2021 ᠣᠨ ᠤ ᠪᠣᠭᠤᠲᠤ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

202 1 ᠣᠨ ᠤ ᠪᠣᠭᠣᠳᠤ᠂ ᠰᠢᠯ ᠢᠩᠬᠣ ᠲᠠ᠂ ᠠᠯ ᠠᠱᠨ ᠂ ᠦᠬᠠᠢ᠂ ᠪ ᠠᠶᠠᠨᠨ ᠠᠭᠤᠷ᠂ ᠤᠯᠠᠭᠠᠨᠴᠠᠪ ᠬ ᠣᠲᠠ ᠶᠢᠨ ᠠᠩᠬ ᠠ ᠲᠣᠮᠳ ᠠᠳᠤ ᠢᠢᠨ ᠰᠣ ᠷᠭᠠᠭ ᠣᠯ ᠢ ᠲᠠᠪᠰᠢᠬ ᠤ ᠰᠢᠯ ᠭᠠᠯᠳᠠ ᠲᠣᠭᠠᠨ ᠤ ᠣᠬ ᠠᠭᠠᠨ ᠨᠢᠭᠡ᠂ ᠰᠣᠩ ᠭᠣᠬ ᠤ ᠰᠡᠲ ᠥᠪ᠃ ( ᠰᠡᠲᠥᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠡᠢ ᠲᠡ 36 ᠬᠤᠪ ᠢ) 1. ᠵᠠᠮ ᠲᠡᠭᠡᠭᠡᠪᠦᠷᠢ ᠶᠢᠨ ᠶᠠᠮᠤᠨ ᠤ ᠮᠡᠳᠡᠭᠡ ᠯᠡᠭᠰᠡᠨ ᠢᠶᠡᠷ᠂ 20 20 ᠣᠨ ᠤ ᠡᠴᠦ ᠰ ᠪᠣᠯᠬᠤ ᠳ ᠤ ᠪᠦᠬᠦ ᠤᠯᠤ ᠰ ᠤᠨ ᠬᠤᠲ ᠠ ᠶᠢᠨ ᠰᠢᠨ ᠡ ᠡᠨᠧᠷᠭ ᠢ ᠶᠢᠨ ᠨᠡᠢᠲᠡ ᠶᠢᠨ ᠮᠠᠰᠢ ᠨ ᠲᠡᠷᠭᠡᠨ ᠦ ᠪᠦᠬᠦ ᠲᠣᠭ ᠠ 4 6.6 1 ᠲᠦᠮᠡ ᠬᠦᠷᠴᠦ᠂ ᠲᠡᠯᠡᠬᠡᠢ ᠳᠦ ᠲᠡᠷᠢᠭᠦᠯᠡᠭᠰᠡᠨ ᠪᠠᠢᠨ ᠠ᠃ 46 .61 ᠲᠦᠮᠡ ᠶᠢ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬᠠ ᠭᠠᠨ ᠤ ᠲᠣᠭ ᠠ ᠲᠡᠮᠲᠡᠭᠯᠡᠬᠦ ᠠᠷ ᠭ ᠠ ᠪᠠᠷ 4.661 × 10 𝑛 ᠭᠡᠵᠦ ᠲᠡᠮᠲᠡᠭᠯᠡ ᠪᠡᠯ᠂ 𝑛 ᠲᠡᠩ ᠴᠡᠭᠦᠦ ( ) A. 6 B. 5 C. 4 D. 3 2. ᠳᠣᠣ ᠷᠠᠬᠢ ᠪᠤᠳᠣᠯᠭ ᠠ ᠤᠳ ᠢ ᠬᠦᠢᠴᠡ ᠳᠬᠡᠭᠰᠡᠨ ᠳᠦᠩ ᠡᠴᠡ ᠬᠠᠷᠢᠴ ᠠᠯᠠᠰᠢᠭᠦᠢ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠢᠶᠡᠷ ᠬᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠬᠡ ᠪ ᠣᠯᠬᠤ ᠨᠢ ( ) A. 1+ (− 4) B. (− 1) 4 C. (− 5) −1 D. √4 3. ᠦᠭᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠂ ᠬᠡᠷᠴᠢᠮ AB = 4᠂ ᠰᠢᠯᠤ ᠭᠤᠨ AB ᠳᠡᠭᠡᠷ ᠡ ᠬᠡᠷᠴᠢᠮ BC ᠶᠢ ᠪᠠᠢᠭᠤᠯᠬᠤ ᠳ ᠤ ᠪᠠᠨ BC = 2 ᠪᠤᠯ ᠭᠠᠨ ᠠ᠃ ᠬᠡᠷᠪ ᠡ D ᠨᠢ ᠬᠡᠷᠴᠢᠮ AC ᠦᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠪᠣᠯ᠂ ᠬᠡᠷᠴᠢᠮ AD ᠦᠨ ᠤᠷᠲ ᠤ ᠨᠢ ( ) A. 1 B. 3 C. 1 ᠪᠤᠶ ᠤ 3 D. 2 ᠪᠤᠶᠤ 3 4. ᠬᠤᠷᠭᠤ ᠨ ᠳ ᠤ ᠠᠳᠠᠯᠢ ᠪᠤᠰᠤ ᠬᠤᠶᠠᠷ ᠵᠦᠢᠯ ᠦᠨ ᠱᠠᠬᠠᠢ ᠪᠤᠢ᠂ ᠬᠡᠷᠪᠡ ᠳᠤᠲ ᠤᠷ ᠠ ᠠᠴᠠ ᠨᠢ ᠲᠠᠰᠢᠷᠠᠮ ᠢᠶᠡᠷ ᠳᠤᠷ ᠠ ᠶᠢᠨ 2 ᠱᠠᠬᠠ ᠢ ᠶᠢ ᠠᠪᠴᠤ ᠭᠠᠷᠭᠠᠬ ᠤ ᠭᠡᠪᠡᠯ᠂ ᠶᠠᠭ ᠰᠠᠢᠬᠠᠨ ᠠᠳᠠᠯᠢᠬᠠᠨ ᠨᠢᠭᠡ ᠵᠦᠢᠷ ᠡᠢ ᠱᠠᠬᠠ ᠢ ᠪᠣᠯᠬᠤ ᠶᠢᠨ ᠲᠣᠬ ᠢᠶᠠᠯᠳᠤᠮᠵᠢ ᠨᠢ ( ) A. 13 B. 14 C. 15 D. 16 5. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ Rt ∆ABC ᠳᠦ᠂ ∠A CB = 90 °,AB = √5 ,𝐵𝐶 = 2᠂ ᠴᠡᠭ A ᠢᠶᠡᠷ ᠲᠣᠭᠣ ᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠣᠯᠭᠠᠵᠤ᠂ AC ᠦᠨ ᠤᠷᠲᠤ ᠪ ᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭᠠᠵᠤ ᠨᠤᠮᠤ ᠵᠢᠷᠤ ᠪᠠᠯ᠂ AB ᠲᠠᠢ ᠴᠡᠭ D ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠵᠤ᠂ AC ᠲᠠᠢ ᠴᠡᠭ C ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ B ᠢ ᠶᠡᠷ ᠲᠣᠭᠣ ᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠣᠯᠭ ᠠᠵᠤ᠂ AC ᠦᠨ ᠤᠷᠲᠤ ᠪᠠᠷ ᠷᠠᠳᠢᠤ ᠰ ᠪᠣᠯᠭᠠᠵᠤ ᠨᠤᠮᠤ ᠵᠢᠷᠤ ᠪᠠᠯ᠂ AB ᠲᠠᠢ ᠴᠡᠭ E ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠ ᠵᠤ᠂ BC ᠲᠠᠢ ᠴᠡᠭ F ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠲᠡᠭᠡᠪᠡᠯ ᠵᠢᠷᠤ ᠭ ᠲᠠᠬᠢ ᠰᠡᠭ ᠦᠳᠡᠷ ᠲᠠᠢ ᠬᠡᠰᠡᠭ ᠦᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ( ) A. 8− π B. 4− π C. 2− π4 D. 1− π4 6. ᠬᠡᠷᠪᠡ 𝑥 = √2 + 1 ᠪᠣᠯ᠂ ᠠᠯᠭ ᠧᠪᠷᠠ ᠶᠢᠨ ᠢᠯᠡᠷᠬᠡᠢ ᠯᠡᠯ𝑥 2− 2𝑥 + 2 ᠦ ᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ) A. 7 B. 4 C. 3 D. 3 − 2√ 2 7. ᠨᠢᠭᠡᠨ ᠰᠢᠨ ᠡ ᠪᠣᠳᠤᠯ ᠭ ᠠ ᠶᠢᠨ ᠳᠦᠷᠢᠮ ︽※ ︾ ᠢ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠮᠡᠲᠦ ᠲᠤᠭ ᠲᠠᠭᠠᠵᠠᠢ᠄ 𝑎※ 𝑏= 𝑎− 2𝑏 ᠂ ᠬᠡᠷᠪᠡ 𝑥 ᠶᠢᠨ ᠲᠤᠬ ᠠᠢ ᠲᠡᠩᠴᠡᠳᠬᠡᠯ ᠪᠤᠰᠤ 𝑥※ 𝑚 3 ᠶᠢᠨ ᠬᠠᠷᠢᠭ ᠤ ᠶᠢᠨ ᠴᠤᠭ ᠯᠠᠷᠠᠯ ᠨᠢ 𝑥 −1 ᠪᠣᠯ 𝑚 ᠤ ᠨ ᠬᠡᠮ ᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ) A. −1 B. −2 C. 1 D. 2 8. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠰᠢᠯᠤᠭᠤ ᠨ 𝑙 1 / /𝑙 2 ᠂ ᠰᠢᠯᠤᠭᠤ ᠨ 𝑙 3 ᠨᠢ 𝑙1 ᠲᠠᠢ ᠴᠡᠭ A ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠵᠤ᠂ 𝑙2 ᠲᠠᠢ ᠴᠡᠭ B ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ B ᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠰᠠᠨ ᠰᠢᠯᠤᠭᠤ ᠨ 𝑙 4 ᠨᠢ 𝑙1 ᠲᠠᠢ ᠴᠡᠭ C ᠳ ᠦ ᠤᠭᠲᠤᠯᠤ ᠯᠴᠠᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ∠ 3= 50° ᠂ ∠1 + ∠2 + ∠3 = 240° ᠪᠤᠯ ᠂ ∠4 = ( ) A. 80° B. 70° C. 60° D. 50° 9. ᠳᠠᠷᠠᠭᠠᠬᠢ ᠦᠭᠦᠯ ᠡᠯ ᠡᠴᠡ ᠵᠥᠪ ᠪᠣᠯᠬᠤ ᠨᠢ ( ) A. ᠹᠥᠩᠺ ᠼ 𝑦= − 12𝑥 ᠳᠦ᠂ 𝑥 0 ᠦᠶᠡᠰ᠂ 𝑦 ᠨᠢ 𝑥 ᠶᠢᠨ ᠨᠡᠮᠡᠭᠳ ᠡᠬᠦ ᠶᠢ ᠳᠠᠭᠠᠯᠳᠤ ᠨ ᠬᠤᠷᠣ ᠭᠳᠠᠨ ᠠ B. ᠬᠡᠷᠪᠡ 𝑎 1− 𝑎 ᠪ ᠣᠯᠤᠨ ᠠ C. ᠷᠠᠳᠢᠤᠰ ᠲᠤ ᠡᠭᠴᠡ ᠪᠣᠯᠬᠤ ᠰᠢᠯᠤ ᠭᠤᠨ ᠪᠣᠯ ᠲᠣᠭᠣᠷ ᠢᠭ ᠤᠨ ᠰᠢᠬᠠᠭᠴᠢ ᠰᠢᠯᠤ ᠭᠤᠨ ᠪᠣᠯᠤ ᠨ ᠠ D. ᠡᠯ ᠡ ᠲ ᠠᠯ ᠠ ᠨᠢ ᠬᠠᠷᠢ ᠯᠴᠠᠨ ᠲᠡᠩᠴᠡᠭᠦᠦ ᠪᠣᠯᠬᠤ ᠲᠣᠭᠣ ᠷᠢᠭ ᠲᠤ ᠪᠠᠭᠲ ᠠᠭᠰᠠᠨ ᠳᠦᠷᠪ ᠡᠯᠵᠢᠨ ᠨᠢ ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪᠡᠯᠵᠢᠨ ᠪᠣᠯᠤ ᠨ ᠠ 10. ᠦᠭᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠂ ᠬᠤᠶᠠᠳᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠥᠩᠺᠼ 𝑦 = 𝑎𝑥 2− 𝑏𝑥 + 𝑐( 𝑎≠ 0) ᠦᠨ ᠵᠢᠷᠤ ᠮᠠᠭ ᠨᠢ ᠨᠢᠭᠡᠳᠦᠭ ᠡᠷ ᠮᠦᠴᠡ ᠳ ᠦ ᠬᠢ ᠴᠡᠭ ( 1, −𝑏 ) ᠶᠢ ᠳᠠᠭᠠᠷᠢᠨ ᠠ᠂ ᠲᠡᠭᠡᠪᠡᠯ ᠨᠢᠭᠡᠳᠦᠭ ᠡᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠥᠩᠺ ᠼ 𝑦 = 𝑏𝑥 − 𝑎𝑐 ᠦᠨ ᠵᠢᠷᠤ ᠮᠠᠭ ᠤ ᠨ ᠳᠠᠭ ᠠᠷᠢᠬᠤ ᠦᠭᠡᠢ ᠮᠦᠴᠡ ᠨᠢ ( ) A. ᠨᠢᠭᠡᠳᠦᠭ ᠡᠷ ᠮᠦ ᠴᠡ B. ᠬᠤᠶᠠᠳᠤ ᠭᠠᠷ ᠮᠦᠴᠡ C. ᠭᠤᠷᠪᠠᠳᠤ ᠭᠠᠷ ᠮᠦᠴᠡ D. ᠳᠦᠷᠪᠡᠳᠦᠭ ᠡᠷ ᠮᠦᠴᠡ 11. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ∆ABC ᠳᠦ᠂ AB = AC ,∆ DBC ᠪᠠ ∆ ABC ᠨᠢ ᠰᠢᠯᠤᠭᠤ ᠨ BC ᠶᠢᠨ ᠬᠤᠪᠢ ᠳᠤ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠲᠡᠢ ᠂ AD ᠶᠢ ᠬᠤᠯᠪᠤᠪᠠᠯ BC ᠲᠠᠢ ᠴᠡᠭ O ᠲᠦ ᠤᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ C ᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠵ ᠤ CE ⊥ CD ᠪᠠᠷ ᠪᠠᠢᠭᠤᠯᠪᠠᠯ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ C ᠂ AD ᠲᠠᠢ ᠴᠡᠭ E ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ AD = 8, BC = 6 ᠪᠣᠯ 2𝑂𝐸 +𝐴𝐸𝐵𝐷 ᠤᠨ ᠬᠡᠮ ᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ) A. 43 B. 34 C. 53 D. 54 12. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦᠭᠲᠦ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ᠋ᠧᠮ ᠳᠦ᠂ ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦᠭᠲᠦ O ABC ᠶᠢᠨ ᠲᠠᠯ ᠠ OA ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠦᠨ ᠡᠶᠡᠷᠭᠦ ᠬᠠᠭᠠᠰ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠤᠷᠤᠰᠢᠵ ᠤ᠂ ᠲᠠᠯ ᠠ OC ᠨᠢ 𝑦 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠡᠶᠡᠷᠭᠦ ᠬᠠᠭᠠᠰ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠳᠡᠭᠡ ᠷ ᠡ ᠤᠷᠤᠰᠢᠨ ᠠ᠂ ᠴᠡᠭ B ᠦ ᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ (4 ,2) ᠂ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢ ᠴᠠᠯᠲᠦ ᠹᠥᠩᠺ ᠼ 𝑦 = 2𝑥 ( 𝑥 0) ᠦᠨ ᠵᠢᠷᠤ ᠮᠠᠭ ᠨᠢ BC ᠲᠠᠢ ᠴᠡᠭ D ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠵᠤ᠂ ᠳᠠᠰᠢᠭᠤ ᠪᠴᠢOB ᠲᠠᠢ ᠴᠡᠭ E ᠲᠦ ᠤᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ᠂ AB ᠲᠠᠢ ᠴᠡᠭ F ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠃ OD ,DE ,EF ,DF ᠢ ᠬᠤᠯᠪᠤᠨ ᠠ᠃ ᠳᠣᠣ ᠷᠠᠬᠢ ᠳᠦᠩᠨᠡᠯᠲᠡ᠄ ① sin ∠𝐷𝑂𝐶 = cos ∠𝐵𝑂𝐶 ; ② OE = BE ; ③ 𝑆∆ 𝐷𝑂𝐸 = 𝑆∆ 𝐵𝐸𝐹 ; ④ OD :DF = 2: 3 ; ᠡᠭᠦᠨ ᠡᠴᠡ ᠵᠥᠪ ᠪᠣᠯᠬᠤ ᠪᠣᠳᠠᠭᠠ ᠯᠢ ᠲᠣᠭ ᠠ ᠨᠢ ( ) A. 4 B. 3 C. 2 D. 1 ᠬᠤᠶᠠᠷ᠂ ᠬᠠᠭᠠᠯᠲ ᠠ ᠲ ᠠᠭᠯᠠᠬ ᠤ ᠰᠡᠲᠥᠪ᠃ ( ᠰᠡᠲᠥ ᠪ ᠪᠦᠷᠢ 3 ᠬᠤ ᠪᠢ᠂ ᠨ ᠡᠢᠲ ᠡ 24 ᠬᠤᠪᠢ) 13. ᠦᠷᠡᠵᠢ ᠭᠳᠡᠭᠦᠨ ᠪᠣᠯ ᠭᠠᠨ ᠵᠠᠳᠠᠯ᠄ 𝑎 𝑥 24 + 𝑎𝑥 + 𝑎= ( ) 14. ᠬᠢᠯᠪᠠᠷᠴᠢᠯᠠ ᠄( 2𝑚𝑚 2− 4 + 12−𝑚 )÷ 1𝑚+2 = ( ) 15. ᠨᠢᠭᠡᠨ ᠡᠶᠡᠷᠭᠦ ᠲᠣᠭ ᠠ 𝑎 ᠶᠢᠨ ᠬᠤᠶᠠᠷ ᠺᠥᠸᠠᠳ᠋ ᠢᠵᠠᠭᠤᠷ ᠨᠢ 2 𝑏− 1 ᠪᠠ 𝑏 + 4 ᠪᠣᠯ᠂ 𝑎 + 𝑏 ᠶᠢᠨ ᠺᠥᠪ ᠢ ᠵᠠᠭᠤᠷ ᠨᠢ ( )᠃ 16. ᠨᠢᠭᠡᠨ ᠬᠦᠮᠦᠨ 5 ᠤᠳᠠᠭᠠᠨ ᠤ ᠫᠤᠤ ᠳᠠᠯᠲᠠ ᠪᠠᠷ ᠳᠠᠮᠵᠢᠵ ᠦ ᠲᠣᠰ ᠲᠣᠰ 5, 10 ,7, 𝑥,10 ᠬᠤᠪᠢ ᠤᠨᠤᠭᠠᠵ ᠠᠢ᠃ ᠬᠡᠷᠪᠡ ᠡᠨᠡ ᠨᠢᠭᠡ ᠪᠥᠯᠥᠭ ᠲᠣᠭᠠᠨ ᠤ ᠳᠤᠮᠳᠠ ᠪᠠᠢᠷᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠨᠢ 8 ᠪᠣᠯ᠂ ᠡᠨᠡ ᠨᠢᠭᠡ ᠪᠥᠯᠥᠭ ᠲᠣ ᠭᠠᠨ ᠤ ᠰ ᠠᠷᠨᠢᠴᠡ ᠨᠢ ( ) ᠪᠣᠯᠤ ᠨ ᠠ᠃ 17. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ Rt ∆ABC ᠳᠦ᠂ ∠A CB = 90 °᠂ ᠴᠡᠭ B ᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ BD ⊥ CB ᠪᠠᠷ ᠪᠠᠢ ᠭᠤᠯᠪᠠᠯ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ B ᠪᠦᠭᠡᠳ BD = 3᠂ CD ᠶᠢ ᠬᠤᠯᠪᠤᠪᠠᠯ AB ᠲᠠᠢ ᠴᠡᠭ M ᠲᠦ ᠤᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ M ᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠤ ᠨ MN ⊥ CB ᠪᠠᠷ ᠪᠠᠢᠭᠤᠯᠪᠠᠯ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤᠷ ᠢ ᠨᠢ N ᠂ ᠬᠡᠷᠪᠡ AC = 2 ᠪᠣᠯ᠂ MN ᠤ ᠤᠷᠲᠤ ᠨᠢ ( ) ᠃ 18. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠵᠢ ᠰᠢᠭᠦᠯᠵᠢᠨ ABC D ᠳ ᠦ᠂ AD = 12 ᠂ AD ᠢᠶᠡᠷ ᠳᠢᠶᠠᠮᠧᠲ᠋ᠷ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ⊙ O ᠪᠠ BC ᠨᠢ ᠴᠡᠭ E ᠲᠦ ᠰᠢᠬᠠᠯᠴᠠᠨ ᠠ᠂ OC ᠶᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ OC = AB ᠪᠣᠯ ᠵᠢᠰᠢᠬᠦᠯᠵᠢᠨ ABC D ᠤ ᠨ ᠬᠦᠷᠢᠶᠠᠨ ᠤ ᠤᠷ ᠲᠤ ᠨᠢ ( ) ᠃ 19. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ BD ᠪᠣᠯ ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪᠡᠯᠵᠢ ᠨ ABC D ᠶᠢᠨ ᠨᠢᠭᠡ ᠳᠠᠰᠢᠭ ᠤᠪᠴᠢ᠂ E ᠪᠣᠯ BD ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ᠂ F ᠪᠣᠯ CB ᠤᠨ ᠦᠷᠭᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ᠂ CE , EF , AF ᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠃ ᠬᠡᠷᠪ ᠡ DE = DC , EF = EC ᠪᠣᠯ ∠ BAF ᠤᠨ ᠭᠷᠠᠳᠦᠰ ᠤᠨ ᠲᠣ ᠭ ᠠ ᠨᠢ ( ) ᠃ 20. ᠦᠭᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠬᠠᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭ ᠤᠮ 𝑦= 𝑥 2 − 2𝑥 − 3 ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ A , B ᠬᠤᠶᠠᠷ ᠴᠡᠭ ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠵᠤ ( ᠴᠡᠭ A ᠨᠢ ᠴᠡᠭ B ᠦᠨ ᠵᠡᠭᠦᠨ ᠲᠠᠯ ᠠ ᠳ ᠤ ᠣᠷᠣ ᠰᠢᠨ ᠠ)᠂ 𝑦 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ C ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ D( 4, 𝑦) ᠨᠢ ᠬᠠᠶᠠᠯᠲᠠ ᠱᠤᠭᠤᠮ ᠳᠡᠭᠡᠷ ᠡ ᠤᠷ ᠤᠰᠢᠨ ᠠ᠂E ᠪᠣᠯ ᠲᠣᠰ ᠬᠠᠶᠠ ᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠤᠨ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦ ᠨ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠬᠦᠳᠡᠯᠬᠦ ᠴᠡᠭ᠂ BE + DE ᠦᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠦᠶᠡᠰ ∆A CE ᠦᠨ ᠲᠡᠯᠡᠪᠡᠢ ᠨᠢ ( ) ᠃ ᠭᠤᠷᠪᠠ᠂ ᠪᠤᠳᠤᠵ ᠤ ᠬᠠᠷᠢᠭᠤᠯᠬᠤ ᠰᠡᠲ ᠥᠪ᠃ ( ᠨᠡ ᠢᠲᠡ6 0 ᠬᠤ ᠪᠢ) 21. (8 ᠬᠤᠪᠢ) ᠳᠤᠮᠳᠠᠳᠤ ᠤᠯᠤ ᠰ ᠤᠨ ᠡᠪ ᠬᠠᠮᠲᠦ ᠨᠠᠮ ᠪᠠᠢᠭᠤᠯᠤ ᠭᠳᠠᠭᠰᠠᠨ 10 0 ᠵᠢᠯ ᠦ ᠨ ᠣᠢ ᠶᠢ ᠪᠡᠯᠭ ᠡᠯᠡᠬᠦ ᠶᠢᠨ ᠲᠥᠯᠥ ᠭᠡ᠂ ᠨᠢᠭᠡᠨ ᠰᠤᠷᠭᠠ ᠭᠤᠯᠢ ᠶᠢᠨ ᠬᠢᠨ ᠨᠠᠮ ᠤ ᠨ ᠲᠡᠦᠬᠡ ᠶᠢ ᠰᠤᠷᠤ ᠯᠴᠠᠭᠰᠠᠨ ᠮᠡᠳᠡᠯᠭᠡ ᠶᠢᠨ ᠤᠷᠤᠯ ᠳᠤᠭᠠᠨ ᠶᠠᠪ ᠤᠭᠳᠠ ᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠪᠠᠢᠨ ᠠ᠃ ᠮᠡᠳᠡᠯᠭᠡ ᠶᠢᠨ ᠤᠷᠤᠯ ᠳᠤᠭᠠᠨ ᠳ ᠤ ᠤᠷ ᠤᠯᠴᠠᠬᠤ ᠰᠤᠷᠤ ᠭᠴᠢᠳ ᠢ ᠠ ᠂ ᠡ ᠬᠤᠶᠠᠷ ᠳᠤᠭᠤᠢᠯᠠᠩ ᠪᠣᠯᠭᠤᠨ ᠬᠤᠪᠢᠶᠠᠵᠤ᠂ ᠳᠤᠭᠤᠢᠯᠠᠩ ᠪᠦᠷᠢ ᠳᠦ ᠴᠦᠮ 20 ᠰᠤᠷᠤ ᠭᠴᠢ ᠲᠣᠬ ᠢᠷᠠᠭ ᠤᠯᠵᠠᠢ᠃ ᠤᠷᠤᠯ ᠳᠤᠭ ᠠᠨ ᠲ ᠡᠭᠦᠰ ᠦᠭᠰᠡᠨ ᠦ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠬᠢᠷᠢ ᠪᠦᠲᠥ ᠮᠵᠢ ᠶᠢ ᠨᠢ ᠦᠨᠳᠦᠰᠦᠯᠡᠵᠦ ᠂ ᠵᠢᠷᠤ ᠭ ᠲᠤ ᠦᠵᠡᠭᠦᠯᠦᠭᠰᠡᠨ ᠮᠡᠲᠦ ᠪᠦᠷᠢᠨ ᠪᠤᠰᠤ ᠪᠦᠷᠢᠳᠬ ᠡᠯ ᠦᠨ ᠵᠢᠷᠤ ᠭ ᠬᠦᠰᠦ ᠨᠦᠭᠲᠦ ᠶᠢ ᠣᠯᠤ ᠭᠰᠠᠨ ᠪᠠᠢᠨ ᠠ᠃ ᠦᠭᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ᠂ ᠤᠷᠤᠯ ᠳᠤᠭᠠᠨ ᠤ ᠬᠢᠷᠢ ᠪᠦᠲᠦ ᠮᠵᠢ ᠶᠢᠨ ᠪᠦᠷᠢᠨ ᠬᠤᠪ ᠢ ᠨᠢ 10 0 ᠬᠤᠪᠢ᠂ ᠪᠦᠷᠢᠳᠬ ᠡᠯ ᠦᠨ ᠬᠦᠰᠦᠨᠦᠭᠲᠦ ᠳ ᠡᠬᠢ 𝑎, 𝑏 ᠨᠢ 𝑏 = 2𝑎 ᠶᠢ ᠬᠠᠩᠭᠠ ᠪᠠᠯ᠂ ᠦᠭᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠴᠢᠮᠡᠭᠡ ᠵᠠᠩᠭᠢ ᠶᠢ ᠦᠨᠳᠦᠰᠦᠯᠡᠵᠦ ᠳᠣᠣ ᠷᠠᠬᠢ ᠠ ᠰᠠᠭᠤᠳ ᠠᠯ ᠢ ᠪᠤ ᠳᠤᠵᠤ ᠬᠠᠷᠢᠭ ᠤᠯ᠄ ᠠ ᠳᠤᠭᠤ ᠢᠯᠠᠩ ᠤᠨ 20 ᠰᠤᠷ ᠤᠭᠴᠢ ᠶ ᠢᠨ ᠣᠷᠣ ᠯᠳᠤᠭᠠᠨ ᠤ ᠬᠢ ᠷᠢ ᠪ ᠦᠲᠦ ᠮᠵᠢ ᠶᠢᠨ ᠬᠦᠰᠦᠨᠦ ᠭᠲᠦ ᠡ ᠳᠤᠭᠤ ᠢᠯᠠᠩ ᠤᠨ 20 ᠰᠤᠷ ᠤᠭᠴᠢ ᠶᠢ ᠨ ᠣᠷᠣ ᠯᠳᠤᠭᠠᠨ ᠤ ᠬᠢ ᠷᠢ ᠪ ᠦᠲᠦ ᠮᠵᠢ ᠶᠢᠨ ᠪᠥᠷᠢ ᠳᠬᠡᠯ ᠦᠨ ᠵᠢᠷᠤ ᠭ (1) ᠪᠦᠷᠢᠳᠬ ᠡᠯ ᠦᠨ ᠬᠦ ᠰᠦᠨᠦᠭᠲᠦ ᠳᠡᠬᠢ 𝑎 , 𝑏 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭᠳ ᠡᠯ ᠢ ᠣᠯ᠃ (2) ᠰᠢᠶᠤᠤ ᠮᠢᠩ ᠠ ᠳᠤᠭᠤᠢᠯᠠᠩ ᠤᠨ 20 ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠤᠷᠤᠯ ᠳᠤᠭᠠ ᠨ ᠤ ᠬᠢᠷᠢ ᠪᠦᠲᠥ ᠮᠵᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ᠬᠤᠪᠢ ᠶᠢ ᠳᠣᠣ ᠷᠠᠬᠢ ᠠ ᠷᠭ ᠠ ᠪᠠᠷ ᠪᠤ ᠳᠤᠵᠠᠢ᠄ (70 + 80 + 90 + 100 )÷ 4= 85 (ᠬᠤᠪᠢ)᠃ ᠰᠣᠷᠴᠤ ᠦᠩ ᠭᠡᠷᠡᠭᠰᠡᠨ ᠪᠦᠷᠢᠳᠬ ᠡᠯ ᠦᠨ ᠮᠡᠳᠡᠯᠭᠡ ᠪᠡᠨ ᠦᠨᠳᠦᠰᠦ ᠯᠡᠵᠦ ᠰᠢᠶᠤᠤ ᠮᠢᠩ ᠦᠨ ᠭᠦᠢᠴᠡᠳᠬᠡᠭᠰ ᠡᠨ ᠪᠤᠳ ᠣᠯᠭ ᠠ ᠶᠢᠨ ᠵᠥᠪ ᠡᠰᠡᠬᠦ ᠶᠢ ᠮᠠᠭᠠᠳᠯᠠᠨ᠂ ᠬᠡᠷᠪᠡ ᠪᠤᠷ ᠤᠭᠤ ᠪᠣᠯ ᠵᠥᠪ ᠪᠣᠳᠤ ᠯᠭ ᠠ ᠶᠢᠨ ᠢᠯᠡᠷ ᠬᠡᠢᠯᠡᠯ ᠪᠢ ᠴᠢᠵᠦ ᠳᠦ ᠩ ᠢ ᠨᠢ ᠪᠤᠳ ᠤᠵᠤ ᠭᠠᠷᠭ ᠠ᠃ (3) ᠬᠡᠷᠪᠡ ᠳᠤᠮᠳᠠᠴᠢ ᠬᠢᠷᠢ ᠪᠦᠲᠦ ᠮᠵᠢ ᠪ ᠡᠷ ᠤᠷ ᠤᠯᠳᠤᠭᠠᠨ ᠤ ᠳᠦᠩ ᠦ ᠨ ᠪᠠᠷᠢ ᠮᠲᠠ ᠪᠣᠯᠭᠠᠪ ᠠᠯ᠂ ᠠᠯᠢ ᠳᠤᠭᠤᠢᠯᠠᠩ ᠤ ᠨ ᠤᠷᠤ ᠯᠳᠤᠭᠠᠨ ᠤ ᠬᠢ ᠷᠢ ᠪ ᠦᠲᠦ ᠮᠵᠢ ᠨᠢ ᠨᠡᠩ ᠰᠠ ᠢᠨ ᠪᠤ ᠢ︖ ᠰᠢᠯᠲᠠᠭᠠᠨ ᠢᠶᠠᠨ ᠲᠣ ᠳᠤᠷᠬᠠᠢᠯᠠ᠃ 22. (8 ᠬᠤᠪᠢ) ᠨᠢᠭᠡᠨ ᠪᠠᠷᠢ ᠯᠭ ᠠ ᠶᠢᠨ ᠳᠤᠭᠤ ᠢᠯᠠᠩ ᠨᠢA ᠡᠴᠡ B ᠲᠡᠰᠢ ᠨᠦᠬᠡᠨ ᠵᠠᠮ ᠵᠠᠰᠠ ᠵᠤ ᠪᠠᠢᠭᠤᠯᠬᠤ ᠪ ᠡᠷ ᠪᠣᠢ᠂ ᠬᠡᠮᠵᠢ ᠭᠴᠢ ᠰᠢᠯᠤᠭᠤ ᠨAB ᠶᠢᠨ ᠢᠵᠢᠯᠢ ᠨᠢᠭᠡᠨ ᠲᠠᠯ ᠠ ᠳ ᠤC , D ᠬᠤᠶᠠᠷ ᠠᠵᠢ ᠭᠯᠠᠯᠲ ᠠ ᠶᠢᠨ ᠴᠡᠭ ᠰᠤᠩ ᠭᠤᠵᠤ ᠲᠣᠭ ᠲᠠᠭᠠᠳ᠂ ᠵᠢᠷᠤ ᠭ9 ᠲᠤ ᠦᠵᠠᠭᠦ ᠯᠦᠭᠰᠡᠨ ᠮᠡᠲᠦ᠂ ᠬᠡᠮᠵᠢᠯᠲ ᠡ ᠪᠡᠷ ᠳᠠᠮᠵᠢᠨ AC ᠶᠢᠨ ᠤᠷᠲᠤ ᠨᠢ 3√ 22 𝑘𝑚 ,𝐶𝐷 ᠶᠢᠨ ᠤᠷᠲᠤ ᠨᠢ 34 ( √2 + √6 )𝑘𝑚 ,𝐵𝐷 ᠶᠢᠨ ᠤᠷᠲᠤ ᠨᠢ 32 𝑘𝑚 ,∠ 𝐴𝐶𝐷 = 60° ,∠ 𝐶𝐷𝐵 = 135° ᠪᠠᠢᠬᠤ ᠶᠢ ᠬᠡᠮᠵᠢᠵᠦ ᠣᠯᠵ ᠠᠢ᠃ ( A , B , C , D ᠨᠢ ᠢᠵᠢᠯᠢ ᠨᠢᠭᠡᠨ ᠳᠦᠪᠰᠢ ᠨ ᠬᠠ ᠪᠲᠠᠭᠠᠢ ᠳᠡᠭᠡᠷ ᠡ ᠤᠷᠤᠰᠢᠨ ᠠ)᠃ (1) A , D ᠬᠤᠶᠠᠷ ᠴᠡᠭ ᠦ ᠨ ᠬᠤᠭᠤᠷᠤ ᠳᠤᠬᠢ ᠵᠠᠢ ᠶᠢ ᠣ ᠯ᠃ (2) ᠨᠦᠬ ᠡᠨ ᠵᠠᠮ AB ᠤᠨ ᠤᠷᠲ ᠤ ᠶᠢ ᠣᠯ᠃ 23. (10 ᠬᠤᠪᠢ) ᠰᠢᠶᠤᠤ ᠭᠠᠩ ᠤ ᠨ ᠭᠡᠷ ᠡᠴᠡ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠬᠦᠷᠲᠡᠯ ᠡ 1800 ᠮᠧᠲ᠋ ᠷ ᠵᠠᠢᠲᠠᠢ᠃ ᠨᠢᠭᠡᠨ ᠥᠷᠯ ᠥᠭᠡ ᠲᠡᠷᠡ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠳᠠᠭᠠᠨ ᠬᠦᠷᠴᠢᠬᠡᠭᠡ ᠳ ᠲᠡᠰᠬᠠᠯ ᠢᠶᠠᠨ ᠭᠡᠷ ᠲᠡᠬ ᠡᠨ ᠮᠠᠷᠲᠠᠭᠰᠠᠨ ᠢᠶᠠ ᠨ ᠮᠡᠳᠡ ᠵᠡᠢ᠃ ᠡᠨᠡ ᠦᠶ ᠡ ᠳᠦ ᠬᠢᠴᠢᠶᠡᠯ ᠣᠷᠣ ᠬᠤ ᠠᠴᠠ ᠪᠠᠰᠠ 20 ᠮᠢᠨᠦᠲ ᠦᠯᠡᠳᠡᠭᠡᠳ ᠪᠠᠢᠵᠠᠢ᠃ ᠰᠢᠶᠤᠤ ᠭᠠ ᠩ ᠳᠠᠷᠤᠢ ᠲᠦᠷᠭ ᠡᠨ ᠢᠷᠡᠭᠰᠡ ᠨ ᠵᠠᠮ ᠢ ᠶᠠᠷ ᠢᠶᠠᠨ ᠭᠡᠷ ᠲᠡᠬ ᠡᠨ ᠭᠦᠶᠦ ᠵᠦ ᠬᠠᠷᠢᠭᠠ ᠳ ᠲᠠᠰᠬᠠᠯ ᠢᠶᠠᠨ ᠠᠪᠤᠩ ᠭᠤᠳᠠ ᠳᠤᠭᠤᠢ ᠲᠡᠷᠭᠡ ᠤᠨᠤᠵ ᠤ ᠲᠠᠬᠢᠨ ᠤᠤᠯ ᠵᠠᠮ ᠢ ᠶᠠᠷ ᠢᠶᠠᠨ ᠰᠤᠷᠭᠠ ᠭᠤᠯᠢ ᠳᠤ ᠪᠠᠨ ᠪᠤᠴᠠᠵᠠᠢ᠃ ᠰᠢᠶᠤᠤ ᠭᠠᠩ ᠤ ᠨ ᠳᠤᠭᠤᠢ ᠲᠡᠷᠭᠡ ᠤᠨᠤᠭᠠᠳ ᠬᠡᠷᠡᠭᠯᠡᠭᠰᠡᠨ ᠬᠤᠭᠤ ᠴᠠᠭ ᠠ ᠨᠢ ᠬᠦᠶᠦᠬᠦ ᠳᠦ ᠬᠡᠷᠡᠭᠯᠡᠭᠰᠡᠨ ᠬᠤᠭᠤᠴᠠᠭ ᠠ ᠠᠴᠠ 4.5 ᠮᠢᠨᠦᠲ ᠲᠤᠲ ᠠᠭᠤ ᠬᠡᠷᠡᠭᠯᠡᠭᠰᠡ ᠨ ᠪᠦᠭᠡᠳ ᠳᠤᠭᠤᠢ ᠤᠨᠤᠬ ᠤ ᠳᠤᠮᠳᠠ ᠴᠢ ᠬᠤᠷᠳᠤ ᠴᠠ ᠨᠢ ᠬᠦᠶᠦᠬᠦ ᠳᠤᠮᠳ ᠠᠴᠢ ᠬᠤᠷᠳᠤ ᠴᠠ ᠶᠢᠨ 1. 6 ᠲᠠᠬ ᠢᠨ ᠪ ᠠᠢᠪᠠ ᠭ ᠡᠵᠦ ᠦᠭᠭᠦᠭᠳᠡᠵ ᠡᠢ᠃ (1) ᠰᠢᠶᠤᠤ ᠭᠠᠩ ᠤ ᠨ ᠬᠦᠶᠦᠬᠦ ᠳᠤᠮᠳᠠᠴᠢ ᠬᠤᠷ ᠳᠤᠴᠠ ᠶᠢ ᠣ ᠯ᠃ (2) ᠬᠡᠷᠪᠡ ᠰᠢᠶᠤᠤ ᠭᠠᠩ ᠭᠡᠷ ᠡᠴᠡ ᠪᠡᠨ ᠲᠠᠰᠬᠠᠯ ᠠᠪᠬ ᠤ ᠪᠠ ᠳᠤᠭᠤᠢ ᠲᠡᠷ ᠭᠡ ᠪᠡᠨ ᠠᠪᠬᠤ ᠳ ᠤ 3 ᠮᠢᠨᠦᠲ ᠬᠡᠷᠡᠭᠯᠡᠭᠰᠡ ᠨ ᠪᠣᠯ ᠲᠡᠷᠡ ᠬᠢᠴᠢᠶᠡᠯ ᠣᠷᠣ ᠬᠣ ᠠᠴᠠ ᠡᠮᠤᠨ ᠡ ᠰᠤᠷᠭᠠ ᠭᠤᠯᠢ ᠳᠤ ᠪᠠᠨ ᠪᠤᠴᠠ ᠵᠤ ᠢᠷᠡᠵᠦ ᠳᠡᠢ ᠯᠤᠨ ᠡ ᠤᠤ ︖ ᠰᠢᠯᠲᠠᠭᠠᠨ ᠢ ᠨᠢ ᠲ ᠣᠳᠤᠷ ᠬᠠᠢ ᠯᠠ᠃ 24. (10 ᠬᠤᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠬᠤᠷᠴ ᠠ ᠦᠨᠴᠦᠭᠲᠦ ᠭᠤ ᠷᠪᠠᠯᠵᠢᠨ ABC ᠳᠤAD ᠪᠣᠯ BC ᠲᠠᠯ ᠠ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠦᠨᠳᠦᠷ᠂ AD ᠢᠶᠡᠷ ᠳᠢᠶᠠᠮᠧᠲ᠋ ᠷ ᠪᠣᠯᠭ ᠠᠭᠰᠠᠨ ⊙ O ᠨᠢ AB ᠲᠠᠢ ᠴᠡᠭ E ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠵᠤ᠂ AC ᠲᠠᠢ ᠴᠡᠭ F ᠳᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ F ᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠵ ᠤ FG ⊥ AB ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠪᠠᠯ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤᠷ ᠢ ᠨᠢ H ᠂ 𝐴𝐸̂ ᠲᠠᠢ ᠴᠡᠭ G ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠ ᠵᠤ᠂ AD ᠲᠠᠢ ᠴᠡᠭ M ᠲᠦ ᠤ ᠭᠲᠤᠯᠤᠯᠴᠠᠨ ᠠ᠃ AG , DE , DF ᠢ ᠬᠤᠯᠪᠤᠨ ᠠ᠃ (1) ᠪᠠᠲᠤ ᠯᠠ᠄ ∠G AD + ∠EDF = 180 ° (2) ᠬᠡᠷᠪᠡ ∠A CB = 45 ° , AD = 4 , tan ∠ABC = 2 ᠪᠣᠯ᠂ HF ᠦᠨ ᠤᠷᠲ ᠤ ᠶᠢ ᠣᠯ᠃ 25. (12 ᠬᠤᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠦᠭᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ∆ ABC ᠪᠣᠯ ᠲᠡᠩᠴᠡᠭ ᠦᠦ ᠲᠠᠯᠠᠲᠤ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ᠂ P ᠪᠣᠯ ∆ABC ᠤᠨ ᠳᠤᠲ ᠤᠭᠠᠳᠤ ᠬᠡᠰᠡ ᠭ ᠲᠦ ᠤᠷᠤᠰᠢᠬ ᠤ ᠨᠢᠭᠡ ᠴᠡᠭ᠂ BP , CP ᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠃ (1) ᠵᠢ ᠷᠤᠭ 11 -1 ᠮᠡᠲᠦ᠂ BC ᠢᠶᠡᠷ ᠳᠢᠶᠠᠮᠧᠲ᠋ᠷ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠬᠠᠭᠠᠰ ᠲᠤᠭᠤ ᠷᠢᠭO ᠨᠢ AB ᠲᠠᠢ ᠴᠡᠭ Q ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠵᠤ᠂ AC ᠲᠠᠢ ᠴᠡᠭ R ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ P ᠨᠢ 𝑄𝑅̂ ᠳᠡᠭᠡᠷ ᠡ ᠤᠷᠤᠰᠢᠬ ᠤ ᠦᠶᠡᠰ᠂ AP ᠶᠢ ᠬᠣᠯᠪᠤᠵ ᠤ᠂ BC ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠳᠣᠣ ᠷ ᠠ ᠲᠠᠯ ᠠ ᠳ ᠤ ∠BCD = ∠B AP , CD = AP ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ DP ᠢ ᠬᠤᠯᠪᠤᠨ ᠠ᠂ ∠CPD ᠦᠨ ᠭᠷᠠᠳᠦ ᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠶᠢ ᠣᠯ᠃ (2) ᠵᠢ ᠷᠤᠭ ᠮᠡᠲᠦ᠂ E ᠪᠣᠯ BC ᠲᠠᠯ ᠠ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠪᠦᠭᠡᠳ EC = 3BE , BP = CP ᠦᠶᠡᠰ᠂ EP ᠢ ᠬᠤᠯᠪ ᠤᠵᠤ ᠦᠷᠭᠦᠯᠵᠢ ᠯᠡᠭᠦᠯᠪᠡᠯ AC ᠲᠠᠢ ᠴᠡᠭ F ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ √ 7𝐴𝐵 = 4𝐵𝑃 ᠪᠣᠯ ᠪᠠᠲ ᠤᠯᠠ᠄ 4EF = 3AB (3) ᠵᠢᠷᠤᠭ ᠮᠡᠲᠦ᠂ M ᠪᠣᠯAC ᠲᠠᠯ ᠠ ᠳᠡᠭ ᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠂AM = 2MC ᠦᠶᠡᠰ MP ᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ ∠CMP = 150 ° , AB = 6𝑎 , MP = √3 𝑎 , ∆𝐴𝐵𝐶 ᠶᠢᠨ ᠲᠠᠯᠠᠪᠠᠢ ᠨᠢ 𝑆 1 , ∆𝐵𝐶𝑃 ᠶᠢᠨ ᠲᠠᠯᠠᠪᠠᠢ ᠨᠢ 𝑆 2 ᠪᠣᠯ 𝑆 1 − 𝑆2 ᠤᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠢ ᠣᠯ᠃ ( 𝑎 ᠠᠭᠤᠯ ᠤᠭᠰᠠᠨ ᠠᠯᠭ ᠧᠪᠷᠠ ᠶᠢᠨ ᠢᠯᠡᠷᠬ ᠡᠢᠯᠡ ᠯ ᠢᠶᠡ ᠷ ᠢᠯᠡᠷᠬ ᠡᠢᠯᠡ ) 26. (1 2 ᠬᠤᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠬᠠᠪᠲᠠ ᠭᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠦᠨ ᠴᠦᠭᠲᠦ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ᠋ ᠧᠮ ᠳᠦ᠂ ᠬᠠᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ 𝑦 = −𝑥 2+ 4𝑥 ᠨᠢ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ᠋ ᠧᠮ ᠦᠨ ᠡᠬᠢᠨ ᠴᠡᠭ ᠢ ᠳᠠᠭ ᠠᠷᠢᠵᠤ ᠂ 𝑥 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠦᠨ ᠡᠶᠡᠷᠭᠦ ᠬᠠᠭᠠᠰ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ A ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠃ ᠴᠡᠭ M( 𝑚, 𝑛) ᠨᠢ ᠬᠠ ᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠣᠭᠣᠮ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠬᠦᠳᠡᠯ ᠬᠦ ᠴᠡ ᠭ᠃ (1) ᠵᠢ ᠷᠤᠭ ᠮᠡᠲᠦ᠂ 𝑚 0 , 𝑛 0 ᠪᠦᠭᠡᠳ 𝑛 = 3𝑚 ᠦᠶᠡᠰ᠂ ① ᠴᠡᠭ M ᠦᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ ② ᠬᠡᠷᠪᠡ ᠴᠡᠭ B( 154 ,𝑦 ) ᠨᠢ ᠲᠣᠰ ᠬᠠᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠳᠡᠭᠡᠷ ᠡ ᠤᠷ ᠤᠰᠢᠭᠠᠳ᠂ OM , BM ᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠂ C ᠪᠣᠯ ᠬᠡᠷᠴᠢᠮ BM ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠬᠦᠳ ᠡᠯᠬᠦ ᠴᠡᠭ ( ᠴᠡᠭ C ᠨᠢ ᠴᠡᠭ M , B ᠲᠠᠢ ᠲᠠᠪᠬᠤᠴᠠᠬᠤ ᠦᠭᠡᠢ) ᠂ ᠴᠡᠭ C ᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠤ ᠨ᠂ CD // MO ᠢᠶᠡᠷ ᠪᠠᠢᠭᠤᠯᠪᠠᠯ 𝑥 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ D ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠬᠡᠷᠴᠢᠮ OD ᠪᠠ MC ᠨᠢ ᠬᠠᠷᠢᠯ ᠴᠠᠨ ᠲᠡᠩᠴᠡᠨ ᠡ ᠤᠤ︖ ᠰᠢᠯᠲᠠᠭᠠᠨ ᠢ ᠨᠢ ᠲ ᠣᠳᠤᠷ ᠬᠠᠢ ᠯᠠ᠃ (2) ᠵᠢᠷᠤᠭ ᠮᠡᠲᠦ᠂ ᠲᠣᠰ ᠬᠠᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠣᠭᠤᠮ ᠤᠨ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦ ᠨ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ K ᠲᠦ ᠤᠭᠲᠤ ᠯᠤᠯᠴᠠᠵᠤ᠂ ᠴᠡᠭ E( 𝑥, 73 ) ᠨᠢ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦ ᠨ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠤᠷ ᠤᠰᠢᠨ ᠠ᠃ 𝑚 2 , 𝑛 0 ᠪᠦᠭᠡᠳ ᠰᠦᠯ ᠤᠭᠤᠨ EM ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠦᠨ ᠰᠦᠭᠡᠷᠭᠦ ᠬᠠᠭᠠᠰ ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ F ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠬᠤ ᠦᠶᠡᠰ᠂ ᠴᠡᠭ A ᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠵ ᠤ 𝑥 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠦᠨ ᠡᠭᠴᠡ ᠰᠢᠯᠤᠭᠤ ᠨ ᠢ ᠪᠠᠢᠭᠤᠯᠪᠠᠯ ᠰᠢᠯᠤᠭ ᠤᠨ EM ᠲᠠᠢ ᠴᠡᠭ N ᠳᠦ ᠤᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ G ᠪᠣᠯ 𝑦 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠨ ᠴᠡᠭ᠂ ᠴᠡᠭ G ᠦᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ ( 0, 185 ) , GF ᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ EF + NF = 2MF ᠪᠣᠯ ᠪᠠᠲ ᠤᠯᠠ᠄ ᠰᠠᠴᠤᠷ ᠠᠭ EF ᠨᠢ ∠AF G ᠶᠢ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠤᠪᠢᠶᠠᠨ ᠠ᠃