2020 ᠣᠨ ᠤ ᠤᠯᠠᠭᠠᠨᠬᠠᠳᠠ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

2020 ᠣᠨ ᠤ ᠤ ᠯᠠᠭᠠ ᠨᠬᠠᠳᠠ ᠬᠣᠲ ᠠ ᠶᠢᠨ ᠠᠩᠬ ᠠ ᠳᠤᠮᠳᠠᠳᠤ ᠶᠢᠨ ᠰᠤᠷᠤ ᠭᠴᠢᠳ ᠤᠨ ᠰᠤᠷᠤ ᠯᠭ ᠠ ᠲᠡᠭ ᠦᠰᠬᠦ ᠂ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠳᠠᠪᠰᠢᠬ ᠤ ᠨᠢᠭᠡᠳᠦᠯ ᠲᠡᠢ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠳᠦᠪ ᠲᠣᠭᠠᠨ ᠤ ᠤᠬᠠᠭᠠ ᠨ ᠨᠢᠭᠡ᠂ ᠰᠣᠩᠭᠣᠬ ᠤ ᠰᠡᠳᠦᠪ᠃ ︵ᠰᠡᠳᠦᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠡᠢ ᠲᠡ 42 ᠬᠤᠪ ᠢ︶ 1. ᠪᠣᠳᠠᠳᠤ ᠲᠣᠭ ᠠ | −5 | , −3 , 0 , √4 ᠠᠴᠠ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠲᠣᠭ ᠠ ᠪᠣᠯᠬᠤ ᠨᠢ ( ) A. |− 5| B. −3 C. 0 D. √4 2. 2020 ᠣ ᠨ ᠤ 6 ᠰᠠᠷ ᠠ ᠶᠢᠨ 23 ᠦ ᠡᠳᠦᠷ ᠦ ᠨ 9᠄ 43 ᠴᠠᠭ ᠲᠤ ᠮᠠᠨ ᠤ ᠤᠯᠤ ᠰ ᠳᠣᠬᠢᠭᠤᠷ ᠰᠢᠰᠲ᠋ ᠧᠮ ᠦᠨ 55 ᠳ᠋ᠤ ᠭᠠᠷ ᠠᠶᠠᠯᠠᠯ ᠵᠠᠯᠠᠬᠤ ᠬᠢᠮᠡᠯ ᠳᠠᠭᠠ ᠭᠤᠯᠢ ᠣᠳᠣ ᠶᠢ ᠠᠮ ᠵᠢᠯᠲ ᠠ ᠲᠠᠢ ᠪᠠᠷ ᠬᠠᠷᠪᠤᠨ ᠭᠠᠷᠭᠠᠪ ᠠ᠂ ᠲᠡᠭᠦᠨ ᠤ ᠴᠠᠭ ᠮᠡᠳᠡᠭᠦᠯᠬ ᠦ ᠨᠠᠷᠢᠪᠴᠢᠯᠠ ᠯ ᠨᠢ ᠳᠡᠯᠡᠬᠡᠢ ᠶᠢᠨ ᠬᠠᠮᠤᠭ ᠥᠨᠳ ᠥᠷ ᠬᠡᠮᠵᠢᠶᠡ ᠨ ᠳ᠋ ᠦ ᠬᠦᠷᠦᠭᠰᠡᠨ ᠪᠥᠭᠡ ᠳ 0.0 00 000009 9 ᠰᠧᠺᠦᠨ᠋ᠳ᠋ ᠡᠴᠡ ᠬᠡᠲᠦᠷ ᠡᠬᠦ ᠦ ᠭᠡᠢ᠃ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢ ᠮᠲᠠ ︽0.0 00 00 0009 9︾ ᠢ ᠰᠢᠨᠵᠢᠯᠡᠬᠦ ᠤᠬ ᠠᠭᠠᠨᠴᠢ ᠲᠣ ᠭ ᠠ ᠲᠡᠮᠳᠡᠭᠯ ᠡᠬᠦ ᠠᠷ ᠭ ᠠ ᠪᠠᠷ ᠢ ᠯᠡᠷᠬᠡᠢᠯᠡ ᠪᠡᠯ ( ) A. 99 × 10 −10 B. 9.9 × 10 −10 C. 9.9 × 10 −9 D. 0.99 × 10 −8 3. ᠳᠠᠷᠠᠭᠠᠬᠢ ᠵᠢᠷᠤ ᠭ ᠳᠦᠷᠰᠦ ᠶᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠢᠶᠡᠷ ᠲᠥ ᠪ ᠪᠣᠯ ᠭᠠᠨ ᠵᠣᠬᠢᠬᠤ ᠥᠨᠴᠥᠭ ᠡᠷᠭᠢ ᠯᠳᠦᠭᠦᠯᠪᠡᠯ ᠪᠦᠷ ᠤᠤᠯ ᠤ ᠨ ᠵᠢ ᠷᠤᠭ ᠳᠦᠷᠰᠦ ᠲᠠᠢ ᠪ ᠠᠨ ᠳᠠᠪᠬᠤ ᠴᠠᠨ ᠠ᠂ ᠲᠡᠳ ᠡᠭᠡᠷ ᠲᠦ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠡᠷᠭᠢ ᠯᠲᠡ ᠶᠢᠨ ᠥᠨᠴᠥᠭ ᠲᠠᠢ ᠵᠢᠷᠤ ᠭ ᠳᠦᠷᠰᠦ ᠨᠢ ( ) A. B. ᠵᠢᠰᠢᠭᠦᠯᠵᠢᠨ C. D. 4. ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠶᠢᠨ ᠤᠩ ᠰᠢᠯᠭ ᠠ ᠶᠢᠨ ᠤᠷᠤᠯ ᠳᠤᠭᠠᠨ ᠳ᠋ᠤ 9 ᠬᠢᠨ᠋ᠠᠭᠴᠢ ᠲᠤᠰ ᠲᠤᠰ ᠣᠷᠣᠯᠴ ᠠᠭᠴᠢ ᠶᠢᠨ ᠠᠩᠬᠠᠨ ᠤ ᠣᠨ᠋ᠣᠭ ᠠ ᠶᠢ ᠥᠭᠭᠦᠭᠰᠡᠨ ᠪᠥᠭᠡᠳ ᠣᠷᠣ ᠯᠴᠠᠭᠴᠢ ᠶᠢᠨ ᠬᠢᠷᠢ ᠪᠦᠲᠦ ᠮᠵᠢ ᠶᠢ ᠦᠨ᠋ᠡᠯᠡᠬᠦ ᠳ᠋ ᠦ ᠪᠡᠨ 9 ᠠᠩ ᠬᠠᠨ ᠤ ᠣᠨ᠋ᠣᠭ ᠠ ᠠᠴᠠ ᠬᠠᠮᠤᠭ ᠤᠨ ᠥᠨᠳᠥᠷ ᠣᠨ᠋ᠣᠭ ᠠ ᠪᠠ ᠬᠠ ᠮᠤᠭ ᠤ ᠨ ᠪᠠᠭ ᠠ ᠣᠨ᠋ᠣᠭ ᠠ ᠶᠢ ᠬᠠᠰᠤᠵ ᠤ᠂ ᠬᠦᠴᠦᠨ ᠲᠠᠢ 7 ᠣᠨ᠋ᠣᠭ ᠠ ᠶᠢ ᠠᠪᠤᠭ ᠰᠠᠨ ᠪᠠ ᠢᠨ ᠠ᠃ ᠡᠨ᠋ᠡ ᠬᠦ ᠬᠦᠴᠦᠨ ᠲᠡᠢ 7 ᠣᠨ᠋ᠣᠭ ᠠ ᠪᠤᠯ ᠤᠨ 9 ᠠᠩᠬᠠᠨ ᠤ ᠣᠨ᠋ᠣᠭ ᠠ ᠶᠢ ᠬᠠᠷᠢᠴᠠ ᠭᠤᠯᠪᠠᠯ ᠬᠤᠪ ᠢᠷᠠᠬᠤ ᠦᠭᠡᠢ ᠲᠣ ᠭ ᠠ ᠪᠠᠷᠢᠮᠲ ᠠ ᠶᠢᠨ ᠣᠨᠴᠠ ᠯᠢᠭ ᠨᠢ ( ) A. ᠳᠤᠮᠳᠠᠴᠢ ᠲᠣ ᠭ ᠠ B. ᠳᠤᠮᠳᠠ ᠪ ᠠᠢᠷᠢ ᠶᠢᠨ ᠲᠣ ᠭ ᠠ C. ᠮᠣᠳ᠋ D. ᠰᠠᠷᠨ᠋ᠢᠴᠡ 5. ᠳᠠᠷᠠᠭᠠᠬᠢ ᠪᠣᠳ ᠣᠯᠭ ᠠ ᠨᠤᠭᠤᠳ ᠠᠴᠠ ᠵᠥᠪ ᠨᠢ ( ) A. 2 3 5 B. 3√ 2− 2√ 2 1 C. 2 3 5 D. 5 3 2 6. ᠲᠡᠩ ᠴᠡᠳᠬ ᠡᠯ ᠪᠤᠰᠤ ᠶᠢᠨ ᠪᠦᠯᠦᠭ { 2 0 −2 4≥ 0 ᠦ ᠨ ᠬᠠ ᠷᠢᠭᠤ ᠴᠤᠭ ᠯᠠᠷᠠᠯ ᠢ ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠳᠡᠭᠡᠷ ᠡ ᠵᠥᠪ ᠢ ᠯᠡᠷᠬᠡᠢᠯᠡ ᠭᠰᠡᠨ ᠨᠢ ( ) A. B. C. D. 7. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ Rt △ ABC ᠳ᠋ᠦ᠂ ∠A CB 90° ᠂ AB 5᠂ AC 3 ᠂ Rt △ ABC ᠶᠢ ᠰᠢᠯᠤᠭᠤ ᠨ BC ᠳᠠᠭᠠᠤ ᠪᠠᠷᠠᠭᠤᠨᠰᠢ ᠵᠡᠷᠭᠡᠴᠡᠭ ᠡ 3 ᠨᠢᠭᠡᠴᠢ ᠤᠷᠲᠤ ᠰᠢᠯᠵᠢᠭᠦᠯᠪᠡ ᠯ △ A′B′ C′ ᠶᠢ ᠣ ᠯᠣᠨ ᠠ᠂ ᠲᠡᠭᠡᠪᠡᠯ ᠳᠥᠷ ᠪᠡᠯᠵᠢᠨ ABC′A′ ᠶᠢᠨ ᠲᠠᠯᠠᠪᠠᠢ ᠨᠢ ( ) A. 15 B. 1 8 C. 2 0 D. 2 2 8. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ △ ABC ᠳ᠋ᠦ ᠴᠡᠭ D , E ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ AB , AC ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠂ ᠴᠡᠭ F ᠨᠢ ᠬᠡᠷᠴᠢᠮ DE ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ᠂ AF ,BF ᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠃ ∠AFB 90° , AB 8 ᠪᠥᠭᠡᠳ BC 14 ᠪᠣᠯ EF ᠤᠨ ᠤᠷ ᠲᠤ ᠨᠢ ( ) A. 2 B. 3 C. 4 D. 5 9. (2 √3 3√ 2) × √ 13 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠢ ᠪᠠᠷᠤ ᠭᠴᠠᠭᠠ ᠯᠠᠪᠠᠯ ( ) A. 4 ᠪᠠ 5 ᠶᠢᠨ ᠬᠣᠭᠣᠷᠣ ᠨᠳᠤ ᠪᠣᠢ B. 5 ᠪᠠ 6 ᠶᠢᠨ ᠬᠣᠭᠣᠷᠣ ᠨᠳᠤ ᠪᠣᠢ C. 6 ᠪᠠ 7 ᠶᠢᠨ ᠬᠣᠭᠣᠷᠣ ᠨᠳᠤ ᠪᠣᠢ D. 7 ᠪᠠ 8 ᠶᠢᠨ ᠬᠣᠭᠣᠷᠣ ᠨᠳᠤ ᠪᠣᠢ 10. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ △ ABC ᠳ᠋ᠦ AB AC ᠂ AD ᠪᠣᠯ ∠B AC ᠶᠢᠨ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ᠂ EF ᠪᠣᠯ AC ᠶᠢᠨ ᠡᠭᠴᠡ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠤᠪᠢᠶᠠᠭᠴᠢ᠂ AD ᠯᠤᠭ ᠠ ᠴᠡᠭ O . ᠳ᠋ ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ OA 3 ᠪᠣᠯ △ ABC ᠶᠢᠨ ᠪᠠᠭᠲ ᠠᠭᠠᠭᠰᠠ ᠨ ᠲᠣᠭᠣ ᠷᠢᠭ ᠤ ᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ( ) A. 3π B. 4π C. 6π D . 9π 11 . ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ⊙ A ᠨᠢ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠲᠦ ᠪᠠᠢᠷᠢᠰ ᠢᠯ ᠤ ᠨ ᠰᠢᠰᠲ᠋ ᠧᠮ ᠦᠨ ᠡᠬᠢᠨ ᠦ ᠴᠡᠭ O ᠶᠢ ᠲᠠᠭ ᠠᠷᠢᠵᠤ ᠂ ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠯᠦᠭᠡ ᠴᠡᠭ B −4 ,0 ᠳ᠋ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠵᠤ᠂ y ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠯᠦᠭᠡ ᠴᠡᠭ C 0,3 ᠳ᠋ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ D ᠨᠢ ᠬᠣᠶᠠᠳᠤ ᠭᠠᠷ ᠮᠥᠴᠡ ᠳ᠋ᠡᠬᠢ ᠲᠣᠭᠣ ᠷᠢᠭ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠨ ᠴ ᠡᠭ ᠪᠣᠯ ∠CDO ᠤᠨ ᠰᠠ ᠶᠢᠨ ᠤ ᠬᠡᠮᠵ ᠢᠭᠳᠡᠯ ᠨᠢ ( ) A. 35 B . − 34 C. 34 D . 45 12 . ᠨᠢᠭᠡᠨ ᠭᠧᠣᠮᠧᠲ᠋ ᠷᠢ ᠶᠢᠨ ᠪᠡᠶ ᠡ ᠶᠢ ᠨ ᠭᠤᠷᠪᠠᠨ ᠬᠠᠷᠠ ᠭᠠᠨ ᠵᠢᠷᠤ ᠭ ᠵᠢᠴᠢ ᠬᠠᠮᠢ ᠶ ᠠ ᠪᠦᠬᠦᠢ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢ ᠮᠲᠠ ( ᠨᠢᠭᠡᠴᠢ᠄ cm ) ᠶᠢ ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ ᠢᠯᠡᠷᠬᠡᠢᠯᠡᠭᠰᠡᠨ ᠪᠣᠯ ᠲᠣᠰ ᠭᠧᠣᠮᠧᠲ᠋ ᠷᠢ ᠶᠢᠨ ᠪᠡᠶ ᠡ ᠶᠢᠨ ᠬᠠᠵᠠᠭᠤ ᠭᠠᠳᠠᠷ ᠭᠤ ᠶᠢᠨ ᠲᠠᠯᠠ ᠪᠠᠢ ᠨᠢ ( ) A. 652 πc m 2 B. 60π cm 2 C. 65π cm 2 D. 130π cm 2 13 . ᠵᠢᠷᠤᠭ ᠮᠡᠲ ᠦ᠂ ᠴᠡᠭ B ᠨᠢ ᠤᠷᠪᠠᠭ ᠤ ᠬᠠᠷᠢ ᠴᠠᠯᠲ ᠤ ᠹᠦᠩᠺ ᠼ y 6𝑥 0 ᠶᠢᠨ ᠵᠢᠷᠤ ᠮᠠᠭ ᠳᠡᠭᠡᠷ ᠡ ᠪᠣᠢ᠂ ᠴᠡᠭ C ᠨᠢ ᠤᠷᠪᠠᠭ ᠤ ᠬᠠᠷᠢ ᠴᠠᠯᠲᠤ ᠹᠦᠩᠺ ᠼ y − 2𝑥 0 ᠤᠨ ᠵᠢᠷᠤ ᠮᠠᠭ ᠳᠡᠭᠡᠷ ᠡ ᠪᠠᠢᠬᠤ ᠪᠥᠭᠡᠳ BC // 𝑦 ᠲᠡᠩᠭᠡᠯᠢ ᠭ᠂ AC ⊥ BC ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ ᠴᠡᠭ C᠂ y ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠯᠦᠭᠡ ᠴᠡᠭ A ᠳ᠋ ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠ ᠪᠠᠯ △ ABC ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ( ) A. 3 B. 4 C. 5 D. 6 14 . ᠵᠢᠷᠤᠭ ᠮᠡᠲᠦ᠂ ᠷᠤᠮᠪ ABC D ᠳ᠋ᠤ᠂ ∠B 60° , AB 2᠂ ᠬᠥᠳᠡ ᠯᠬᠦ ᠴᠡᠭ P ᠨᠢ ᠴᠡᠭ B ᠡᠴᠡ ᠰᠧᠺ ᠦᠨ᠋ᠳ᠋ ᠪᠦᠷᠢ 1 ᠨᠢᠭᠡᠴᠢ ᠤᠷᠲᠤ ᠶᠢᠨ ᠬᠤᠷᠳᠤ ᠴᠠ ᠪ ᠠᠷ ᠲᠠᠬᠢᠷ ᠱᠤᠭᠤᠮ BA → AC ᠴᠢᠭᠯᠡᠯ ᠳᠠᠭᠠᠤ ᠬᠥᠳ ᠡᠯᠭᠡᠭᠡᠨ ᠬᠢᠵᠦ ᠴᠡᠭ C ᠳ᠋ ᠦ ᠬᠦᠷᠦᠨ ᠡ᠂ ᠡᠭᠦᠨ ᠦ ᠵᠡᠷᠭᠡ ᠪᠡᠷ ᠬᠥᠳᠡᠯ ᠬᠦ ᠴ ᠡᠭ Q ᠨᠢ ᠴᠡᠭ A ᠡᠴᠡ ᠠᠳᠠᠯᠢ ᠬᠤᠷᠳᠤᠴᠠ ᠪᠠᠷ ᠲᠠᠬᠢᠷ ᠱᠤᠭᠤᠮ AC → CD ᠴᠢᠭᠯᠡᠯ ᠳᠠᠭᠠᠤ ᠬᠥᠳᠡᠯᠭᠡᠭ ᠡᠨ ᠬᠢᠵᠦ ᠴᠡᠭ D ᠳ᠋ᠦ ᠬᠦᠷᠦᠨ ᠡ᠃ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠨᠢ ᠵᠣᠭᠰᠣᠬᠤ ᠦᠶᠡᠰ ᠲᠦ ᠨᠥᠭᠥᠭᠡ ᠨᠢᠭᠡ ᠴᠡᠭ ᠨᠢ ᠳᠠᠭᠠᠯᠳᠤ ᠨ ᠵᠣᠭᠰᠣᠨ ᠠ᠃ △ APQ ᠦᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠶᠢ 𝑦 ᠭᠡᠵᠦ᠂ ᠬᠥᠳ ᠡᠯᠭᠡᠭᠡᠨ ᠬᠢᠭᠰᠡᠨ ᠬᠤᠭᠤᠴ ᠠᠭ ᠠ ᠶᠢ ᠰᠧᠺᠦᠨ᠋ᠳ᠋ ᠭᠡᠵᠦ ᠵᠢᠱᠢᠪᠡᠯ᠂ ᠳᠠᠷᠠᠭᠠ ᠬᠢ ᠵᠢᠷᠤ ᠭ ᠨᠤᠭᠤᠳ ᠠᠴᠠ ᠶᠡᠬᠡ ᠲᠥᠯᠥ ᠪ 𝑦 ᠪᠣᠯᠣ ᠨ ᠤᠨ ᠹᠦᠩᠺ ᠼ ᠦᠨ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠢ ᠲᠤ ᠰᠬᠠᠭᠰᠠᠨ ᠨ ᠢ ( ) A. B. C. D. ᠬᠣᠶ ᠠᠷ᠂ ᠬᠣᠭᠣᠰᠤᠨ ᠳᠤ ᠲᠠᠭᠯᠠᠬᠤ ᠰᠡᠳᠦᠪ ᠃ ( ᠬᠠ ᠷᠢᠭᠤᠯᠲ ᠠ ᠪᠠᠨ ᠰᠡᠳᠦᠪ ᠬᠠᠷᠢᠭᠤ ᠯᠬᠤ ᠺᠠᠷᠲ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠬᠠᠷᠠ ᠭᠠᠯᠵᠠᠭᠰᠠᠨ ᠪᠠ ᠢᠷᠢᠨ ᠳ᠋ ᠤ ᠪᠢ ᠴᠢ᠃ ᠰᠡ ᠳᠦᠪ ᠪᠦᠷ ᠢ 3 ᠬᠤ ᠪᠢ᠂ ᠨ ᠡᠢᠲ ᠡ 1 2 ᠬᠤ ᠪᠢ) 15 . ᠨᠢᠭᠡᠨ ᠵᠥᠪ ᠥᠨᠴᠥᠭᠲᠦ ᠶᠢᠨ ᠳᠣᠲ ᠣᠭᠠᠳᠤ ᠥᠨᠴᠥᠭ ᠦᠨ ᠨᠡᠢ ᠯᠡᠪᠦᠷᠢ ᠨᠢ ᠲᠡᠭᠦᠨ ᠦ ᠭᠠᠳᠠ ᠭᠠᠳᠤ ᠥᠨᠴᠥᠭ ᠦᠨ ᠨᠡᠢᠯᠡ ᠪᠦᠷᠢ ᠶᠢᠨ 4 ᠳᠠᠬᠢ ᠨ ᠪᠣᠯ ᠃ 16 . ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠣᠭᠲ ᠠᠷᠭᠤᠢ ᠠᠴᠠ ᠰᠡᠭᠦᠳᠡᠷᠯᠡ ᠬᠦ ᠨᠢ ᠰᠬᠡᠯ A ᠪᠠᠢᠷᠢ ᠠᠴᠠ ᠨᠢᠭᠡᠨ ᠪᠠᠷᠢ ᠯᠭ ᠠ ᠶ ᠢᠨ ᠣᠷᠣ ᠢ C ᠶᠢᠨ ᠥᠯᠢᠢ ᠭᠰᠡᠨ ᠥᠨᠴᠥᠭ ᠨᠢ 30° ᠂ ᠰᠠᠭᠤᠷ ᠢ B ᠶᠢᠨ ᠰᠢᠷᠭᠤᠭ ᠰᠠᠨ ᠥᠨᠴᠥᠭ ᠨᠢ 60° ᠪᠣᠯᠬᠤ ᠶᠢ ᠬᠡᠮᠵᠢᠨ ᠣᠯᠪᠠ ᠂ ᠡ ᠨ᠋ᠡ ᠦᠶᠡᠰ ᠲᠦ ᠣᠭᠲ ᠠᠷᠭᠤᠢ ᠠᠴᠠ ᠰᠡᠭᠦᠳᠡᠷᠯᠡᠬᠦ ᠨᠢᠰᠬᠡ ᠯ ᠪᠣᠯᠣ ᠨ ᠲᠤᠰ ᠪᠠᠷᠢ ᠯᠭ ᠠ ᠶᠢᠨ ᠬᠡᠪᠲᠡᠭᠡ ᠵᠠᠢ AD ᠨᠢ 9 ᠮᠧᠲ᠋ ᠷ ᠪᠣ ᠯ ᠲᠣᠰ ᠪᠠᠷᠢ ᠯᠭ ᠠ ᠶᠢᠨ ᠥᠨᠳᠥᠷ BC ᠨᠢ __ ____ ᠮᠧᠲ᠋ ᠷ ︵ ᠳ᠋ᠦᠩ ᠳ᠋ ᠦ ᠢᠵᠠᠭᠤᠷ ᠤ ᠨ ᠲᠡᠮ ᠳᠡᠭ ᠢᠶᠡᠨ ᠦᠯᠡᠳᠡᠭᠡ ︶᠃ 17 . ᠨᠢᠭᠡᠨ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠨᠢ ᠳᠣᠯᠣ ᠳᠤᠭᠠᠷ ᠣᠨ ᠤ ᠰᠤᠷᠤ ᠭᠴᠢᠳ ᠤᠨ ᠪᠡᠶ ᠡ ᠮᠠᠬᠠ ᠪᠤᠳ ᠤᠨ ᠬᠡᠪ ᠴᠢᠨ᠋ᠠᠷ ᠤᠨ ᠪᠠᠢᠳᠠᠯ ᠢ ᠣᠢᠯᠠᠭᠠ ᠬᠤ ᠶᠢᠨ ᠲᠥᠯᠥ ᠭᠡ᠂ ᠳᠣᠯᠣ ᠳᠤᠭᠠᠷ ᠣᠨ ᠤ ᠡᠯ ᠡ ᠠᠩ ᠭᠢ ᠠ ᠴᠠ ᠲᠠᠰᠢᠷᠠᠮ ᠢᠶᠠᠷ ᠠᠳᠠᠯᠢ ᠲᠣᠭᠠ ᠨ ᠤ ᠬᠦᠦ ᠰᠤᠷᠤ ᠭᠴᠢ ᠪᠠ ᠬᠡᠦᠬᠡᠨ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠᠪᠴᠤ᠂ ᠪᠠᠭᠲ ᠠᠭᠠᠮᠵᠢ ᠨᠢ 60 ᠪᠣᠯᠬᠤ ᠶᠠᠩᠵᠤ ᠶ ᠢ ᠪᠦᠷᠢᠯ ᠳᠦᠭᠦᠯᠵᠦ᠂ ᠡᠯ ᠡ ᠵᠦᠢᠯ ᠦᠨ ᠲᠠᠮᠢᠷ ᠤᠨ ᠲᠥᠷᠥ ᠯ ᠢ ᠰᠢᠯᠭᠠᠨ ᠪᠠᠢᠴᠠ ᠭᠠᠪᠠ᠃ ᠳᠠᠷ ᠠᠭᠠᠬᠢ ᠬᠦᠰᠦᠨ᠋ᠦᠭᠲ ᠦ ᠪᠣᠯ ᠶᠠᠩᠵ ᠤ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢ ᠮᠲᠠ ᠶᠢ ᠡᠮᠬᠢ ᠯᠡᠬᠦ ᠪᠡᠷ ᠳᠠᠮᠵᠢᠨ ᠪᠣᠳᠠᠭᠠᠯᠢ ᠪᠡᠶ ᠡ ᠨᠢᠭᠡ ᠪᠦᠷᠢ ᠶᠢᠨ ᠬᠢᠷᠢ ᠪᠦᠲᠦ ᠮᠵᠢ ᠶᠢᠨ ᠬᠡᠰᠡ ᠭ ᠪᠦᠷᠢᠳᠬ ᠡᠯ ᠦᠨ ᠬᠦᠰ ᠦᠨ᠋ᠦᠭᠲᠦ ᠪ ᠣᠯᠣᠨ ᠠ᠃ ᠨᠢᠭᠡᠨ ᠰᠤᠷᠭᠠ ᠭᠤᠯᠢ ᠶᠢᠨ 60 ᠨᠡᠷ ᠡ ᠰᠤᠷᠤ ᠭᠴᠢ ᠶᠢᠨ ᠲᠠᠮᠢᠷ ᠤᠨ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠬᠢᠷᠢ ᠪᠦᠲᠦ ᠮᠵᠢ ᠶᠢᠨ ᠳᠠᠪᠲᠠᠭᠳ ᠠᠭᠰᠠᠨ ᠲ ᠣᠭ ᠠ ᠶᠢᠨ ᠲᠠ ᠷᠬᠠᠯᠲ ᠠ ᠶᠢᠨ ᠬᠦᠰ ᠦᠨ᠋ᠦᠭᠲᠦ ᠬᠡᠷᠪᠡ ᠲᠤᠰ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠶᠢᠨ ᠳᠣᠯᠣ ᠳᠤᠭᠠᠷ ᠣᠨ ᠳ᠋ ᠤ ᠨᠡᠢ ᠲᠡ 300 ᠰᠤᠷᠤᠭᠴᠢ ᠲᠠᠢ ᠪᠣᠯ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢ ᠮᠲᠠ ᠶᠢ ᠦᠨᠳᠦᠰᠦᠯᠡᠵᠦ᠂ ᠲᠤᠰ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠶᠢᠨ ᠳᠣᠯᠣ ᠳᠤᠭᠠᠷ ᠣᠨ ᠤ ᠰᠤᠷ ᠤᠭᠴᠢᠳ ᠤ ᠨ ᠪᠡᠶ ᠡ ᠮᠠᠬᠠ ᠪᠤᠳ ᠤᠨ ᠬᠡᠪ ᠴᠢᠨ᠋ᠠᠷ ᠨᠢ ᠰᠠᠢᠨ ᠵᠢᠴᠢ ᠰᠠᠢᠨ ᠠᠴ ᠠ ᠳᠡᠭᠡᠭᠰᠢᠬᠢ ᠰᠤ ᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠶᠢ ᠪᠠᠷᠤ ᠭᠴᠠᠭᠠᠯᠠᠪᠠᠯ __ __ __ ᠬᠥᠮᠥ ᠨ᠃ 18.ᠲᠣᠭᠠᠨ ᠤ ᠲᠡᠩ ᠭᠡᠯᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠨᠢ ᠭᠡ ᠡᠯᠧᠺᠲ᠋ᠷᠣ ᠨ ᠨᠣᠬᠠᠢ ᠶᠢᠨ ᠪᠥᠭᠡᠰᠦ ᠦᠰᠦᠷᠴᠦ ᠪᠠᠢᠨ ᠠ᠃ ᠨᠢᠭᠡᠳᠦᠭ ᠡᠷ ᠤᠳᠠᠭ ᠠ ᠡᠬᠢᠨ ᠦ ᠴᠡᠭ O ᠡᠴᠡ ᠡᠬᠢᠯᠡᠨ ᠦᠰᠦᠷᠴᠦ᠂ ᠴᠡᠭ 𝐴1 ᠬᠦᠷᠦᠨ ᠡ᠂ ᠴᠡᠭ 𝐴1 ᠨᠢ ᠲᠣᠭ ᠠ 1 ᠶᠢ ᠢᠯᠡᠷᠬᠡᠢᠯᠡᠨ ᠡ︔ ᠬᠣᠶᠠᠳᠤ ᠭᠠᠷ ᠤᠳᠠ ᠭ ᠠ 𝐴1 ᠠᠴᠠ ᠦᠰᠦᠷᠴᠦ᠂ 𝑂𝐴 1 ᠶᠢᠨ ᠳᠤᠮᠳ ᠠ ᠴᠡᠭ 𝐴2 ᠬᠦᠷᠦᠨ ᠡ︔ ᠭᠤᠷᠪᠠ ᠳᠤᠭᠠᠷ ᠤᠳᠠᠭ ᠠ 𝐴 2 ᠴᠡᠭ ᠡᠴᠡ ᠦᠰᠦᠷᠴᠦ᠂ 𝑂𝐴 2 ᠤ ᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ 𝐴3 ᠬᠦᠷ ᠦᠨ ᠡ ︔ ᠡᠨ᠋ᠡ ᠮᠡᠲᠦ ᠪᠡᠷ ᠦᠷᠭᠦᠯᠵᠢ ᠯᠡᠨ ᠦᠰᠦᠷᠦ ᠨ ᠡ ᠁ ᠬᠠᠮᠤᠭ ᠤᠨ ᠡᠴᠦᠰ ᠦ ᠨ ᠤᠳᠠᠭ ᠠ ᠦᠰᠦᠷᠦ ᠭᠡᠳ 𝑂𝐴 2019 ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ 𝐴20 20 ᠬᠦᠷᠴ ᠡᠢ᠃ ᠲᠡᠭᠡᠪᠡᠯ ᠴᠡᠭ 𝐴20 20 ᠦ ᠨ ᠢᠯᠡᠷᠬ ᠡᠢᠯᠡ ᠬᠦ ᠲᠣᠭ ᠠ ᠨᠢ __ __ __ ᠃ ᠭᠤᠷᠪᠠ᠂ ᠪᠣᠳᠣ ᠵᠤ ᠬᠠᠷᠢᠭᠤᠯᠬᠤ ᠰᠡᠳᠦᠪ ᠃ ︵ ᠬᠠᠷᠢ ᠭᠤᠯᠲ ᠠ ᠪᠠᠨ ᠰᠡᠳᠦᠪ ᠬᠠ ᠷᠢᠭᠤ ᠯᠬᠤ ᠺᠠᠷᠲ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠬᠠᠷᠠ ᠭᠠᠯᠵᠠᠭᠰᠠᠨ ᠪᠠᠢᠷᠢ ᠳ᠋ᠤ ᠪᠢ ᠴᠢ᠃ ᠬᠠᠷᠢᠭᠤ ᠯᠬᠤ ᠳ᠋ ᠤ ᠪᠠᠨ ᠡᠷᠬᠡᠪᠰᠢ ᠦᠰᠦᠭ ᠪᠢᠴᠢᠭ ᠦᠨ ᠲᠣᠳᠣᠷᠬᠠᠢ ᠯᠠᠯ᠂ ᠪᠠᠲᠤᠯ ᠠᠬᠤ ᠶᠠᠪᠤᠴᠠ ᠪᠤᠶᠤ ᠲᠥ ᠰᠥᠭᠡᠯᠡᠨ ᠪᠣᠳᠣᠭ ᠰᠠᠨ ᠠᠯᠬᠤ ᠮ ᠪᠠᠢᠬᠤ ᠬᠡᠷᠡᠭᠲᠡᠢ᠃ ᠨᠡᠢ ᠲᠡ 8 ᠰᠡᠳᠦᠪ᠂ ᠪᠦᠷᠢᠨ ᠬᠤᠪᠢ 96 ᠬᠤᠪᠢ ︶ 1 9. ︵ 10 ᠬᠤᠪᠢ ︶ ᠤᠷ ᠢᠳᠠᠪᠠᠷ ᠬᠢᠯᠪᠠᠷᠴᠢᠯᠠᠵᠤ ᠳ ᠠᠷᠠᠭ ᠠ ᠨᠢ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠢ ᠨᠢ ᠣᠯ᠄ ᠡᠭᠦᠨ ᠳ᠋ᠦ m ᠨᠢ m 2− m − 1 0 ᠢ ᠬᠠᠩᠭᠠ ᠨ ᠠ᠃ 20 . ︵ 10 ᠬᠤ ᠪᠢ ︶ ᠰᠤᠷᠤ ᠭᠴᠢ ᠴᠡᠴᠡᠭ ᠠᠪᠤ ᠡᠵᠢ ᠲᠡᠢ ᠪ ᠡᠨ ᠬᠠ ᠮᠲᠤ ᠡᠮᠡᠭᠡ ᠡᠵᠢ ᠶᠢᠨ ᠢᠶᠡᠨ ᠲᠥᠷᠥ ᠭᠰᠡᠨ ᠡᠳᠦᠷ ᠢ ᠪᠡᠯᠭ ᠡᠯᠡᠬᠦ ᠪᠡᠷ ᠪᠣᠢ᠂ ᠲᠡᠳᠡ ᠡᠮᠡᠭᠡ ᠡᠵᠢ ᠳ᠋ᠦ ᠪᠡᠨ ᠵᠢᠷᠤ ᠭ ᠲᠤ ᠢᠯᠡᠷᠬᠡᠢᠯᠡᠭᠰ ᠡᠨ ᠮᠡᠲᠦ ᠲᠡᠭᠰᠢ ᠳᠥᠷᠪᠡᠯᠵᠢ ᠨ ᠬᠡᠯᠪᠡᠷᠢᠲᠦ ᠪᠣᠭᠣᠷᠰᠣ ᠭ ᠪᠡᠯᠡᠳᠬ ᠡᠪᠡ᠂ ᠪᠣᠭᠣᠷᠰᠣ ᠭ ᠤᠨ ᠲᠠᠯ ᠠ ᠪᠦᠷᠢ ᠳ᠋ᠦ ᠵᠢᠭᠳᠡ ᠪᠡᠷ 4 ᠴᠣᠺᠣᠯᠠᠨᠳ᠋ ᠰᠢᠬᠢ ᠷ ᠢ ᠰᠢᠭᠢᠳᠬ ᠡᠪᠡ᠃ ᠴᠡᠴᠡᠭ ᠦᠨ ᠠᠪᠤ ᠨᠢ ᠪᠣ ᠭᠣᠷᠰᠣ ᠭ ᠢ ᠬᠣᠶᠠᠷ ᠤᠳᠠᠭ ᠠ ᠬᠡᠷᠴᠢᠵ ᠦ ᠳᠤᠮᠳᠠᠴᠢ ᠪ ᠠᠷ 4 ᠬᠤᠪᠢ ᠬᠤᠪᠢᠶᠠᠵᠤ᠂ ᠬᠥᠮᠥ ᠨ ᠪᠦᠷᠢ ᠳ᠋ᠦ ᠨᠣᠭᠳᠠᠭᠰᠠᠨ ᠪᠣᠭᠣᠷᠰᠣ ᠭ ᠪᠠ ᠴᠤᠺᠤᠯᠠᠨᠳ᠋ ᠰᠢᠬᠢ ᠷ ᠦᠨ ᠲᠣᠭ ᠠ ᠬᠠ ᠷᠢᠯᠴᠠᠨ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠪᠠᠢᠯ ᠭᠠᠬᠤ ᠶᠢ ᠱᠠᠭᠠᠷᠳᠠᠭᠰᠠᠨ ᠪᠠ ᠢᠨ ᠠ᠃ ︵1︶ ᠲᠠ ᠵᠢᠷᠤ ᠭ 1 ᠳ᠋ᠤ ᠨᠢᠭᠡᠨ ᠵᠦᠢᠯ ᠦᠨ ᠬᠤᠪᠢᠶᠠᠬᠤ ᠠᠷᠭ ᠠ ᠶᠢ ᠵᠢᠷᠤ᠃ ︵ ᡂᠢ ᠭᠣᠷᠲᠢᠭ ᠬᠡᠷᠡᠭᠯᠡᠬ ᠦ ᠶᠢ ᠱᠠᠭᠠᠷᠳᠠᠬᠤ ᠦᠭᠡᠢ ︶︔ ︵2︶ ᠵᠢᠷᠤᠭ 2 ᠮᠡᠲᠦ᠂ ᠴᠡᠴᠡᠭ ᠲᠡᠭᠰᠢ ᠳᠥᠷᠪᠡᠯᠵᠢ ᠨ ᠦ ᠲᠥᠪ ᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠤ ᠨ ᠨᠢᠭᠡᠳᠦᠭ ᠡᠷ ᠤᠳᠠᠭ ᠠ ᠬᠡᠷᠴᠢᠪᠡ᠃ ᠲᠠ ᡂᠢ ᠭᠣᠷᠲ ᠢᠭ ᠢ ᠬᠡᠷᠡ ᠭᠯᠡᠨ ᠲᠡᠭᠦ ᠨ ᠢ ᠬᠠᠪᠰᠤ ᠷᠤᠨ ᠬᠣᠶᠠᠳᠤ ᠭᠠᠷ ᠤᠳᠠᠭ ᠠ ᠬᠡᠷ ᠴᠢᠵᠦ᠂ ᠬᠡᠷᠴᠢᠭᠰᠡᠨ ᠮᠥᠷ ᠦᠨ ᠣᠷᠣ ᠰᠢᠵᠤ ᠪᠠᠢᠭ ᠠ ᠰᠢᠯᠤᠭ ᠤᠨ ᠢ ᠵᠢᠷᠤ ᠵᠤ ᠪᠠᠢᠭᠤᠯ᠃ ︵ ᠪᠠᠢ ᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠠᠷᠭ ᠠ ᠪᠠᠨ ᠪᠢᠴᠢᠬᠦ ᠦᠭᠡᠢ᠂ ᠵᠢᠷᠤ ᠭ ᠪᠠᠢᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠤᠯᠠ ᠮᠥᠷ ᠢᠶᠡᠨ ᠦᠯᠡᠳᠡᠭᠡ ︶ 21 . ︵ 12 ᠬᠤᠪᠢ ︶ ᠵᠢᠷᠤ ᠭ 1 ᠮᠡᠲᠦ᠂ ᠨᠢᠭᠡᠨ ᠵᠢᠭᠳᠡ ᠪᠣᠯᠬᠤ ᠵᠥᠪ ᠳᠥᠷᠪᠡᠨ ᠭᠠᠳᠠᠷ ᠭᠠᠲᠤ ᠪᠡᠶ ᠡ ᠬᠡᠯᠪᠡᠷᠢᠲᠦ ᠱᠣ ᠳ᠋ᠤ ᠳᠥᠷᠪᠡᠨ ᠭᠠᠳᠠᠷᠭ ᠤ ᠲᠠᠢ᠂ ᠭᠠᠳᠠᠷ ᠭᠤ ᠳᠡᠭᠡᠷ ᠡ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 1, 2, 3, 4 ᠭᠡᠬᠦ ᠴᠡᠭ ᠦ ᠨ ᠲᠣᠭ ᠠ ᠶᠢ ᠰᠡᠢ ᠯᠵᠦ ᠪᠣᠢ︔ ᠵᠢᠷᠤ ᠭ 2 ᠮᠡᠲᠦ᠂ ᠲᠡᠩᠴᠡᠭ ᠦᠦ ᠲᠠᠯᠠᠲᠤ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ABC ᠶᠢᠨ ᠭᠤᠷᠪᠠᠨ ᠣᠷᠣ ᠢ ᠶᠢᠨ ᠴᠡᠭ ᠲᠦ ᠲᠤ ᠰ ᠪᠦᠷ ᠢ ᠨᠢᠭ ᠡ ᠲᠣᠭᠣ ᠷᠢᠭ ᠪᠠᠢᠨ ᠠ᠃ ᠭᠤ ᠸ ᠠ ᠪᠠ ᠪᠠᠭᠠᠲᠤ ᠷ ᠲᠣᠭᠣᠷ ᠢᠭ ᠦᠰᠦᠷ ᠬᠦ ᠲᠣᠭᠯ ᠠᠭᠠᠮ ᠨᠠᠭᠠᠳᠬᠤ ᠪᠠᠷ ᠪᠣᠢ᠂ ᠲᠣᠭ ᠯᠠᠭᠠ ᠮ ᠤᠨ ᠳᠦᠷᠢᠮ ᠨᠢ᠄ ᠲᠣᠭᠯᠠᠭᠠ ᠮ ᠳ᠋ᠤ ᠣᠷᠣ ᠯᠴᠠᠭᠴᠢ ᠲᠣᠭᠣ ᠷᠢᠭ ᠠᠴᠠ ᠡᠬᠢᠯᠡᠨ ᠱᠣ ᠣᠷᠬ ᠢᠬᠤ ᠪᠦᠷᠢ ᠱᠤ ᠶᠢᠨ ᠭᠠᠵᠠ ᠷ ᠲᠠᠢ ᠬᠦᠷᠦᠯᠴᠡᠭ ᠰᠡᠨ ᠭᠠᠳᠠᠷᠭ ᠤ ᠳᠡᠭᠡᠷᠡ ᠬᠢ ᠴᠡᠭ ᠦ ᠨ ᠲᠣᠭ ᠠ ᠨᠢ ᠬᠡᠳᠦ ᠪᠣᠯ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠲᠠᠯ ᠠ ᠳᠠ ᠭᠠᠤ ᠴᠠᠭ ᠤ ᠨ ᠵᠡᠭ ᠦᠦ ᠶᠢᠨ ᠡᠰᠡᠷ ᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ ᠦᠷᠭᠦᠯᠵᠢ ᠯᠡᠨ ᠬᠡᠳᠦ ᠨ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠲᠤ ᠦᠰᠦᠷᠦ ᠨ ᠡ᠃ ᠵᠢᠱᠢᠶᠡᠯᠡᠪᠡᠯ᠄ ᠬᠡᠷᠪᠡ ᠨᠢᠭᠡᠳᠦᠭ ᠡᠷ ᠤᠳᠠ ᠭ ᠠ ᠣᠷᠬ ᠢᠭᠠᠳ ᠣᠯᠣ ᠭᠰᠠᠨ ᠴᠡᠭ ᠦᠨ ᠲᠣᠭ ᠠ ᠨᠢ 2 ᠪᠣ ᠯ ᠴᠠᠭ ᠤ ᠨ ᠵᠡᠭ ᠦᠦ ᠶᠢᠨ ᠡᠰᠡᠷᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡ ᠷ 2 ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠲᠤ ᠦᠰᠦᠷᠴᠦ᠂ ᠲᠣᠭᠣ ᠷᠢᠭ C ᠳ᠋ ᠤ ᠬᠦᠷᠦᠨ ᠡ︔ ᠬᠡᠷᠪᠡ ᠬᠣᠶᠠᠳᠤ ᠭᠠᠷ ᠤᠳᠠᠭ ᠠ ᠣᠷᠬ ᠢᠭᠠᠳ ᠣᠯᠣᠭᠰᠠᠨ ᠴᠡᠭ ᠦ ᠨ ᠲᠣᠭ ᠠ ᠨᠢ 4 ᠪᠣᠯ ᠲᠣᠭᠣ ᠷᠢᠭ C ᠠᠴᠠ ᠡᠬᠢᠯᠡᠨ ᠴᠡᠭ ᠦᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠡ ᠰᠡᠷᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ ᠦᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠨ 4 ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷ ᠲᠤ ᠦᠰᠦ ᠷᠴᠦ᠂ ᠲᠣᠭᠣ ᠷᠢᠭ A ᠲᠤ ᠬᠦᠷᠦᠨ ᠡ᠃ ︵1︶ ᠭᠣᠸ ᠠ ᠲᠠᠰᠢᠷ ᠠᠮ ᠢ ᠶᠠᠷ ᠨᠢᠭᠡ ᠤᠳᠠᠭ ᠠ ᠱᠤ ᠣᠷᠬ ᠢᠪᠠ᠂ ᠲᠡᠷᠡ ᠦᠰ ᠦᠷᠦᠭᠡᠳ ᠲᠣᠭᠣ ᠷᠢᠭ A ᠲᠤ ᠬᠦᠷᠬᠦ ᠲᠣᠬ ᠢᠶᠠᠯᠳᠤᠮᠵᠢ ᠨᠢ___ ___ ︔ ︵2︶ ᠭᠣᠸ ᠠ ᠪᠠ ᠪᠠᠭᠠᠲᠤ ᠷ ᠬᠠ ᠮᠲᠤ ᠪ ᠠᠷ ᠲᠣᠭᠣ ᠷᠢᠭ ᠦᠰ ᠦᠷᠬᠦ ᠲᠣᠭ ᠯᠠᠭᠠ ᠮ ᠨᠠᠭ ᠠᠳᠬ ᠤ ᠪᠠᠷ ᠪᠣᠯᠵᠠ ᠢ᠂ ᠭᠣᠸ ᠠ ᠲᠠᠰᠢᠷᠠᠮ ᠢᠶᠠᠷ ᠨᠢᠭᠡ ᠤᠳᠠᠭ ᠠ ᠱᠣ ᠣᠷᠬ ᠢᠵᠤ᠂ ᠪᠠᠭᠠᠲᠤ ᠷ ᠲᠠᠰᠢᠷᠠᠮ ᠢᠶᠠᠷ ᠬᠣᠶ ᠠᠷ ᠤᠳᠠᠭ ᠠ ᠱᠣ ᠣᠷᠬ ᠢᠵᠤ᠂ ᠬᠠᠮᠤᠭ ᠤᠨ ᠡᠴᠦᠰ ᠲᠦ ᠪᠡᠨ ᠲᠣᠭᠣ ᠷᠢᠭ A ᠲᠤ ᠬᠦᠷᠴᠦ ᠢᠷᠡᠭᠰᠡᠨ ᠨᠢ ᠳᠡᠢ ᠯᠦᠨ ᠡ ᠭᠡᠵᠦ ᠲᠣᠭ ᠲᠠᠪᠠ᠃ ᠡᠨ᠋ᠡ ᠬᠦ ᠲᠣᠭ ᠯᠠᠭᠠ ᠮ ᠨᠢ ᠡᠷᠬᠡ ᠲᠡᠭᠰᠢ ᠦᠦ ︖ ᠰᠢᠯᠲᠠᠭᠠᠨ ᠢᠶᠠᠨ ᠲᠣᠳᠣ ᠷᠬᠠᠢᠯᠠ᠃ 22 . ( 12 ᠬᠤ ᠪᠢ ) ᠠ ᠡ ᠬᠣᠶᠠᠷ ᠪᠠᠷᠢ ᠯᠭ ᠠ ᠶᠢᠨ ᠠᠩ ᠭᠢ ᠨᠢ ᠬᠣᠶᠠᠳᠤ ᠭᠠᠷ ᠲᠡᠰ ᠦᠨ ᠵᠠᠰᠠ ᠮᠠᠯ ᠵᠠᠮ ᠳᠡᠪᠢᠰᠬ ᠦ ᠪᠡᠷ ᠪᠣᠢ᠂ ᠠ ᠠᠩᠭᠢ ᠶᠢᠨ ᠡᠳᠦᠷ ᠪᠦᠷᠢ ᠶᠢᠨ ᠳᠡᠪᠢ ᠰᠬᠦ ᠵᠠᠮ ᠤ ᠨ ᠤᠷ ᠲᠤ ᠨᠢ ᠡ ᠠᠩᠭᠢ ᠶᠢᠨ 2 ᠳᠠᠬᠢᠨ᠂ ᠬᠣᠶᠠᠷ ᠠᠩᠭᠢ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 50 0 m ᠵᠠᠰᠠᠮᠠᠯ ᠵᠠᠮ ᠳᠡᠪᠢᠰᠬ ᠦ ᠳ᠋ᠦ ᠠ ᠠᠩᠭ ᠢ ᠨᠢ ᠡ ᠠᠩ ᠭᠢ ᠠ ᠴᠠ ᠳᠤᠲ ᠠᠭᠤ 5 ᠡᠳᠦᠷ ᠬᠡᠷᠡᠭᠯᠡᠬᠦ ᠶᠢ ᠮᠡᠳᠡ ᠪᠡ᠃ ︵1︶ ᠠ ᠡ ᠬᠣᠶᠠ ᠷ ᠪᠠᠷᠢᠯ ᠭ ᠠ ᠶᠢᠨ ᠠᠩ ᠭᠢ ᠡᠳᠦᠷ ᠪᠦᠷ ᠢ ᠬᠡᠳᠦᠢ ᠮᠧᠲ᠋ᠷ ᠵᠠᠮ ᠳᠡᠪᠢᠰᠬᠦ ᠪᠣᠢ ︖ ︵2︶ ᠮᠠᠨ ᠤ ᠬᠣᠲ ᠠ ᠤᠷ ᠲᠤ ᠨᠢ 36 00 m ᠪᠣᠯᠬᠤ ᠬᠣᠶᠠᠳᠤ ᠭᠠᠷ ᠳᠡᠰ ᠦ ᠨ ᠵᠠᠰᠠᠮᠠᠯ ᠵᠠᠮ ᠳᠡᠪᠢ ᠰᠬᠦ ᠪᠡᠷ ᠲᠥᠯᠥ ᠪᠯᠡᠵᠦ ᠪᠣᠢ᠂ ᠪᠠᠷᠢ ᠯᠭ ᠠ ᠶᠢᠨ ᠬᠡᠷᠡᠭᠴᠡᠭᠡ ᠡᠴᠡ ᠪᠣᠯᠵᠤ ᠠ ᠡ ᠬᠣᠶᠠ ᠷ ᠪᠠᠷᠢ ᠯᠭ ᠠ ᠶᠢᠨ ᠠᠩᠭ ᠢ ᠪᠡᠶᠡᠯᠡᠭᠦᠯᠬᠦ ᠪᠡᠷ ᠪᠣᠯᠵᠠᠢ᠃ ᠬᠡᠷᠪᠡ ᠠ ᠠᠩᠭᠢ ᠶᠢᠨ ᠡᠳᠦᠷ ᠪᠦᠷᠢ ᠶᠢᠨ ᠵᠠᠷᠤᠴᠠᠭᠤᠯᠬᠤ ᠰᠦᠢᠳ ᠬᠡᠯ ᠨᠢ 1.2 ᠲᠦ ᠮᠡᠨ ᠲᠥᠭᠥᠷᠢᠭ ᠂ ᠡ ᠠᠩᠭᠢ ᠶᠢᠨ ᠡᠳᠦᠷ ᠪᠦᠷᠢ ᠵᠠᠷᠤᠴᠠᠭᠤᠯᠬ ᠤ ᠰᠦᠢᠳᠬᠡᠯ ᠨᠢ 0.5 ᠲᠦᠮᠡᠨ ᠲᠥᠭ ᠥᠷᠢᠭ ᠪᠣᠯ ᠶᠡᠷᠦᠩᠬ ᠡᠢ ᠵᠠᠷᠤᠳ ᠠᠯ ᠨᠢ 40 ᠲᠦᠮᠡᠨ ᠲᠥᠭᠥᠷᠢ ᠭ ᠡᠴᠡ ᠬᠡᠲᠦᠷ ᠡᠬᠦ ᠦᠭᠡᠢ ᠪᠠᠢᠳᠠᠯ ᠳᠣᠣ ᠷ ᠠ᠂ ᠡ ᠠᠩᠭᠢ ᠶᠢ ᠠᠳᠠᠭ ᠲᠤ ᠪ ᠠᠨ ᠬᠡᠳᠦᠢ ᠡᠳᠦᠷ ᠲᠣᠬ ᠢᠷᠠᠭᠤᠯᠬᠤ ᠴᠢᠬᠤ ᠯᠠ ᠲᠠᠢ ᠪᠣᠢ ︖ 23. ︵ 12 ᠬᠤᠪᠢ ︶ ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ AB ᠪᠣᠯ ⊙ O ᠶᠢᠨ ᠳ᠋ᠢᠶᠠᠮᠧᠲ᠋ ᠷ᠂ AC ᠪᠣᠯ ⊙ O ᠶᠢᠨ ᠨᠢᠭᠡᠨ ᠬᠥᠪᠴᠢ᠂ ᠴᠡᠭ P ᠨᠢ ⊙ O ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠪᠥᠭᠡᠳ PA PC ,PD // AC ᠂ BA ᠶᠢᠨ ᠦᠷ ᠭᠦᠯᠵᠢᠯᠡᠯ ᠱᠤᠭᠤ ᠮ ᠯᠤ ᠭ ᠠ ᠴᠡᠭ D ᠳ᠋ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠃ ︵1︶ ᠪᠠᠲᠤᠯ ᠠᠭᠳᠠᠬ ᠤ ᠨᠢ᠄ PD ᠨᠢ ⊙ O ᠶᠢᠨ ᠰᠢᠬᠠᠭᠴ ᠢ ᠰᠢᠯᠤᠭᠤ ᠨ︔ ︵2︶ ᠬᠡᠷᠪᠡ tan∠P AC 23 AC 12 ᠪᠣᠯ ᠳ᠋ᠢᠶᠠᠮᠧᠲ᠋ ᠷ AB ᠤᠨ ᠤᠷ ᠲᠤ ᠶᠢ ᠣᠯ᠃ 24 . ︵ 12 ᠬᠤᠪᠢ ︶ ᠤᠩ ᠰᠢᠵᠤ ᠣᠢ ᠯᠠᠭᠠ ᠬᠤ᠄ ᠮᠠᠲ᠋ᠧᠷᠢᠶ ᠠᠯ ᠨᠢᠭᠡ ᠄ ᠬᠡᠷᠪᠡ ᠭᠤᠷᠪᠠᠨ ᠲᠡᠭ ᠪᠤᠰᠤ ᠪᠣᠳᠠᠲᠤ ᠲᠣᠭ ᠠ ,𝑦 ,𝑧 ᠨᠢ ᠵᠥᠪᠬᠡᠨ ᠲᠡᠭᠦᠨ ᠦ ᠳᠣᠲ ᠣᠷᠠᠬᠢ ᠨᠢᠭᠡ ᠲᠣᠭᠠᠨ ᠤ ᠲᠣᠩ ᠭᠣᠷᠠᠭᠣ ᠲᠣᠭ ᠠ ᠨᠢ ᠨᠥᠭᠥᠭᠡ ᠬᠣᠶᠠᠷ ᠲᠣᠭᠠ ᠨ ᠤ ᠲᠣᠩ ᠭᠤᠷᠠᠭᠤ ᠲᠣᠭᠠ ᠨ ᠤ ᠨᠡᠢ ᠯᠡᠪᠦᠷᠢ ᠲᠠᠢ ᠲᠡᠩᠴᠡᠭᠦᠦ ᠭᠡᠳᠡᠭ ᠢ ᠬᠠ ᠩᠭᠠ ᠪᠠᠯ᠂ ᠡᠨ᠋ᠡ ᠬᠦ ᠭᠤᠷᠪᠠᠨ ᠲᠣᠭ ᠠ ,𝑦 ,𝑧 ᠨᠢ ︽ ᠡᠶ ᠡ ᠵᠣᠬᠢᠴᠠᠩᠭᠤ ᠢ ᠭᠤᠷᠪᠠᠨ ᠲ ᠣᠭᠠᠨ ᠤ ᠳᠤᠭᠤᠢᠯᠠᠩ ︾ ᠢ ᠪᠦᠷᠢᠯᠳ ᠦᠭᠦᠯᠦᠨ ᠡ᠃ ᠮᠠᠲ᠋ᠧᠷᠢᠶ ᠠᠯ ᠬᠣᠶᠠᠷ᠄ ᠬᠡᠷᠪᠡ ᠤ ᠨ ᠲᠤᠬ ᠠᠢ ᠨᠢᠭᠡ ᠥᠯᠥ ᠮᠡᠳᠡ ᠭᠳᠡᠭᠴᠢᠲᠡᠢ ᠬᠣᠶᠠᠳᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ 2 𝑏 𝑐 0 ≠ 0 ᠦᠨ ᠬᠣᠶᠠᠷ ᠢᠵᠠᠭᠤᠷ ᠨᠢ ᠲᠤᠰ ᠪᠦᠷᠢ 1 , 2 ᠪᠣᠯ 1 2 − 𝑏𝑎 , 1 ⋅ 2 𝑐𝑎 ᠪᠣᠯᠣ ᠨ ᠠ᠃ ᠠᠰᠠᠭᠤᠳᠠᠯ ᠢ ᠰᠢᠢᠳ ᠪᠦᠷᠢᠯᠡᠬᠦ᠄ ︵1︶ ᠲᠠ ︽ ᠡᠶ ᠡ ᠵᠣᠬᠢᠴᠠᠩᠭᠤ ᠢ ᠭᠤᠷᠪᠠᠨ ᠲᠣᠭᠠ ᠨ ᠤ ᠳᠤᠭᠤᠢᠯᠠᠩ ︾ ᠪᠦᠷᠢᠯᠳ ᠦᠭᠦᠯᠵᠦ ᠳᠡᠢᠯᠬᠦ ᠭᠤᠷᠪᠠᠨ ᠪᠣᠳᠠᠲᠤ ᠲᠣ ᠭ ᠠ ᠶᠢ ᠪᠢ ᠴᠢᠪᠡᠯ __ ____ ︔ ︵2︶ ᠬᠡᠷᠪᠡ 1 , 2 ᠨᠢ ᠦ ᠨ ᠲᠤᠬ ᠠᠢ ᠲᠡᠭᠰᠢᠳᠬ ᠡᠯ 2 𝑏 𝑐 0 ︵ , 𝑏, 𝑐 ᠪᠦᠷ 0 ᠲᠡᠢ ᠲᠡᠩ ᠴᠡᠬᠦ ᠦᠭᠡᠢ ︶ ᠦᠨ ᠬᠣᠶᠠᠷ ᠢᠵᠠᠭᠤᠷ᠂ 3 ᠨᠢ ᠤ ᠨ ᠲᠤᠬ ᠠᠢ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ 𝑏 𝑐 0 ︵b ,c ᠪᠦᠷ 0 ᠲᠡᠢ ᠲᠡᠩ ᠴᠡᠬᠦ ᠦᠭᠡᠢ ︶ ᠦᠨ ᠬᠠᠷᠢᠭ ᠤ ᠮᠥᠨ᠃ ᠪᠠᠲᠤᠯ ᠠᠭᠳᠠᠬᠤ ᠨᠢ᠄ 1 , 2 , 3 ᠨᠢ ︽ ᠡᠶ ᠡ ᠵᠣᠬᠢᠴᠠᠩᠭᠤ ᠢ ᠭᠤᠷ ᠪᠠᠨ ᠲᠣ ᠭᠠᠨ ᠤ ᠳᠤ ᠭᠤᠢᠯᠠᠩ ︾ ᠪᠦᠷᠢᠯ ᠳᠦᠭᠦᠯᠵᠦ ᠪᠣᠯᠣ ᠨ ᠠ︔ ︵3︶ ᠬᠡᠷᠪᠡ ᠭᠤᠷᠪᠠᠨ ᠴᠡᠭ 𝐴 , 𝑦1 , 𝐵 1, 𝑦2 , 𝐶 3, 𝑦3 ᠨᠢ ᠴᠥᠮ ᠤᠷᠪᠠᠭ ᠤ ᠬᠠᠷᠢ ᠴᠠᠯᠲᠤ ᠹᠦᠩᠺ ᠼ 𝑦 4𝑥 ᠶᠢᠨ ᠵᠢᠷᠣ ᠮᠠᠭ ᠳᠡᠭᠡᠷ ᠡ ᠣᠷᠣ ᠰᠢᠬᠤ ᠪᠥᠭᠡᠳ ᠭᠤᠷᠪᠠ ᠨ ᠴᠡᠭ ᠦ ᠨ ᠭᠤᠯᠳᠤ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ ᠶᠠᠭ ᠰᠠᠢ ᠬᠠᠨ ︽ ᠡᠶ ᠡ ᠵᠣᠬ ᠢᠴᠠᠩᠭᠤ ᠢ ᠭᠤᠷᠪᠠᠨ ᠲᠣᠭᠠᠨ ᠤ ᠳᠤᠭᠤ ᠢᠯᠠᠩ ︾ ᠪᠦᠷᠢᠯ ᠳᠦᠭᠦᠯᠪᠡᠯ ᠪᠣᠳᠠᠲᠤ ᠲᠣ ᠭ ᠠ ᠤᠨ ᠬᠡᠮᠵᠢᠭᠳ ᠡᠯ ᠢ ᠣᠯ᠃ 25. ︵ 14 ᠬᠤᠪᠢ ︶ ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠥᠭᠭᠦ ᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠬᠣᠶᠠᠳ ᠤᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺ ᠼ 𝑦 2 𝑏 𝑐 ≠ 0 ᠶᠢᠨ ᠵᠢᠷᠤ ᠮᠠᠭ ᠨᠢ ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠯᠦᠭᠡ A 1,0 ,B 4 ,0 ᠬᠣᠶᠠᠷ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠵᠤ᠂ y ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠯᠦᠭᠡ ᠴᠡᠭ C ᠳ᠋ ᠤ ᠣ ᠭᠲᠣᠯᠣᠯᠴᠠᠨ᠂ ᠰᠢᠯᠤᠭᠤ ᠨ 𝑦 − 12 2 ᠨᠢ B, C ᠬᠣᠶᠠᠷ ᠴᠡᠭ ᠢ ᠳᠠ ᠭᠠᠷᠢ ᠨ ᠠ᠃ ︵1︶ ᠬᠣᠶᠠᠳᠤ ᠭᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺ ᠼ ᠦ ᠨ ᠠᠨ᠋ᠠᠯᠢᠰ ᠢᠯᠡᠷᠬᠡᠢᠯᠡᠯ ᠢ ᠰᠢᠭᠤ ᠳ ᠪᠢᠴᠢᠨ ᠭᠠᠷᠭ ᠠᠪᠠᠯ __ ____ ︔ ︵2︶ ᠰᠢᠯᠤᠭᠤ ᠨ BC ᠶᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵᠢᠭᠦᠯᠦᠨ ᠡ᠂ ᠰᠢᠯᠤᠭᠤ ᠨ BC ᠨᠢ ᠬᠠᠶᠠ ᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠯᠤᠭ ᠠ ᠴᠤᠷ ᠤ ᠨ ᠭᠠ ᠭᠴᠠ ᠶ ᠡᠷᠦᠩ ᠬᠡᠢ ᠴᠡᠭ Q ᠪᠠᠢᠬᠤ ᠦᠶ ᠡᠰ᠂ ᠴᠡᠭ Q ᠦᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ ︔ ︵3︶ ᠳᠡᠭᠡᠷᠡᠬᠢ ︵2︶ ᠲᠠᠬᠢ ᠴᠡᠭ Q ᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠤ ᠨ QE // y ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠢ ᠪᠠᠢᠭᠤᠯᠪᠠᠯ x ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠯᠦᠭᠡ ᠴᠡᠭ E ᠳ᠋ ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠ ᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ᠴᠡᠭ M ᠨᠢ ᠬᠠ ᠶᠠᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠬᠥᠳᠡᠯ ᠬᠦ ᠴᠡᠭ᠂ ᠴᠡᠭ N ᠨᠢ x ᠲᠡᠩᠭᠡᠯᠢᠭ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠬᠥᠳᠡᠯᠬᠦ ᠴᠡᠭ ᠪᠣᠯ △ BOC ᠲᠠᠢ ᠲᠥᠰᠲ ᠡᠢ ᠪᠣᠯᠬ ᠤ E, M, N ᠭᠤᠷᠪᠠᠨ ᠴᠡᠭ ᠢᠶᠡ ᠷ ᠣᠷᠣ ᠢ ᠶᠢᠨ ᠴᠡᠭ ᠪᠣᠯᠭᠠᠭᠰᠠ ᠨ ᠲ ᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠲᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ︵ ᠡᠭᠦᠨ ᠳ᠋ᠦ M ᠨᠢ ᠲᠡᠭᠰᠢ ᠥᠨ ᠴᠥᠭ ᠦᠨ ᠣᠷᠣ ᠢ ᠶᠢᠨ ᠴᠡᠭ ︶ ᠣᠷᠣ ᠰᠢᠨ ᠠ ᠦᠦ ︖ ᠬᠡᠷᠪᠡ ᠣᠷᠣ ᠰᠢᠪᠠᠯ ᠨᠥᠬᠥᠴᠡᠯ ᠢ ᠬᠠᠩᠭᠠᠬ ᠤ ᠴᠡᠭ M ᠦ ᠨ ᠪᠣᠳᠠ ᠭᠠᠯᠢ ᠲᠣᠭ ᠠ ᠪᠠ ᠲᠡᠭᠦᠨ ᠦ ᠨᠥᠬᠥᠴᠡᠯ ᠳ᠋ ᠦ ᠲᠣᠬ ᠢᠷᠠᠭᠰᠠᠨ ᠴᠡᠭ M ᠦᠨ ᠪᠠᠢᠷᠢ ᠰᠢᠯ ᠠᠴᠠ ᠨᠢᠭᠡ ᠶᠢ ᠰᠢᠭᠤᠳ ᠪ ᠢᠴᠢᠵ ᠦ ᠭᠠ ᠷᠭ ᠠ ︔ ᠬᠡᠷᠪᠡ ᠣ ᠷᠣᠰᠢᠬᠤ ᠦᠭᠡᠢ ᠪᠣᠯ ᠰᠢᠯᠲᠠᠭᠠᠨ ᠢ ᠲᠣᠳ ᠣᠷᠬᠠᠢ ᠯᠠ᠃ 26 . ︵ 14 ᠬᠤᠪᠢ ︶ ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠲ ᠦ ABC D ᠳ᠋ᠦ ᠴᠡᠭ P ᠨᠢ ᠲᠠᠰᠢᠭᠤ ᠪᠴᠢ AC ᠶᠢᠨ ᠣᠷᠣ ᠰᠢᠬᠤ ᠰᠢᠯᠤᠭᠤ ᠨ ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠬᠥ ᠳᠡᠯᠬᠦ ᠴᠡᠭ᠂ PD ᠶᠢ ᠬᠣᠯᠪᠣᠵᠤ᠂ ᠴᠡᠭ P ᠶᠢ ᠳᠠᠭᠠᠷᠢᠭᠤᠯ ᠤᠨ PE ⊥ PD ᠶᠢ ᠪᠠᠢᠭᠤᠯᠤ ᠨ ᠰᠢᠯᠤᠭᠤ ᠨ AB ᠯᠤᠭ ᠠ ᠴᠡᠭ E ᠳ᠋ ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ P ᠶᠢ ᠳᠠᠭᠠᠷᠢᠭᠤᠯ ᠤᠨ MN ⊥ AB ᠶᠢ ᠪᠠᠢᠭᠤᠯᠤ ᠨ ᠰᠢᠯᠤᠭᠤ ᠨ CD ᠯᠤᠭ ᠠ ᠴᠡᠭ M ᠳ᠋ ᠤ ᠣᠭᠲ ᠣᠯᠣᠯ ᠴᠠᠵᠤ᠂ ᠰᠢᠯᠤᠭᠤ ᠨ AB ᠯᠤᠭ ᠠ ᠴᠡ ᠭ N ᠳ᠋ᠤ ᠣᠭᠲ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠃ AB 4√ 3, AD 4 ᠃ ︵1︶ ᠵᠢᠷᠤ ᠭ 1 ᠮᠡᠲᠦ᠂ ① ᠴᠡᠭ P ᠨᠢ ᠬᠡᠷᠴᠢᠮ AC ᠳᠡᠭᠡᠷ ᠡ ᠪᠠᠢᠬᠤ ᠦᠶᠡᠰ ᠲᠦ ∠PDM ᠪᠠ ∠EPN ᠦᠨ ᠲᠣᠭ ᠠ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨ ᠦ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ᠄ ∠PDM ∠EPN ︔ ② DPPE ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ __ ____ ︔ ︵2︶ ᠵᠢᠷᠤᠭ 2 ᠮᠡᠲᠦ᠂ ᠴᠡᠭ P ᠨᠢ CA ᠶᠢᠨ ᠦᠷᠭᠦᠯᠵᠢᠯ ᠡᠯ ᠱᠤᠭᠤᠮ ᠳᠡᠭᠡᠷ ᠡ ᠪᠠᠢᠬᠤ ᠦᠶᠡᠰ ᠲᠦ᠂ ︵1︶ ᠳ᠋ᠡᠬᠢ ᠳ᠋ᠦᠩ ② ᠪᠠᠢᠭᠤᠯ ᠤᠭᠳᠠᠨ ᠠ ᠦᠦ ︖ ᠬᠡᠷᠪᠡ ᠪᠠᠢᠭᠤᠯᠤ ᠭᠳᠠᠪᠠᠯ᠂ ᠪᠠᠲᠤᠯ ᠠᠵᠤ ᠥᠭᠭᠦᠭᠡᠷᠡᠢ ᠬᠡᠷᠪᠡ ᠪᠠᠢᠭᠤᠯᠤ ᠭᠳᠠᠬᠤ ᠦᠭᠡᠢ ᠪ ᠣᠯ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠲᠣ ᠳᠣᠷᠬᠠᠢᠯᠠ ︔ ︵3︶ ᠵᠢᠷᠤ ᠭ 3 ᠮᠡᠲᠦ᠂ ᠬᠡᠷᠴᠢᠮ PD, PE ᠪᠠᠷ ᠵᠡᠷᠭᠡᠯᠡᠳᠡᠭᠡ ᠲᠠᠯ ᠠ ᠪᠣᠯᠭᠠᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠲᠦ PEFD ᠶᠢ ᠪᠠᠢ ᠭᠤᠯᠤ ᠨ ᠠ᠃ PM ᠦᠨ ᠤᠷᠲᠤ ᠶᠢ x ᠭᠡᠵᠦ᠂ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠲᠦ PEFD ᠶᠢᠨ ᠲᠠᠯᠠᠪᠠᠢ ᠶᠢ y ᠭᠡᠪᠡᠯ 𝑦 ᠪᠣᠯᠣ ᠨ ᠤᠨ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠤ ᠬᠢ ᠹᠦ ᠩᠺᠼ ᠦᠨ ᠬᠣᠯᠪᠣᠭᠳ ᠠᠯ ᠤ ᠨ ᠢᠯ ᠡᠷᠬᠡᠢᠯᠡᠯ ᠵᠢᠴᠢ y ᠶᠢᠨ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠢ ᠰᠢ ᠭᠤᠳ ᠪ ᠢᠴᠢᠨ ᠭᠠᠷᠭ ᠠ᠃