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❜✐➸✉ t❤ù❝ b2 a : √ a+ a+b a3 − a − 2b − P = b + a a 1− a3 + a2 + ab + a2 b b + a2 − b2 a−b ✈ỵ✐ a > 0, b > 0, a = b, a + b = a2✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ P = a − b✳ ❜✮ ❚➻♠ a, b ❜✐➳t r➡♥❣ P = ✈➔ a3 − b3 = 7✳ ❇➔✐ ✸✸ ✭❙P ✷✵✶✼✱ ✈á♥❣ ✶✮✳ ●✐↔ sû x, y ❧➔ ❤❛✐ sè t❤ü❝ ♣❤➙♥ ❜✐➺t t❤ä❛ ♠➣♥✿ x2 1+ + y2 1+ = xy 2+ ✳ ❍➣② t➼♥❤ S= + + x2 + y + xy + ❈❤♦ ♣❛r❛❜♦❧ (P ) : y = x2 ✈➔ ✤÷í♥❣ t❤➥♥❣ d : y = −2ax − 4a✱ ✈ỵ✐ a ❧➔ t❤❛♠ sè✳ ❛✮ ❚➻♠ tå❛ ✤ë ❣✐❛♦ ✤✐➸♠ ❝õ❛ ❞ ✈➔ (P ) ❦❤✐ a = − 21 ✳ ❜✮ ❚➻♠ t➜t ❝↔ ❝→❝ ❣✐→ trà ❝õ❛ a ✤➸ ✤÷í♥❣ t❤➥♥❣ ❞ ❝➢t ♣❛r❛❜♦❧ (P ) t↕✐ ❤❛✐ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝â ❤♦➔♥❤ ✤ë x1, x2 t❤ä❛ ♠➣♥ |x1| + |x2| = 3✳ ❇➔✐ ✸✺ ✭❙P ✷✵✶✺✱ ✈á♥❣ ✷✮✳ ❈❤♦ a ≥ 0, a = 1✳ ❘ót ❣å♥ ❜✐➸✉ t❤ù❝ ❇➔✐ ✸✹ ✭❙P ✷✵✶✼✱ ✈á♥❣ ✶✮✳ S= √ √ − 20 + 14 2+ ❇➔✐ ✸✻ ✭❙P ✷✵✶✺✱ ✈á♥❣ ✷✮✳ 1, < y < ✈➔ (a + 3) √ a − 3a − : a−1 √ −1 ( a − 1) ❈❤♦ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ♠➣♥ < x < ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ x y + = 1✳ 1−x 1−y x2 − xy + y P =x+y+ ✻ ❇➔✐ ✸✼ ✭❙P ✷✵✶✽✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❜✐➸✉ t❤ù❝ √ 2x − x+1 x−1 √ √ P = (x + 1) x + + (x − 1) x − √ −√ x−1 x+1 √ ❛✮ ❘ót ❣å♥ ❜✐➸✉ t❤ù❝ P ✳ ❜✮ ❚➻♠ x ✤➸ P = x − 1✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤✿ x3 − x − = 0✳ ●✐↔ sû x0 ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ t➻♥❤ ✤➣ ❝❤♦✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ x0 > 0✳ x20 − ❜✮ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ M = x3 2x20 + 3x0 + 2✳ ❇➔✐ ✸✾ ✭❑❍❚◆ ✷✵✶✺✱ ✈á♥❣ ✶✮✳ ●✐↔ sû a, b ❧➔ ❤❛✐ sè t❤ü❝ ♣❤➙♥ ❜✐➺t t❤ä❛ ♠➣♥ a2 + 3a = b2 + 3b = 2✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❛✮ a + b = −3 ❜✮ a3 + b3 = −45 ❇➔✐ ✹✵ ✭❝❤✉②➯♥ ♥❣ú ✷✵✵✺✮✳❈❤♦ ❝→❝ sè t❤ü❝ x, y, z = t❤ä❛ ♠➣♥✿ ❇➔✐ ✸✽ ✭❙P ✷✵✶✽✱ ✈á♥❣ ✶✮✳ (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 ) ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ = 3abc✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✹✸ ✭❆♠s ✷✵✶✻✱ ❝❤✉②➯♥ ❚✐♥✮ a+b+c=0 ✈➔ a 3 +b +c a2 b2 c2 P = + + b + c2 c2 + a2 a2 + b2 ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ = 3abc ✈➔ a, b, c = 0✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ ❇➔✐ ✹✹ ✭❆♠s ✷✵✶✻✱ ❝❤✉②➯♥ ❚♦→♥✮✳ ♥❤❛✉ t❤ä❛ ♠➣♥✿ a t❤ù❝ P = 3 +b +c ab2 bc2 ca2 + + a2 + b2 − c2 b2 + c2 − a2 c2 + a2 − b2 ✼ ❇➔✐ ✹✺ ✭❙P ✷✵✶✻✱ ✈á♥❣ ✷✮✳ ❚➻♠ t➜t ❝↔ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ♠➣♥ √ x2 − y − x−1+ + +8=4 x y ❇➔✐ ✹✻ ✭❆♠s ✷✵✶✽✱ ❝❤✉②➯♥ ❚♦→♥✮✳ xyz = ✈➔ y−1 ❱ỵ✐ x, y, z ❧➔ ❝→❝ sè t❤ü❝ t❤ä❛ ♠➣♥ (xy + x + 1)(yz + y + 1)(zx + z + 1) = ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 1 + + =1 xy + x + yz + y + zx + z + ❇➔✐ ✹✼ ✭❙P ✷✵✵✽✱ ✈á♥❣ ✶✮✳ ❦❤→❝ ♥❤❛✉ ✈➔ t❤ä❛ ♠➣♥✿ ❈❤♦ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ x, , y, z ✤æ✐ ♠ët (x + z)(z + y) = ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✿ 1 ≥4 + + (x − y) (z + x) (z + y)2 ❇➔✐ ✹✽ ✭❑❍❚◆ ✷✵✵✾✱ ✈á♥❣ ✶✮✳ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❱ỵ✐ a, b ❧➔ ♥❤ú♥❣ sè t❤ü❝ ❞÷ì♥❣✱ t➻♠ a+b a (4a + 5b) + b (4b + 5a) P = ❇➔✐ ✹✾ ✭❑❍❚◆ ✷✵✵✾✱ ✈á♥❣ ✶✮✳ ♠✐♥❤ r➡♥❣ ❱ỵ✐ a, b ❧➔ ♥❤ú♥❣ sè t❤ü❝ ❞÷ì♥❣✳ ❈❤ù♥❣ b2 c2 a+b+c a2 √ +√ +√ ≥ 3a2 + 8b2 + 14ab 3b2 + 8c2 + 14bc 3c2 + 8a2 + 14ca ❇➔✐ ✺✵ ✭❙P ✷✵✵✾✱ ✈á♥❣ ✶✮✳ x+ √ ❝❤♦ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ♠➣♥ ✤➥♥❣ t❤ù❝✿ + x2 y+ + y2 = ❈❤ù♥❣ ♠✐♥❤✿ x + y = 0✳ ❈❤♦ a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥✿ a + b + c + ab + bc + ca = 6✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❇➔✐ ✺✶ ✭❝❤✉②➯♥ ♥❣ú ✷✵✶✵✮✳ a3 b c + + ≥ a2 + b2 + c2 ≥ b c a ✽ ❱ỵ✐ a, b ❧➔ ❝→❝ sè t❤ü❝ t❤ä❛ ♠➣♥✿ (1 + a)(1 + b) = ✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✺✷ ✭❑❍❚◆ ✷✵✶✵✱ ✈á♥❣ ✶✮✳ P = √ 1+ a4 + √ + b4 ●✐↔ sû x, y, z ❧➔ ♥❤ú♥❣ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥✿ x + y + z = 1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❇➔✐ ✺✸ ✭❑❍❚◆ ✷✵✶✵✱ ✈á♥❣ ✷✮✳ √ xy + z + 2x2 + 2y ≥1 √ + xy ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✿ ❇➔✐ ✺✹ ✭❙P ✷✵✶✵✱ ✈á♥❣ ✶✮✳ x2 − = (x − 1)2 x2 − 5x + ❇➔✐ ✺✺ ✭❝❤✉②➯♥ ♥❣ú ✷✵✶✶✮✳ sè y= ❇➔✐ ✺✻ ✭❙P ✷✵✶✶✱ ✈á♥❣ ✶✮✳ √ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ✈➔ ♥❤ä ♥❤➜t ❝õ❛ ❤➔♠ √ x+3+ √ 6−x ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ 1 1 √ +√ √ +√ √ + + √ √ >4 1+ 3+ 5+ 79 + 80 ❈❤♦ x, y ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥✿ √ x2 y √ + ✳ ( x + 1) y + ≥ 4✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ P = y x ❇➔✐ ✺✽ ✭❑❍❚◆ ✷✵✶✷✱ ✈á♥❣ ✷✮✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❤➔♠ sè ❇➔✐ ✺✼ ✭❑❍❚◆ ✷✵✶✷✱ ✈á♥❣ ✶✮✳ √ y = 2x − + x − 4x2 ✈ỵ✐ √ ≤x≤ ✳ 2 ❈❤♦ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ x, y t❤ä❛ ♠➣♥✿ xy (x − y) = x + y ✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ P = x + y ✳ ❇➔✐ ✻✵ ✭❑❍❚◆ ✷✵✶✸✱ ✈á♥❣ ✷✮✳ ❱ỵ✐ x, y ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥✿ x + y ≤ 1✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✺✾ ✭❙P ✷✵✶✷✱ ✈á♥❣ ✶✮✳ √ 1 + + x2 y x y P = ✈ỵ✐ a, b ❧➔ ❤❛✐ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ + 4b ✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✻✶ ✭❆♠s ✷✵✶✸✱ ❝❤✉②➯♥ ❚♦→♥✮✳ ♠➣♥✿ a + b + 4ab = 4a 2 A = 20 a3 + b3 − a2 + b2 + 2013 ✾ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❇➔✐ ✻✷ ✭❙P ✷✵✶✹✱ ✈á♥❣ ✶✮✳ x x2 − 56 21x + 22 − =4 − 7x x +2 ❈❤♦ a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥ abc = 1✳ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ❇➔✐ ✻✸ ✭❙P ✷✵✶✹✱ ✈á♥❣ ✷✮✳ 1 + + ≤ ab + a + bc + b + c ca + c + ❇➔✐ ✻✹ ✭❆♠s ✷✵✶✹✱ ❝❤✉②➯♥ ❚✐♥✮✳ ♠➣♥✿ x + y + z = 1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❱ỵ✐ ❜❛ sè t❤ü❝ ❞÷ì♥❣ x, y, z t❤ä❛ − y2 − z2 − x2 + + ≥6 x + yz y + zx z + xy ❱ỵ✐ ❜❛ sè ❞÷ì♥❣ x, y, z t❤ä❛ ♠➣♥ x + y + z = 1✱ t➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✻✺ ✭❆♠s ✷✵✶✹✱ ❝❤✉②➯♥ ❚♦→♥✮✳ Q= x y z √ √ √ + + x + x + yz y + y + zx z + z + xy ●✐↔ sû x, y ❧➔ ♥❤ú♥❣ sè t❤ü❝ ❦❤æ♥❣ ➙♠ + y ✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ✈➔ ♥❤ä ♥❤➜t ❝õ❛ ❇➔✐ ✻✻ ✭❑❍❚◆ ✷✵✶✹✱ ✈á♥❣ ✷✮✳ t❤ä❛ ♠➣♥✿ x ❜✐➸✉ t❤ù❝ 3 + y + xy = x √ √ 1+ x 2+ x P = √ + √ 2+ y 1+ y ❈❤♦ ❤❛✐ sè t❤ü❝ ❞÷ì♥❣ x, y t❤ä❛ ♠➣♥ x + y ≥ 3✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✻✼ ✭❝❤✉②➯♥ ♥❣ú ✷✵✶✺✮✳ P = 2x2 + y + 28 + x y ❈❤♦ ❜❛ sè t❤ü❝ ❞÷ì♥❣ a, b, c t❤ä❛ ♠➣♥✿ (a + b)(b + c)(c + a) = 1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❇➔✐ ✻✽ ✭❆♠s ✷✵✶✺✱ ❝❤✉②➯♥ ❚♦→♥✮✳ ab + bc + ca ≤ ❇➔✐ ✻✾ ✭❙P ✷✵✶✺✱ ✈á♥❣ ✶✮✳ a2 + b + 4 ❚➻♠ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ a, b, c t❤ä❛ ♠➣♥ b2 + a + ✶✵ = 2a + 2b + ❈❤♦ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ♠➣♥ ✤➥♥❣ + 2y + = 0✱ t➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ✈➔ ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✼✵ ✭❑❍❚◆ ✷✵✶✺✱ ✈á♥❣ ✶✮✳ t❤ù❝✿ x y 2 P = xy 3y + ❈❤♦ a, b, c ❧➔ ❜❛ sè t❤ü❝ ❦❤æ♥❣ ➙♠ ✈➔ t❤ä❛ ♠➣♥✿ a + b + c = 1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❇➔✐ ✼✶ ✭❙P ✷✵✶✻✱ ✈á♥❣ ✶✮✳ √ 5a + + ❇➔✐ ✼✷ ✭❙P ✷✵✶✼✱ ✈á♥❣ ✶✮✳ ♠➣♥✿ √ 5b + + √ 5c + ≥ ❈❤♦ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ x1, x2, , x9 t❤ä❛ x1 + x2 + + x9 = 10 x1 + 2x2 + + 9x9 = 18 ❈❤ù♥❣ ♠✐♥❤✿ 1.19x1 + 2.18x2 + + 9.11x9 ≥ 270✳ ✣➥♥❣ t❤ù❝ ①➞② r❛ ❦❤✐ ♥➔♦❄ ❇➔✐ ✼✸ ✭❙P ✷✵✶✽✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❝→❝ sè ♥❣✉②➯♥ x1 , x2 , , x9 t❤ä❛ ♠➣♥✿ (1 + x1 )(1 + x2 ) (1 + x9 ) = (1 − x1 )(1 − x2 ) (1 − x9 ) = x ❚➼♥❤ P = x.x1.x2 x9✳ ❈❤♦ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ x, y, z t❤ä❛ = 3✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❇➔✐ ✼✹ ✭❆♠s ✷✵✶✼✱ ❝❤✉②➯♥ ❚♦→♥✮✳ ♠➣♥ x 2 +y +z x y z + + ≤ − yz − zx − xy ❇➔✐ ✼✺ ✭❆♠s ✷✵✶✼✱ ❝❤✉②➯♥ ❚✐♥✮✳ ✤ê✐ ❧✉æ♥ t❤ä❛ ♠➣♥ ❈❤♦ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ a, b, c t❤❛② 1 + + =3 a2 b2 c2 ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ P = 1 + + (2a + b + c) (2b + c + a) (2c + a + b)2 ❱ỵ✐ x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤❛② ✤ê✐ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x1 + y1 + z1 = 3✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✼✻ ✭❆♠s ✷✵✶✽✱ ❝❤✉②➯♥ ❚♦→♥✮✳ P = + 2x2 + y + 1 +√ 2z + x2 + 2y + z + ✶✶ ợ x, y, z số tỹ ữỡ t ✤ê✐ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ xyz ≥ 1✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✼✼ ✭❆♠s ✷✵✶✽✱ ❝❤✉②➯♥ ❚✐♥✮✳ P =√ 1 +√ +√ xy + x + yz + y + zx + z + ▼ët ♥❤➔ ♠→② ❝❤✉②➯♥ s↔♥ ①✉➜t ♠ët ❧♦↕✐ s↔♥ ♣❤➞♠✳ ◆➠♠ ✷✵✶✺✱ ♥❤➔ ♠→② s↔♥ ①✉➜t ✤÷đ❝ ✺✵✵✵ s↔♥ ♣❤➞♠✳ ❉♦ ↔♥❤ ❤÷ð♥❣ ❝õ❛ t❤à tr÷í♥❣ t✐➯✉ t❤ư ♥➯♥ s↔♥ ❧÷đ♥❣ ❝õ❛ ♥❤➔ ♠→② tr♦♥❣ ❝→❝ ♥➠♠ ✷✵✶✻ ✈➔ ✷✵✶✼ ✤➲✉ ❣✐↔♠✳ ❈ö t❤➸✿ sè ❧÷đ♥❣ s↔♥ ♣❤➞♠ ♥❤➔ ♠→② s↔♥ ①✉➜t ✤÷đ❝ ♥➠♠ ✷✵✶✻ x% s ợ s ữủ s s↔♥ ①✉➜t ✤÷đ❝ tr♦♥❣ ♥➠♠ ✷✵✶✺✱ sè ❧÷đ♥❣ s↔♥ ♣❤➞♠ ♥❤➔ ♠→② s↔♥ ①✉➜t ✤÷đ❝ tr♦♥❣ ♥➠♠ ✷✵✶✼ ❝ơ♥❣ ❣✐↔♠ x% s ợ số ữủ s s ①✉➜t ✤÷đ❝ tr♦♥❣ ♥➠♠ ✷✵✶✻✳ ❇✐➳t r➡♥❣ sè ❧÷đ♥❣ s↔♥ ♣❤➞♠ ♥❤➔ ♠→② s↔♥ ①✉➜t ✤÷đ❝ tr♦♥❣ ♥➠♠ ✷✵✶✼ ❣✐↔♠ 51% s ợ số ữủ s s ①✉➜t ✤÷đ❝ tr♦♥❣ ♥➠♠ ✷✵✶✺✳ ❚➻♠ x✳ ❇➔✐ ✼✾ ✭❙P ✷✵✶✼✱ ✈á♥❣ ✶✮✳ ❆♥❤ ◆❛♠ ✤✐ ①❡ ✤↕♣ tø A ✤➳♥ C ✳ ❚r➯♥ q✉➣♥❣ ✤÷í♥❣ AB ❜❛♥ ✤➛✉ ✭B ♥➡♠ ❣✐ú❛ A ✈➔ C ✮ ❛♥❤ ◆❛♠ ✤✐ ✈ỵ✐ ✈➟♥ tè❝ ❦❤ỉ♥❣ ✤ê✐ ❧➔ a ✭km/h✮ ✈➔ t❤í✐ ❣✐❛♥ ✤✐ tø A ✤➳♥ B ❧➔ 1, ❣✐í✳ ❚r➯♥ q✉➣♥❣ ✤÷í♥❣ BC ❝á♥ ❧↕✐✱ ❛♥❤ ◆❛♠ ✤✐ ❝❤➟♠ ❞➛♥ ✤➲✉ ✈ỵ✐ ✈➟♥ tè❝ t↕✐ t❤í✐ ✤✐➸♠ t ✭t➼♥❤ ❜➡♥❣ ❣✐í✮ ❦➸ tø B ❧➔ v = −8t + a ✭km/h✮✳ ◗✉➣♥❣ ✤÷í♥❣ ✤✐ ✤÷đ❝ tø B ✤➳♥ t❤í✐ ✤✐➸♠ t ✤â ❧➔ S = −4t2 + at✳ ❚➼♥❤ q✉➣♥❣ ✤÷í♥❣ AB ❜✐➳t r➡♥❣ ✤➳♥ C ①❡ ❞ø♥❣ ❤➥♥ ✈➔ q✉➣♥❣ ✤÷í♥❣ BC ❞➔✐ 16❦♠✳ ❇➔✐ ✽✵ ✭❙P ✷✵✶✻✱ ✈á♥❣ ✶✮✳ ▼ët ♥❣÷í✐ ✤✐ ①❡ ♠→② tø ✤à❛ ✤✐➸♠ A ✤➳♥ ✤à❛ ✤✐➸♠ B ❝→❝❤ ♥❤❛✉ ✶✷✵❦♠✳ ❱➟♥ tè❝ tr➯♥ 43 q✉➣♥❣ ✤÷í♥❣ AB ✤➛✉ ❦❤ỉ♥❣ ✤ê✐✱ ✈➟♥ tè❝ tr➯♥ 41 q✉➣♥❣ ✤÷í♥❣ AB s❛✉ ❜➡♥❣ 21 ✈➟♥ tè❝ tr➯♥ 34 q✉➣♥❣ ✤÷í♥❣ AB ✤➛✉✳ ❑❤✐ ✤➳♥ B ✱ ♥❣÷í✐ ✤â ♥❣❤➾ ✸✵ ♣❤ót ✈➔ trð ❧↕✐ A ✈ỵ✐ ✈➟♥ tố ợ ỡ tố tr 43 q ữớ AB ✤➛✉ t✐➯♥ ❧ó❝ ✤✐ ❧➔ 10km/h✳ ❚❤í✐ ❣✐❛♥ ❦➸ tø ❧ó❝ ①✉➜t ♣❤→t t↕✐ A ✤➳♥ ❦❤✐ ①❡ trð ✈➲ A ❧➔ 8, ❣✐í✳ ❚➼♥❤ ✈➟♥ tè❝ ❝õ❛ ①❡ ♠→② tr➯♥ q✉➣♥❣ ✤÷í♥❣ ♥❣÷í✐ ✤â ✤✐ tø B ✈➲ A✳ ❇➔✐ ✽✶ ✭❙P ✷✵✶✺✱ ✈á♥❣ ✷✮✳ ▼ët ①❡ t↔✐ ❝â ❝❤✐➲✉ rë♥❣ ❧➔ 2, 4m ✈➔ ❝❤✐➲✉ ❝❛♦ ❧➔ 2, 5m ♠✉è♥ ✤✐ q✉❛ ♠ët ❝→✐ ❝ê♥❣ ❤➻♥❤ ♣❛r❛❜♦❧✳ ❇✐➳t ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ❤❛✐ ❝❤➙♥ ❝æ♥❣ √❧➔ 4m ✈➔ tứ ỗ r tợ ộ ❝ê♥❣ ❧➔ 5m ✭❜ä q✉❛ ✤ë ❞➔✐ ❝õ❛ ❝ê♥❣✮✳ ❛✮ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë Oxy✱ ❣å✐ ♣❛r❛❜♦❧ (P ) : y = ax2 ✈ỵ✐ a < ❧➔ ❤➻♥❤ ❜✐➸✉ ❞✐➵♥ ❝ê♥❣ ♠➔ ①❡ t↔✐ ♠✉è♥ ✤✐ q✉❛✳ ❈❤ù♥❣ ♠✐♥❤ a = −1✳ ❜✮ ❍ä✐ ①❡ t↔✐ ❝â t❤➸ ✤✐ q✉❛ ❝ê♥❣ ✤÷đ❝ ❦❤ỉ♥❣❄ ❚↕✐ s❛♦❄ ❇➔✐ ✽✷ ✭❙P ✷✵✶✹✱ ✈á♥❣ ✶✮✳ ❈❤♦ q✉➣♥❣ ✤÷í♥❣ AB ❞➔✐ 120❦♠✳ ▲ó❝ ✼ ❣✐í ❇➔✐ ✼✽ ✭❙P ✷✵✶✽✱ ✈á♥❣ ✶✮✳ ✶✷ s→♥❣✱ ♠ët ①❡ ♠→② ✤✐ tø A ✤➳♥ B ✳ ✣✐ ✤÷đ❝ 34 q✉➣♥❣ ✤÷í♥❣ ①❡ ❜à ❤ä♥❣ ♣❤↔✐ ❞ø♥❣ sỷ t út rỗ t B ✈ỵ✐ ✈➟♥ tè❝ ♥❤ä ❤ì♥ ✈➟♥ tè❝ ❧ó❝ ✤➛✉ 10km/h✳ ❇✐➳t ①❡ ♠→② ✤➳♥ B ❧ó❝ ✶✶ ❣✐í ✹✵ ♣❤ót tr÷❛ ❝ị♥❣ ♥❣➔②✳ ●✐↔ sû ✈➟♥ tè❝ ❝õ❛ ①❡ ♠→② tr➯♥ 43 q✉➣♥❣ ✤÷í♥❣ ❜❛♥ ✤➛✉ ❦❤ỉ♥❣ t❤❛② ✤ê✐ ✈➔ ✈➟♥ tè❝ ❝õ❛ ①❡ ♠→② tr➯♥ 41 q✉➣♥❣ ✤÷í♥❣ ❝á♥ ❧↕✐ ❝ơ♥❣ ❦❤ỉ♥❣ t❤❛② ✤ê✐✳ ❍ä✐ ①❡ ♠→② ❜à ❤ä♥❣ ❧ó❝ ♠➜② ❣✐í❄ ❇➔✐ ✽✸ ✭❙P ✷✵✶✷✱ ✈á♥❣ ✶✮✳ ❚r➯♥ q✉➣♥❣ ✤÷í♥❣ AB ❞➔✐ 210km✱ t↕✐ ❝ị♥❣ ♠ët t❤í✐ ✤✐➸♠✱ ♠ët ①❡ ♠→② ❦❤ð✐ ❤➔♥❤ tø A ✤✐ ✈➲ B ✈➔ ♠ët æ tæ ❦❤ð✐ ❤➔♥❤ tø B ✤✐ ✈➲ A✳ ❙❛✉ ❦❤✐ ❣➦♣ ♥❤❛✉✱ ①❡ ♠→② ✤✐ t✐➳♣ ✹ ❣✐í ♥ú❛ t❤➻ ✤➳♥ B ✈➔ ỉ tỉ ✤✐ t✐➳♣ ✷ ❣✐í ✶✺ ♣❤ót ♥ú❛ t❤➻ ✤➳♥ A✳ ❇✐➳t r➡♥❣ ①❡ ♠→② ✈➔ æ tæ ❦❤æ♥❣ t❤❛② ✤ê✐ ✈➟♥ tè❝ tr➯♥ s✉èt ❝❤➦♥❣ ✤÷í♥❣✳ ❚➼♥❤ ✈➟♥ tè❝ ❝õ❛ ①❡ ♠→② ✈➔ æ tæ✳ ❇➔✐ ✽✹ ✭❙P ✷✵✶✶✱ ✈á♥❣ ✶✮✳ ▼ët ♥❤â♠ ❝æ♥❣ ♥❤➙♥ ✤➦t ❦➳ ❤♦↕❝❤ s↔♥ ①✉➜t ✷✵✵ s↔♥ ♣❤➞♠✳ ❚r♦♥❣ ✹ ♥❣➔② ✤➛✉ ❤å t❤ü❝ ❤✐➺♥ ✤ó♥❣ ♠ù❝ ✤➲ r❛✱ ♥❤ú♥❣ ♥❣➔② ❝á♥ ❧↕✐ ❤å ✤➣ ❧➔♠ ✈÷đt ♠ù❝ ♠é✐ ♥❣➔② ✶✵ s↔♥ ♣❤➞♠✱ ♥➯♥ ✤➣ ❤♦➔♥ t❤➔♥❤ ❦➳ ❤♦↕❝❤ sỵ♠ ✷ ♥❣➔②✳ ❍ä✐ t❤❡♦ ❦➳ ❤♦↕❝❤ ♠é✐ ♥❣➔② ♥❤â♠ ❝æ♥❣ ♥❤➙♥ ❝➛♥ s↔♥ ①✉➜t ❜❛♦ ♥✐➯✉ s↔♥ ♣❤➞♠✳ ❇➔✐ ✽✺ ✭❙P ✷✵✵✾✱ ✈á♥❣ ✶✮✳ ❍❛✐ ♥❣÷í✐ ❝ỉ♥❣ ♥❤➙♥ ❝ị♥❣ ❧➔♠ ♠ët ❝ỉ♥❣ ✈✐➺❝ tr♦♥❣ ✶✽ ❣✐í t❤➻ ①♦♥❣✳ ◆➳✉ ♥❣÷í✐ t❤ù ♥❤➜t ❧➔♠ ✻ ❣✐í ✈➔ ♥❣÷í✐ t❤ù ❤❛✐ ❧➔♠ ✶✷ ❣✐í t❤➻ ❝❤➾ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ 50% ❝ỉ♥❣ ✈✐➺❝✳ ❍ä✐ ♥➳✉ ❧➔♠ r✐➯♥❣ t❤➻ ♠é✐ ♥❣÷í✐ ❤♦➔♥ t❤➔♥❤ ❝æ♥❣ ✈✐➺❝ ✤â tr♦♥❣ ❜❛♦ ❧➙✉✳ ❇➔✐ ✽✻ ✭❙P ✷✵✶✽✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❤➻♥❤ ❝❤ú ♥❤➟t ABCD ✈ỵ✐ BC = a, AB = b✳ ●å✐ M, N ❧➛♥ ❧÷đt ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ❝→❝ ❝↕♥❤ AB, CD✳ ◗✉❛ ✤✐➸♠ M ❞ü♥❣ ✤÷í♥❣ t❤➥♥❣ ❝➢t ✤÷í♥❣ ❝❤➨♦ AC ❝õ❛ ❤➻♥❤ ❝❤ú ♥❤➟t ABCD t↕✐ ✤✐➸♠ P ✈➔ ❝➢t ✤÷í♥❣ t❤➥♥❣ BC t↕✐ ✤✐➸♠ Q s❛♦ ❝❤♦ B ♥➡♠ ❣✐ú❛ C ✈➔ Q✳ ✶✳ ❑❤✐ M P ⊥ AC ✱ ❤➣②✿ ❛✮ ❚➼♥❤ P Q t❤❡♦ a ✈➔ b✳ ❜✮ ❈❤ù♥❣ ♠✐♥❤ a.BP = b.P N ✳ ✷✳ ❈❤ù♥❣ ♠✐♥❤ M N P = M N Q ✭❦❤æ♥❣ ♥❤➜t t❤✐➳t M P ✈➔ AC ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ ♥❤❛✉✮✳ ❇➔✐ ✽✼ ✭❙P ✷✵✶✼✱ ✈á♥❣ ✶✮✳ ❈❤♦ ✤÷í♥❣ trá♥ (O) ❜→♥ ❦➼♥❤ R ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABC ❝â ❜❛ ❣â❝ ♥❤å♥✳ ❈→❝ t✐➳♣ t✉②➳♥ ❝õ❛ ✤÷í♥❣ trá♥ (O) t↕✐ B ✈➔ C ❝➢t ♥❤❛✉ t↕✐ P ✳ ●å✐ D, E t÷ì♥❣ ù♥❣ ❧➔ ❝❤➙♥ ❝→❝ ✤÷í♥❣ ✈✉ỉ♥❣ ❣â❝ ❤↕ tø P ①✉è♥❣ ❝→❝ ✤÷í♥❣ t❤➥♥❣ AB, AC ✈➔ M ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ BC ✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ M EP = M DP ✳ ❜✮ ●✐↔ sû B, C ❝è ✤à♥❤ ✈➔ A ❝❤↕② tr➯♥ ✤÷í♥❣ trá♥ (O) s❛♦ ❝❤♦ t❛♠ ❣✐→❝ ABC ❧✉æ♥ ❧➔ t❛♠ ❣✐→❝ ❝â ❜❛ ❣â❝ ♥❤å♥✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷í♥❣ t❤➥♥❣ DE ❧✉ỉ♥ ✶✸ ✤✐ q✉❛ ♠ët ✤✐➸♠ ❝è ✤à♥❤✳ ❝✮ ❑❤✐ t❛♠ ❣✐→❝ ABC ❧➔ t❛♠ ❣✐→❝ ✤➲✉✱ ❤➣② t➼♥❤ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ ADE t❤❡♦ R✳ ❇➔✐ ✽✽ ✭❙P ✷✵✶✻✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❜❛ ✤✐➸♠ A, B, M ♣❤➙♥ ❜✐➺t✱ t❤➥♥❣ ❤➔♥❣ ✈➔ M ♥➡♠ ❣✐ú❛ A, B ✳ ❚r➯♥ ❝ò♥❣ ♠ët ♥û❛ ♠➦t ♣❤➥♥❣ ❜í ❧➔ ✤÷í♥❣ t❤➥♥❣ AB ✱ ❞ü♥❣ ❤❛✐ t❛♠ ❣✐→❝ ✤➲✉ AM C ✈➔ BM D✳ ●å✐ P ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ AD ✈➔ BC ✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ AM √ P C ✈➔ BM √ P D ❧➔ ❝→❝ tù ❣✐→❝ ♥ë✐ t✐➳♣✳ ❜✮ ❈❤ù♥❣ ♠✐♥❤ CP.CB + DP.DA = AB ✳ ❝✮ ✣÷í♥❣ t❤➥♥❣ ♥è✐ t➙♠ ❝õ❛ ❤❛✐ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ ❝→❝ tù ❣✐→❝ AM P C ✈➔ BM P D ❝➢t P A, P B t÷ì♥❣ ù♥❣ t↕✐ E, F ✳ ❈❤ù♥❣ ♠✐♥❤ tù ❣✐→❝ CDF E ❧➔ ❤➻♥❤ t❤❛♥❣✳ ❇➔✐ ✽✾ ✭❙P ✷✵✶✺✱ ✈á♥❣ ✶✮✳ ❈❤♦ t❛♠ ❣✐→❝ ABC ❝â ❝→❝ ❣â❝ ABC, ACB ♥❤å♥ ✈➔ BAC = 600✳ ❈→❝ ✤÷í♥❣ ♣❤➙♥ ❣✐→❝ tr♦♥❣ BB1, CC1 ❝õ❛ t❛♠ ❣✐→❝ ABC ❝➢t ♥❤❛✉ t↕✐ I ✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ tù ❣✐→❝ AB1IC1 ♥ë✐ t✐➳♣✳ ❜✮ ●å✐ K ❧➔ ❣✐❛♦ ✤✐➸♠ t❤ù ❤❛✐ ✭❦❤→❝ B ✮ ❝õ❛ ữớ t BC ợ ữớ trỏ t t BC1I ✳ ❈❤ù♥❣ ♠✐♥❤ tù ❣✐→❝ CKIB1 ♥ë✐ t✐➳♣✳ ❝✮ ❈❤ù♥❣ ♠✐♥❤ AK ⊥ B1C1✳ ❇➔✐ ✾✵ ✭❙P ✷✵✶✹✱ ✈á♥❣ ✶✮✳ ❈❤♦ tù ❣✐→❝ ABCD ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ (O) ✤÷í♥❣ ❦➼♥❤ AC = 2R✳ ●å✐ K ✈➔ M t❤❡♦ t❤ù tü ❧➔ ❝❤➙♥ ❝→❝ ✤÷í♥❣ ✈✉ỉ♥❣ ❣â❝ ❤↕ tø A ✈➔ C ①✉è♥❣ BD✱ E ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ AC ✈➔ BD✱ ❜✐➳t K t❤✉ë❝ ✤♦↕♥ BE ✭K ❦❤→❝ B ✈➔ E ✮✳ ✣÷í♥❣ t❤➥♥❣ q✉❛ K s♦♥❣ s♦♥❣ ✈ỵ✐ BC ❝➢t AC t↕✐ P ✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ tù ❣✐→❝ AKP D ♥ë✐ t✐➳♣✳ ❜✮ ❈❤ù♥❣ ♠✐♥❤ KP ⊥ P M ✳ ❝✮ ❇✐➳t ABD = 600 ✈➔ AK = x✳ ❚➼♥❤ BD t❤❡♦ R ✈➔ x✳ ❇➔✐ ✾✶ ✭❙P ✷✵✶✸✱ ✈á♥❣ ✶✮✳ ❈❤♦ t❛♠ ❣✐→❝ ABC ❦❤æ♥❣ ❝➙♥✱ ❝â ❜❛ ❣â❝ ♥❤å♥✱ ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ (O)✳ ❈→❝ ✤÷í♥❣ ❝❛♦ AA1, BB1, CC1 ❝õ❛ t❛♠ ❣✐→❝ ABC ❝➢t ♥❤❛✉ t↕✐ H ✱ ❝→❝ ✤÷í♥❣ t❤➥♥❣ A1C1 ✈➔ AC ❝➢t ♥❤❛✉ t↕✐ D✳ ●å✐ X ❧➔ ❣✐❛♦ ✤✐➸♠ tự ữớ t BD ợ ữớ trỏ (O) ❛✮ ❈❤ù♥❣ ♠✐♥❤ DX.DB = DC1.DA1✳ ❜✮ ●å✐ M ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ❝↕♥❤ AC ✳ ❈❤ù♥❣ ♠✐♥❤ DH ⊥ BM ✳ ❇➔✐ ✾✷ ✭❙P ✷✵✶✷✱ ✈á♥❣ ✶✮✳ ❈❤♦ t❛♠ ABC ữớ trỏ (O) t ú ợ ❝↕♥❤ AB, AC t÷ì♥❣ ù♥❣ t↕✐ K, L✳ ❚✐➳♣ t✉②➳♥ d ❝õ❛ ✤÷í♥❣ trá♥ (O) t↕✐ ✤✐➸♠ E t❤✉ë❝ ❝✉♥❣ ♥❤ä KL✱ ❝➢t ❝→❝ ✤÷í♥❣ t❤➥♥❣ AL, AK t÷ì♥❣ ù♥❣ t↕✐ M, N ✳ ✣÷í♥❣ t❤➥♥❣ KL ❝➢t OM t↕✐ P ✈➔ ❝➢t ON t↕✐ Q✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ M ON = 900 − 21 BAC ✳ ❜✮ ❈❤ù♥❣ ♠✐♥❤ ❝→❝ ✤÷í♥❣ t❤➥♥❣ M Q, N P ✈➔ OE ❝ị♥❣ ✤✐ q✉❛ ♠ët ✤✐➸♠✳ ✶✹ ❝✮ ❈❤ù♥❣ ♠✐♥❤ KQ.P L = EM.EN ✳ ❇➔✐ ✾✸ ✭❆♠s ✷✵✶✺✱ ❝❤✉②➯♥ ❚✐♥✮✳ ❈❤♦ ✤÷í♥❣ trá♥ (O)✱ ✤÷í♥❣ ❦➼♥❤ AB ✳ ●å✐ I ❧➔ ✤✐➸♠ ❜➜t ❦➻ tr➯♥ ✤♦↕♥ t❤➥♥❣ AO ✭I ❦❤→❝ A O ữớ t q I ổ õ ợ AB ❝➢t ✤÷í♥❣ trá♥ (O) t↕✐ ❝→❝ ✤✐➸♠ C ✈➔ D✳ ●å✐ E ❧➔ ✤✐➸♠ tr➯♥ ✤÷í♥❣ trá♥ (O) s❛♦ ❝❤♦ D ❧➔ ✤✐➸♠ ❝❤➼♥❤ ❣✐ú❛ ❝õ❛ ❝✉♥❣ AE ✳ K ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ AE ✈➔ CD✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ ✤÷í♥❣ t❤➥♥❣ OK ✤✐ q✉❛ tr✉♥❣ ✤✐➸♠ ❝õ❛ CE ✳ ❜✮ ✣÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ I ✈➔ s♦♥❣ s♦♥❣ ợ CE t AE, BE ữủt t P Q✳ ❈❤ù♥❣ ♠✐♥❤ tù ❣✐→❝ DP EQ ❧➔ ❤➻♥❤ ❝❤ú ♥❤➟t✳ ❝✮ ❚➻♠ ✈à tr➼ ❝õ❛ ✤✐➸♠ I tr➯♥ ✤♦↕♥ t❤➥♥❣ AO s❛♦ ❝❤♦ KC = KA + KO✳ ❇➔✐ ✾✹ ✭❑❍❚◆ ✷✵✶✺✱ ✈á♥❣ ✷✮✳ ❈❤♦ t❛♠ ❣✐→❝ ♥❤å♥ ABC ❦❤ỉ♥❣ ❝➙♥ ✈ỵ✐ AB < AC ✳ ●å✐ M ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ✤♦↕♥ t❤➥♥❣ BC ✳ H ❧➔ ❤➻♥❤ ❝❤✐➳✉ ✈✉æ♥❣ ❣â❝ ❝õ❛ B tr➯♥ ✤♦↕♥ t❤➥♥❣ AM ✳ ❚r➯♥ t✐❛ ✤è✐ ❝õ❛ t✐❛ AM ❧➜② ✤✐➸♠ N s❛♦ ❝❤♦ AN = 2M H ✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ BN = AC ✳ ❜✮ ●å✐ Q ❧➔ ✤✐➸♠ ✤è✐ ự ợ A q N ữớ t AC t BQ t↕✐ D✳ ❈❤ù♥❣ ♠✐♥❤ ❜è♥ ✤✐➸♠ B, D, N, C ❝ị♥❣ t❤✉ë❝ ♠ët ✤÷í♥❣ trá♥✱ ❣å✐ ✤÷í♥❣ trá♥ ♥➔② ❧➔ (O)✳ ❝✮ ✣÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ AQD ❝➢t (O) t↕✐ G ❦❤→❝ D✳ ❈❤ù♥❣ ♠✐♥❤ N G//BC ✳ ❇➔✐ ✾✺ ✭❆♠s ✷✵✶✷✱ ❝❤✉②➯♥ ❚♦→♥✮✳ ❈❤♦ ✤÷í♥❣ trá♥ (O; R) ✈➔ ❞➙② ❝✉♥❣ BC ❝è ✤à♥❤ (BC < 2R)✳ ✣✐➸♠ A ❞✐ ✤ë♥❣ tr➯♥ ✤÷í♥❣ trá♥ (O) s❛♦ ❝❤♦ t❛♠ ❣✐→❝ ABC ❧➔ t❛♠ ❣✐→❝ ♥❤å♥✳ ●å✐ AD ❧➔ ✤÷í♥❣ ❝❛♦ ✈➔ H ❧➔ trü❝ t➙♠ ❝õ❛ t❛♠ ❣✐→❝ ABC ✳ ❛✮ ✣÷í♥❣ t❤➥♥❣ ❝❤ù❛ ♣❤➙♥ ❣✐→❝ ♥❣♦➔✐ ❝õ❛ BHC ❝➢t AB, AC ❧➛♥ ❧÷đt t↕✐ M, N ✳ ❈❤ù♥❣ ♠✐♥❤ t❛♠ ❣✐→❝ AM N ❝➙♥✳ ❜✮ ●å✐ E, F ❧➛♥ ❧÷đt ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❝õ❛ D ❧➯♥ ❝→❝ ✤÷í♥❣ t❤➥♥❣ BH, CH ✳ ❈❤ù♥❣ ♠✐♥❤ OA ⊥ EF ✳ ❝✮ ✣÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ AM N ❝➢t ✤÷í♥❣ ♣❤➙♥ ❣✐→❝ tr♦♥❣ ❝õ❛ BAC t↕✐ K ✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷í♥❣ t❤➥♥❣ HK ❧✉ỉ♥ ✤✐ q✉❛ ♠ët ✤✐➸♠ ❝è ✤à♥❤✳ ❇➔✐ ✾✻ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✾✱ ✈á♥❣ ✷✮✳ ❈❤♦ t❛♠ ❣✐→❝ ABC ✈ỵ✐ AB < AC ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥ (O)✳ ❚✐❛ ♣❤➙♥ ❣✐→❝ ❝õ❛ BAC ❝➢t (O) t↕✐ D ❦❤→❝ A✳ M ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ AD✳ ●✐↔ sû ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABM ❝➢t ✤♦↕♥ t❤➥♥❣ AC t↕✐ N ❦❤→❝ A✳ ❛✮ ❈❤ù♥❣ t BM D BN C ỗ ❞↕♥❣✳ ❜✮ ●å✐ ❣✐❛♦ ✤✐➸♠ t❤ù ❤❛✐ ❝õ❛ CM ✈➔ (O) ❧➔ Q✳ ❍❛✐ t✐➳♣ t✉②➳♥ t↕✐ B ✈➔ M ❝õ❛ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ABM ❝➢t ♥❤❛✉ t↕✐ P ✳ ❈❤ù♥❣ ♠✐♥❤ ❜❛ ✤✐➸♠ Q, P, D t❤➥♥❣ ❤➔♥❣✳ ❝✮ ✣÷í♥❣ trá♥ (P ) ✤✐ q✉❛ B ✈➔ M ❝➢t (O) t↕✐ R✳ ❈❤ù♥❣ ♠✐♥❤ RM ✈➔ ữớ t q N ổ õ ợ AC t ♥❤❛✉ tr➯♥ ✤÷í♥❣ trá♥ (O)✳ ✶✻ ... ❝➢t ❝→❝ ✤÷í♥❣ t❤➥♥❣ AL, AK t÷ì♥❣ ù♥❣ t↕✐ M, N ✳ ✣÷í♥❣ t❤➥♥❣ KL ❝➢t OM t↕✐ P ✈➔ ❝➢t ON t↕✐ Q✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ M ON = 900 − 21 BAC ✳ ❜✮ ❈❤ù♥❣ ♠✐♥❤ ❝→❝ ✤÷í♥❣ t❤➥♥❣ M Q, N P ✈➔ OE ❝ị♥❣ ✤✐ q✉❛ ♠ët ✤✐➸♠✳