2020 ᠣᠨ ᠤ ᠲᠦᠩᠯᠢᠶᠣᠣ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

20 20 ᠣᠨ ᠤ ᠲᠦᠩ ᠯᠢᠶᠣᠤ ᠬᠤᠳ ᠠ ᠶᠢᠨ ᠠ ᠩᠬ ᠠ ᠲ ᠣᠮᠳᠠᠳᠤ ᠶᠢᠨ ᠰᠤᠷ ᠤᠭᠴᠢ ᠶᠢᠨ ᠰᠤ ᠷᠭᠠᠭ ᠤᠯᠢ ᠲᠠᠪᠰᠢ ᠬᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠲᠣᠭᠠᠨ ᠤ ᠤᠬᠠᠭᠠ ᠨ ᠨᠢᠭᠡ ᠂ ᠰᠣᠩ ᠭᠤᠬ ᠤ ᠰᠡᠳᠤ ᠪ᠃ ( ᠪᠠᠭ ᠠ ᠰᠡᠳᠤᠪ ᠪᠦᠷᠢ 3 ᠬ ᠤᠪᠢ᠂ ᠨᠠᠢᠳᠠ 3 0 ᠬᠤ ᠪᠢ) 1. 2020 ᠣᠨ ᠤ ᠮᠠᠨ ᠤ ᠬᠣᠲ ᠠ ᠶᠢᠨ ᠠᠩᠬ ᠠ ᠳᠤᠮ ᠳᠠᠳᠤ ᠲᠡᠭᠦᠰ ᠬᠦ ᠰᠤ ᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ 30000 ᠬᠦᠮᠦᠨ ᠢ ᠳᠠᠪᠠ ᠪᠠ᠂ ᠡᠭᠦᠨ ᠳᠦ 30000 ᠶᠢ ᠰᠢᠨᠵᠢᠯᠡ ᠬᠦ ᠤᠬᠠ ᠭᠠᠨᠴᠢ ᠲᠣᠭ ᠠ ᠲᠡᠮᠲᠡᠭᠯᠡᠬᠦ ᠠᠷᠭ ᠠ ᠪᠠᠷ ᠢ ᠯᠡᠷᠬᠡ ᠢᠯᠡᠪ ᠡᠯ ( ) A. 0.3 × 10 6 B. 3 × 10 4 C. 30 × 10 3 D. 3 ᠲᠦᠮᠡ 2. ᠳᠠᠷᠠᠭᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠶᠢᠨ ᠪᠤᠷᠤᠭᠤ ᠨᠢ ( ) A. 2𝑎 ᠪᠣᠯ 2 ᠲᠣᠭ ᠠ 𝑎 ᠶᠢᠨ ᠨᠡᠢᠯᠡᠪᠦᠷ ᠢ B. 2𝑎 ᠪᠣᠯ 2 ᠪᠠ ᠲᠣᠭ ᠠ 𝑎 ᠶᠢᠨ ᠦᠷᠡᠵᠢ ᠪᠦᠷᠢ C. 2𝑎 ᠪᠣᠯ ᠨᠢᠭᠡᠨᠭᠡᠰᠢ ᠭᠦᠨᠲᠦ D. 2𝑎 ᠪᠣᠯ ᠲᠡᠭᠰᠢ ᠲᠣ ᠭ ᠠ 3. ᠳᠠᠷᠠᠭᠠᠬᠢ ᠶᠠᠪᠤ ᠳᠠᠯ ᠤ ᠳ ᠤ ᠨ ᠳᠣᠲᠣᠷ ᠠ ᠪᠣᠯᠣᠯᠴᠠᠭᠠᠭᠦᠢ ᠶᠠ ᠪᠤᠳᠠᠯ ᠪᠣᠯᠬᠤ ᠨᠢ ( ) A. ᠮᠤᠳᠤ ᠰᠠᠬᠢᠨ ᠲᠠᠤᠯᠠᠢ ᠬᠦᠯᠢᠶᠡᠬ ᠦ B. ᠪᠤᠳ ᠣᠩ ᠲᠤ ᠲᠤᠷ ᠠ ᠠᠴᠠ ᠵᠢᠭ ᠠᠰᠤ ᠪ ᠠᠷᠢᠬᠤ C. ᠤᠰᠤ ᠨ ᠠᠴᠠ ᠰᠠᠷ ᠠ ᠰᠢᠭᠦᠬᠦ D. ᠵᠠᠭᠤ ᠨ ᠠᠯᠬᠤ ᠮ ᠤᠨ ᠦᠵ ᠦᠭᠦᠷ ᠡᠴᠡ ᠵᠤ ᠭᠤᠰᠤ ᠨ ᠨᠦᠬᠡ ᠶᠢ ᠣ ᠨᠣᠭᠠᠬᠤ 4. ᠵᠢᠷᠤ ᠭ ᠮᠠᠲᠦ᠂ ᠨᠢᠭᠡ ᠢᠵᠢ ᠭᠤᠷᠪᠠᠯ ᠵᠢᠨ ᡂᠢ ᠶᠢ ᠳᠠᠷᠠᠭᠠ ᠬᠢ ᠮᠡᠲᠦ ᠪᠠᠢ ᠷᠢ ᠪᠠᠷ ᠦᠷᠦ ᠵᠦ ᠲᠠᠯᠪᠢᠪᠠ᠂ ᠡᠭᠦᠨ ᠳᠦ ∠α ᠪᠠ ∠β ᠨᠢ ᠬᠠᠷᠢᠯ ᠴᠠᠨ ᠦᠯᠡᠳᠡ ᠪᠦᠷᠢ ᠪᠣᠯᠬᠤ ᠪᠠᠢᠷᠢ ᠪᠠᠷ ᠲᠠᠯᠪᠢᠭᠰᠠᠨ ᠨᠢ ( ) A. B. C. D. 5. 𝑥 ᠤ ᠨ ᠲᠤᠬ ᠠᠢ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ 𝑘 𝑥 2 − 6𝑥 + 9= 0 ᠲᠤ ᠪᠣᠳᠠᠳᠤ ᠲᠣᠭᠠ ᠨ ᠤ ᠢᠵᠠᠭᠤᠷ ᠲᠠᠢ᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑘 ᠦᠨ ᠬᠡᠮ ᠵᠢᠭᠳᠡ ᠯ ᠠᠪᠬᠤ ᠬᠡᠪᠴᠢ ᠶ ᠡ ᠨᠢ ( ) A. 𝑘 1 ᠪᠦᠭᠡᠳ 𝑘≠ 0 B. 𝑘 1 C. 𝑘 ≤ 1 ᠪᠦᠭᠡᠳ 𝑘 ≠ 0 D. 𝑘 ≤ 1 6. ᠭᠣᠷᠲ ᠢᠭ ᠢᠶᠠᠷ ᠵᠢᠷᠤ ᠭ ᠪᠠᠢᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠤᠷᠤ ᠮᠦᠷ ᠢ ᠦᠨᠳᠦᠰᠦᠯᠡᠵᠦ᠂ ᠠᠰ ᡂᠢ ᠶᠢ ᠬᠡᠷᠡᠭᠯᠡᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠲᠣᠳᠤ ᠭᠠᠳᠤ ᠲᠥᠪ ᠢ ᠪᠦ ᠲᠦᠮᠵᠢᠲ ᠡᠢ ᠣᠯᠵᠤ ᠳᠡᠢᠯᠬᠦ ᠨᠢ ( ) A. B. C. D. 7. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ 𝑃𝐴 , 𝑃𝐵 ᠨᠢ ᠲᠣᠰ ᠲᠣᠰ ⊙ 𝑂ᠲᠠᠢ ᠬᠤᠶᠠ ᠷ ᠴᠡᠭ 𝐴 , 𝐵 ᠳᠤ ᠰᠢᠬᠠᠯᠴᠠᠨ ᠠ᠂ ∠𝑃 = 72 ° ᠲᠡᠭᠡᠪᠡᠯ ∠ 𝐶 = ( ) A. 108° B. 72° C. 54° D. 36° 8. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ 𝐴𝐷 ᠪᠣᠯ ∆ 𝐴𝐵𝐶 ᠤᠨ ᠳᠣᠮᠳᠠ ᠱᠤᠭᠤᠮ ᠂ ᠳᠦᠷᠪᠡᠯᠵᠢ ᠨ𝐴𝐷𝐶𝐸 ᠪᠣᠯ ᠵᠢᠰᠢᠭᠦᠯᠵᠢᠨ᠂ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠨᠦᠬᠦᠴᠡᠯ ᠡᠴᠡ ᠨᠢᠭᠡ ᠶᠢ ᠨᠠᠮᠠᠪᠡᠯ ᠂ ᠵᠢᠰᠢᠭᠦᠯᠵᠢᠨ 𝐴𝐷𝐶𝐸 ᠶᠢᠨ ᠷᠣᠮᠪ ᠪᠣᠯᠬ ᠤ ᠶᠢ ᠮᠠᠭᠠᠳᠯ ᠠᠵᠤ ᠳᠡᠢ ᠯᠬᠦ ᠨᠢ ( ) A. ∠B AC = 90° B. ∠D AE = 90° C. AB = AC D. AB = AE 9. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ OC ᠨᠢ ᠬᠣᠣ ᠰ ᠮᠤᠷᠤ ᠢ ᠱᠤᠭᠤᠮ 𝑦= 𝑘𝑥 ᠲᠠᠢ ᠴᠡᠭ A ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠ ᠬᠤ ᠪᠦᠭᠡ ᠳ OC :OA = 5: 3 ᠂ ᠬᠡᠷᠪᠡ ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦᠭ ᠲᠦ ABC D ᠶᠢᠨ ᠲᠠᠯᠠᠪᠠᠢ ᠨᠢ 8 ᠪᠦᠭᠡᠳ AB // 𝑥 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠪᠣᠯᠪᠠᠯ᠂ 𝑘 ᠶᠢᠨ ᠬᠡᠮᠵ ᠢᠭᠳᠡᠯ ᠨᠢ ( ) A.18 B.50 C.12 D. 2009 10. ᠳᠠᠷᠠᠭᠠᠬᠢ ᠦᠭᠦᠯᠡ ᠯ ᠳᠣᠲ ᠣᠷ ᠠ ᠠᠴᠠ ᠲᠠᠰᠢᠷᠠᠮ ᠢᠶᠠᠷ ᠨᠢᠭᠡ ᠶᠢ ᠰᠤᠭᠤᠯ ᠵᠤ ᠠᠪᠬᠤ ᠳ ᠤ ᠵᠢᠩᠬ ᠢᠨᠢ ᠦᠭᠦᠯᠡᠯ ᠪᠣᠯᠬᠤ ᠲᠣᠬ ᠢᠶᠠᠯᠳᠤᠮᠵᠢ ᠨᠢ ( ) (1) ᠶᠤᠰᠤ ᠭᠦᠢ ᠲᠣᠭ ᠠ ᠪᠣᠯ ᠴᠦᠮ ᠬᠢᠵᠠᠭᠠ ᠷᠬᠦᠢ ᠠᠷ ᠪᠠᠲᠤ ᠶᠢᠨ ᠪᠣᠲᠠᠷᠬ ᠠᠢ ᠃ (2) ᠦᠷᠠᠵᠢ ᠭᠳᠡᠭᠦᠨ ᠪᠣᠯᠭᠠᠨ ᠵᠠᠳᠠᠯᠪᠠᠯ᠄ 𝑎𝑥 2− 𝑎= 𝑎( 𝑥+ 1) (𝑥 − 1) ᠃ (3) ᠢᠷ ᠮᠡᠭ ᠦᠨ ᠤᠷᠲᠤ ᠨᠢ 1𝑐𝑚 ᠪᠣᠯᠬᠤ ᠲᠡᠭᠰᠢ ᠳᠦᠷ ᠪᠡᠯᠵᠢᠨ ᠪᠡᠶ ᠡ ᠶᠢ ᠨ ᠭᠠᠳᠠᠷᠭ ᠤ ᠶᠢᠨ ᠳᠡᠯᠭᠡᠮᠡᠯ ᠵᠢᠷᠤ ᠭ ᠤᠨ ᠬᠦᠷᠢᠶᠡ ᠨ ᠦ ᠤᠷᠲᠤ ᠨᠢ ᠯᠠᠪᠲᠠᠢ 14 𝑐𝑚 ᠃ (4) ᠨᠤᠮᠤ ᠶᠢᠨ ᠤᠷᠲᠤ ᠨᠢ 20 𝜋᠂ ᠲᠡᠯᠠᠪ ᠠᠢ ᠨᠢ 240 𝜋𝑐𝑚 2 ᠪᠣᠯᠬᠤ ᠳᠡᠪᠢᠭᠦᠷ ᠬᠡᠯᠪᠡᠷᠢᠲᠦ ᠶᠢᠨ ᠲ ᠥᠪ ᠦᠨᠴᠦᠭ ᠨᠢ 120° A. 14 B. 12 C. 34 D. 1 ᠬᠤᠶᠠᠷ᠂ ᠬᠠᠭᠠᠯᠲ ᠠ ᠲᠠᠭᠯᠠᠬᠤ ᠰ ᠡᠲᠥᠪ ᠃ ( ᠰᠡᠲᠥ ᠪ ᠪᠦᠷᠢ 3 ᠬᠤ ᠪᠢ᠂ ᠨᠡᠢᠲᠡ 21 ᠬᠤᠪᠢ) 11. ᠪᠣᠳᠣ᠄ (1) (3 .14 − π) 0 = ( )᠂ (2) 2cos 45° = ( ) ᠂ (3) − 1 2 = ( ) 12. ᠬᠡᠷᠪᠡ ᠨᠢᠭᠡ ᠪᠦᠯᠦᠭ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢ ᠮᠲᠠ 3, 𝑎,3 ,5, 3 ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠢ ᠲᠣᠭ ᠠ ᠨᠢ 3 ᠪᠣᠯ᠂ ᠡᠨᠡ ᠪᠦᠯᠦᠭ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢ ᠮᠲᠠ ᠶᠢᠨ᠄ ( 1) ᠮᠤᠳ᠋ ᠨᠢ ( ) ᠂ (2) 𝑎 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ ( )᠂ (3) ᠰᠠᠷᠨᠢᠴᠡ ᠨ ᠢ ( )᠃ 13. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ 𝑂 ᠪᠣᠯ ᠰᠢᠯᠣᠭᠤ ᠨAB ᠳᠡᠭᠡ ᠷᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠂ ∠A OC = 50°17′28′′ ᠪᠣᠯᠤ ᠨ ᠠ᠂ ᠳᠡᠭᠡ ᠪᠡᠯ ∠BOC ᠦᠨ ᠭᠷᠠᠳᠦᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠨᠢ ( ) 14. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠶᠡᠬᠡ ᠪᠠᠭ ᠠ ᠨᠢ ᠠᠳᠠᠯᠢᠬᠠᠨ ᠵᠢᠵᠢᠭ ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪᠡᠯᠵᠢ ᠨ ᠢᠶᠡᠷ ᠶᠡᠬᠡ ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪᠡᠯᠵᠢ ᠨ ᠢ ᠵᠦᠢᠨ ᠠ᠂ ᠨᠢᠭᠡᠳᠦᠭ ᠡᠷ ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪᠡᠯᠵᠢ ᠨ ᠢ ᠵᠦᠢᠬᠦ ᠳ ᠦ 4 ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪᠡᠯᠵᠢ ᠨ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ᠂ ᠬᠤᠶᠠᠳᠤᠭᠠᠷ ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪᠡᠯᠵᠢ ᠨ ᠢ ᠵᠦᠢᠬᠦ ᠳ ᠦ 9 ᠵᠢᠵᠢᠭ ᠲᠡᠭᠰᠢ ᠳᠦᠷ ᠪᠡᠯᠵᠢᠨ ᠬᠡᠷᠡᠭᠯᠡᠨ ᠡ᠁᠂ ᠡᠢᠮ ᠦ ᠠᠷ ᠭ ᠠ ᠪᠠᠷ ᠵᠦᠢᠭᠰᠡᠨ (𝑛 + 1) ᠳᠦᠭᠡᠷ ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪᠡᠯᠵᠢ ᠨ ᠨᠢ 𝑛 ᠳᠦᠭᠡᠷ ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪᠡᠯᠵᠢ ᠨ ᠡᠴᠡ ( ) ᠵᠢᠵᠢᠭ ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪᠡᠯᠵᠢᠨ ᠢᠯᠡᠭᠦᠦ ᠪᠣᠯᠤ ᠨ ᠠ᠃ 15. ᠨᠢᠭᠡᠨ ᠬᠦᠮᠦᠨ ᠰᠢᠨ ᠡ ᠲᠢᠳᠢᠮᠯᠢᠭ ᠸᠢᠷᠦ ᠰ ᠦᠨ ᠠᠭᠤᠰᠬ ᠢᠨ ᠦᠷᠪᠢ ᠰᠦᠯ ᠳ ᠦ ᠨᠡᠷᠪᠡᠭ ᠳᠡᠵᠦ ᠂ ᠬᠤᠶᠠᠷ ᠡᠭᠡᠯᠵᠢ ᠬᠠ ᠯᠳᠠᠪᠤᠷᠢ ᠯᠠᠭᠰᠠᠨ ᠤ ᠳᠠᠷᠠᠭ ᠠ ᠨᠡᠢ ᠲᠡ 169 ᠬᠦᠮᠦᠨ ᠰᠢᠨ ᠡ ᠲᠢᠳᠢᠮ ᠯᠢᠭ ᠸᠢᠷᠦ ᠰ ᠦᠨ ᠠᠭᠤᠰᠬ ᠢᠨ ᠦᠷᠪᠢ ᠰᠦᠯ ᠳ ᠦ ᠨᠡᠷᠪᠡᠭ ᠳᠡᠵᠡᠢ᠂ ᠡᠭᠡᠯ ᠵᠢ ᠪᠦᠷᠢ ᠶᠢᠨ ᠬᠠᠯᠳᠠᠪᠤᠷᠢᠯ ᠠᠯ ᠳ ᠤ ᠳᠤᠮᠳᠠᠴᠢ ᠪ ᠠᠷ ᠨᠢᠭᠡ ᠬᠦᠮᠦᠨ ᠨᠢ ( ) ᠬᠦᠮᠦᠨ ᠳᠦ ᠬᠠᠯᠳᠠᠪᠤ ᠷᠢᠯᠠᠭᠤᠯᠵᠠᠢ᠃ 16. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠬᠡᠷᠪᠡ ∆ ABC ᠳᠦ᠂ ∠A CB = 90° , AC = BC ᠂ ᠴᠡᠭ P ᠪᠣᠯ ᠵᠢᠰᠢ ᠭᠦᠦ ᠲᠠᠯ ᠠ AB ᠳᠡᠭᠡᠷ ᠡ ᠪᠣᠢ᠂ PC ᠢᠶᠡᠷ ᠨᠢᠭᠡ ᠡᠭᠴᠡ ᠲᠠᠯ ᠠ ᠪᠣᠯᠭᠠᠨ ᠠᠳᠠᠯᠢ ᠬᠠᠵᠠᠭᠤ ᠲᠤ ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦ ᠭᠲᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ PC Q ᠢ ᠪᠠᠢᠭᠤᠯᠤ ᠨ ᠠ᠂ ∠ PC Q = 90° ᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑃𝐴 2, 𝑃𝐵 2, 𝑃𝐶 2 ᠭᠤᠷᠪᠠ ᠶᠢᠨ ᠬᠤᠭᠤ ᠷᠤᠨᠳᠤᠬ ᠢ ᠲᠣᠭ ᠠ ᠬᠡᠮᠵᠢᠭᠳᠡᠭᠦ ᠨ ᠦ ᠬᠣᠯᠪᠤ ᠭᠳᠠᠯ ᠨᠢ ( ) 17. ᠵᠢᠷᠤ ᠭ ① ᠮᠡᠲᠦ᠂ ∆ABC ᠳᠦ᠂ AB = AC ,∠ BA C= 120° , E ᠪᠣᠯ AB ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠳᠣᠮᠳ ᠠ ᠴᠡᠭ᠂ P ᠪᠣᠯ BC ᠲᠠᠯ ᠠ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠬᠦᠳᠡᠯᠬ ᠦ ᠴ ᠡᠭ᠂ PC = 𝑥 , PA + PE = 𝑦 ᠭᠡᠵᠦ ᠵᠢᠱᠢᠨ ᠡ᠂ ᠵᠢᠷᠤ ᠭ ② ᠪᠣᠯ 𝑦 ᠦ ᠨ𝑥 ᠶᠢᠨ ᠲᠤᠬ ᠠᠢ ᠹᠥᠩᠺ ᠼ ᠦᠨ ᠵᠢᠷᠤ ᠮᠠᠭ᠂ ᠡᠭᠦᠨ ᠳᠦ H ᠪᠣᠯ ᠵᠢᠷᠤ ᠮᠠᠭ ᠳᠡᠭᠡᠷᠡᠬᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ᠳᠣᠣ ᠷᠠᠳᠤ ᠴᠡᠭ᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑎 + 𝑏 ᠦ ᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠨᠢ ( ) ᠭᠤᠷᠪᠠ ᠂ ᠪᠤ ᠳᠤᠵᠤ ᠬᠠᠷᠢᠭᠤ ᠯᠬᠤ ᠰ ᠡᠲᠥᠪ ( ᠨᠡᠢᠲᠡ69 ᠬᠤᠪᠢ) 18. (5 ᠬᠤᠪᠢ) ᠲᠡᠭᠰ ᠢᠳᠬᠡᠯ ᠢ ᠪᠤ ᠳᠤ᠄ 2𝑥−2 = 3𝑥 19. (6 ᠬᠤᠪᠢ) A ᠭᠠᠵᠠᠷ ᠠᠴᠠ ᠨᠢᠭᠡ ᠠᠰᠠᠷ ᠤ ᠨ ᠤᠷ ᠤᠢ ᠶᠢᠨ ᠬᠡᠰᠡ ᠭ ᠢ ᠬᠠᠷᠠᠬᠤ ᠳᠤ ᠦᠯᠡᠭᠰᠡᠨ ᠦᠨᠴᠦᠭ ᠨᠢ α ᠂ ᠡᠨᠡ ᠠᠰᠠᠷ ᠤ ᠨ ᠰᠠᠭ ᠤᠷᠢ ᠶᠢᠨ ᠬᠡᠰᠡ ᠭ ᠢ ᠬᠠᠷᠠᠬᠤ ᠳ ᠦ ᠰᠢᠷᠭᠤᠭ ᠰᠠᠨ ᠦ ᠨᠴᠦᠭ ᠨᠢ β ᠂ A ᠭᠠᠵᠠ ᠷ ᠠᠴᠠ ᠠᠰᠠ ᠷ ᠬᠦᠷᠬᠦ ᠳᠦᠪ ᠰᠢᠨ ᠵᠠᠢ AD ᠨᠢ 90 𝑚 ᠪᠤᠯ ᠤᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ tan 𝛼 = 0.27 ,tan 𝛽 = 2.73 ᠪᠣᠯ᠂ ᠡᠨᠡ ᠠᠰᠠ ᠷ ᠤᠨ ᠦᠨᠳᠦᠷ ᠢ ᠣ ᠯ᠃ 20. (6 ᠬᠤᠪᠢ) ᠨᠢᠭᠡ ᠵᠦᠢᠯ ᠦᠨ ᠰᠢᠨ ᠡ ᠪᠤᠳ ᠣᠯᠭ ᠠ ᠶᠢ ︽※ ︾ ᠢ ᠶᠡᠷ ᠲᠣᠳ ᠤᠷᠬᠠᠢ ᠯᠠᠯᠲ ᠠ ᠬᠢᠨ ᠡ᠄ ᠳᠤᠷ ᠠ ᠶᠢᠨ ᠪᠣᠳᠠ ᠲᠤ ᠲᠣᠭ ᠠ𝑚 ᠪᠠ 𝑛 ᠳ ᠤ᠂ 𝑚 ※𝑛 = 𝑚 2𝑛 − 𝑚𝑛 − 3𝑛 ᠭᠡᠵᠦ ᠲᠣᠭ ᠲᠠᠭᠠᠵᠠᠢ᠂ ᠵᠢᠱᠢ ᠶᠡᠯᠡᠪᠠᠯ ᠄1 ※2 = 1 2 × 2− 1× 2− 3× 2= −6 . (1) (− 2) ※√3 ᠶᠢ ᠣ ᠯ᠃ (2) ᠬᠡᠷᠪᠡ 3 ※𝑚 ≥ −6 ᠪᠣᠯ᠂ 𝑚 ᠦᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠠᠪᠬᠤ ᠬᠡᠪᠴᠢᠶ ᠡ ᠶᠢ ᠣᠯᠵ ᠤ᠂ ᠦᠭᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠲᠣᠭᠠᠨ ᠤ ᠲ ᠡᠩᠬᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠬᠠ ᠷᠢᠭᠤ ᠴᠣ ᠭᠯᠠᠷᠠᠯ ᠢ ᠢ ᠯᠡᠷᠬᠡᠢ ᠯᠡ᠃ 21. (7 ᠬᠤᠪᠢ) ᠠ ᠤᠭᠤᠲ ᠠ ᠳᠣᠲᠣᠷ ᠠ ᠠᠳᠠᠯᠢᠬᠠᠨ 2 ᠵᠢᠵᠢᠭ ᠪᠦᠮᠪᠦᠭᠡ ᠶᠢ ᠰᠠᠪᠠᠯ ᠠᠵᠤ ᠪᠣᠢ᠂ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠳᠡᠭᠡᠷ ᠡ ᠲᠣᠰ ᠲᠣᠰ ᠲᠣᠭ ᠠ1᠂ 2 ᠢ ᠪᠢᠴᠢ ᠵᠦ ᠪᠠᠢᠨ ᠠ᠂ ᠡ ᠤᠭᠤᠲ ᠠᠨ ᠳᠣᠲ ᠣᠷ ᠠ ᠠᠳᠠᠯᠢᠬᠠᠨ 3 ᠵᠢᠵᠢᠭ ᠪᠦᠮᠪᠦᠭᠡ ᠶᠢ ᠰᠠᠪ ᠠᠯᠠᠵᠤ ᠪᠤᠢ᠂ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠳᠡᠭᠡᠷ ᠡ ᠲᠣᠰ ᠲᠣᠰ ᠲᠣᠭ ᠠ 3᠂ 4᠂ 5 ᠢ ᠪᠢᠴᠢᠵᠦ ᠪᠠᠢᠨ ᠠ᠂ ᠢ ᠤᠭᠤᠲ ᠠᠨ ᠳᠣᠲ ᠣᠷ ᠠ 2 ᠠᠳᠠᠯᠢᠬᠠᠨ ᠵᠢᠵᠢᠭ ᠪᠦᠮᠪᠦᠭᠡ ᠶᠢ ᠰᠠᠪ ᠠᠯᠠᠵᠤ ᠪᠤᠢ᠂ ᠲᠡᠳᠡᠭᠡᠷ ᠦᠨ ᠳᠡᠭᠡᠷ ᠡ ᠲᠣᠰ ᠲᠣᠰ 6᠂ 7 ᠢ ᠪᠢᠴᠢᠵᠦ ᠪᠠᠢᠨ ᠠ᠃ ᠭᠤᠷᠪᠠᠨ ᠤᠭᠤᠲ ᠠᠨ ᠠᠴᠠ ᠲᠣᠰ ᠲᠣᠰ 1 ᠵᠢᠵᠢᠭ ᠪᠦᠮᠪᠦᠭᠡ ᠶᠢ ᠲᠠᠰᠢᠷᠠᠮ ᠢᠶᠠᠷ ᠠᠪᠣᠨ ᠠ᠃ ᠮ ᠤᠳᠤᠯ ᠵᠢᠨ ᠵ ᠢᠷᠤᠭ ᠵᠢᠷᠤᠬᠤ ᠪᠤ ᠶᠤ ᠬᠦᠰᠦᠨᠦᠭᠵᠢᠭᠦᠯᠬᠦ ᠠᠷᠠᠭ ᠠ ᠶᠢ ᠬᠡᠷᠡᠭᠯᠡᠵ ᠦ ᠣᠯ᠄ (1)ᠠ ᠪᠤᠭᠰᠠᠨ 3 ᠵᠢᠵᠢᠭ ᠪᠦᠮᠪᠦᠭᠡᠨ ᠳᠡᠭᠡᠷ ᠡ ᠶᠠᠭ ᠰᠠᠢᠬᠠᠨ ᠨᠢᠭᠡ ᠲᠡᠭᠰᠢ ᠲᠣᠭ ᠠ ᠪᠠᠢᠬᠤ ᠶᠢᠨ ᠲᠣᠬ ᠢᠶᠠᠯᠳᠤᠮᠵᠢ ; (2) ᠠᠪᠤᠭ ᠰᠠᠨ 3 ᠵᠢᠵᠢᠭ ᠪᠦᠮᠪᠦᠭᠡᠨ ᠳᠡᠭᠡᠷ ᠡ ᠴᠦᠮ ᠰᠤᠨᠳᠤ ᠭᠤᠢ ᠲᠣᠭ ᠠ ᠪᠠᠢᠬ ᠤ ᠶᠢᠨ ᠲᠣᠬ ᠢᠶᠠᠯᠳᠤᠮᠵᠢ ᠃ 22. (7 ᠬᠤᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ⊙ O ᠦ ᠨ ᠳᠢ ᠶᠠᠮᠧᠲ ᠷ AB ᠨᠢ ᠬᠦᠪᠴᠢ ( ᠳᠢᠶᠠᠮᠧᠲ ᠷ ᠪᠢ ᠰᠢ) CD ᠲᠠᠢ ᠴᠡᠭ P ᠲᠦ ᠤ ᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ ᠪᠦᠭᠡᠳ 𝑃𝐶 2= PA ∙PB . ᠪᠠᠲ ᠤᠯᠠ᠄ CD ⊥ AB 23. (8 ᠬᠤᠪᠢ) ᠨᠢᠭᠡᠨ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠰᠤᠷᠤ ᠭᠴᠢᠳ ᠤᠨ ᠬᠢᠴᠢᠶᠡᠯ ᠦ ᠨ ᠭᠠ ᠳᠠᠨᠠᠬᠢ ᠲᠣᠷ ᠠ ᠪᠠᠬ ᠠ ᠶᠢᠨ ᠪᠠᠢᠳᠠᠯ ᠢ ᠰᠤᠳᠤ ᠯᠬᠤ ᠶᠢᠨ ᠲᠥᠯᠥ ᠭᠡ᠂ ᠶᠠᠩᠵᠤ ᠰᠤᠭᠤᠯ ᠬᠤ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠠᠷᠭ ᠠ ᠶᠢ ᠠᠪᠴᠤ᠂ ᠨᠣᠮ ᠤᠩ ᠰᠢᠬᠤ᠂ ᠲᠠᠮᠢᠷ ᠤᠨ ᠬᠦᠳᠡᠯᠭᠡᠭᠡᠨ ᠬᠢᠬᠦ᠂ ᠨᠠᠭᠠᠳᠤᠨ ᠲᠣᠭ ᠯᠠᠬᠤ᠂ ᠰᠦᠯᠵᠢᠶ ᠠᠨ ᠳᠤ ᠭᠠᠷᠬᠤ ᠵᠡᠷᠭᠡ ᠳᠦᠷᠪᠡᠨ ᠲᠠᠯ ᠠ ᠪᠠᠷ ᠡᠳᠦᠢ ᠲᠡᠳᠦᠢ ᠰᠤᠷᠤ ᠭᠴᠢᠳ ᠤᠨ ᠳᠤᠷ ᠠ ᠪᠠᠬ ᠠ ᠶᠢᠨ ᠪᠠᠢᠳᠠ ᠯ ᠳᠤ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠲᠠ ᠬᠢᠪᠡ᠂ ᠪᠠᠢᠴ ᠠᠭᠠᠯᠲ ᠠ ᠶᠢᠨ ᠦᠷ ᠡ ᠳᠦᠩ ᠢ ᠶᠡᠨ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠵᠢᠷᠤ ᠭ ᠮᠠᠲᠦ ᠬᠤᠶᠠᠷ ᠵᠦᠢᠯ ᠦᠨ ᠪᠦᠷᠢᠨ ᠪᠤᠰᠤ ᠪᠦᠷᠢᠳᠬ ᠡᠯ ᠦᠨ ᠵᠢᠷᠤ ᠭ ᠪᠣ ᠯᠭᠠᠨ ᠵᠢᠷᠤ ᠪᠠ᠂ ᠲᠠ ᠵᠢᠷᠤ ᠭ ᠲ ᠠᠬᠢ ᠮᠡᠳᠡᠭᠡ ᠵᠠᠩᠭ ᠢ ᠶᠢ ᠦᠨᠳᠦᠰᠦᠯᠡᠵ ᠦ ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠠᠰᠠᠭᠤᠯᠲ ᠠ ᠶᠢ ᠬᠠ ᠷᠢᠭᠤ ᠯ᠄ (1) ᠲᠤᠰ ᠤᠳᠠᠭᠠᠨ ᠤ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠲᠠ ᠳᠤ ᠨᠡᠢ ᠲᠡ ᠬᠡᠳᠦᠢ ᠰᠤᠷᠤ ᠭᠴᠢ ᠳᠤ ᠪ ᠠᠢᠴᠠᠭ ᠠᠯᠲᠠ ᠬᠢᠭᠰᠡ ᠨ ᠪᠣ ᠢ︖ (2) ᠰᠠᠪ ᠠᠭᠠᠨ ᠪᠦᠷᠢᠳᠬ ᠡᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠢ ᠨᠦᠬᠦᠪ ᠦᠷᠢᠯ ᠡᠵᠦ ᠪᠦᠷᠢᠨ ᠪᠣᠯᠭ ᠠ᠃ (3) ᠬᠡᠷᠪᠡ ᠲᠤᠰ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠳᠤ ᠲᠠᠮᠢᠷ ᠤᠨ ᠬᠦᠳᠡᠯᠭᠡᠭᠡᠨ ᠬᠢᠬᠦ ᠳᠤᠷ ᠠᠲᠠᠢ ᠰᠤᠷᠤ ᠭᠴᠢ 800 ᠪᠠᠢᠭᠰᠠᠨ ᠪᠣᠯ᠂ ᠲᠤᠰ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠶᠢᠨ ᠪᠦᠬᠦ ᠰᠣᠷᠣ ᠭᠴᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠨᠢ ᠪᠠᠷᠤ ᠭ ᠬᠡᠳᠦᠢ ᠪᠠᠢᠬᠤ ᠪᠤᠢ ︖ 24. ᠨᠢᠭᠡᠨ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠤ ᠳᠡᠯᠭᠡᠭᠦᠷ A, B ᠬᠤᠶᠠᠷ ᠬᠡᠯᠪᠡᠷᠢ ᠶᠢᠨ ᠬᠤᠪᠴᠠᠰᠤ ᠶᠢ ᠤᠷᠤᠭᠤ ᠯᠬᠤ ᠪ ᠠᠷ ᠲᠥᠯᠥ ᠪᠯᠡᠵᠦ ᠪᠠᠢᠨ ᠠ᠃ ᠦᠭᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ 2 ᠵᠠᠬ ᠠ A ᠬᠡᠯᠪᠡᠷᠢ ᠶᠢᠨ ᠬᠤᠪᠴᠠᠰᠤ ᠪᠠ 3 ᠵᠠᠬ ᠠ B ᠬᠡᠯᠪᠡᠷᠢ ᠶᠢᠨ ᠬᠤᠪᠴ ᠠᠰᠤ ᠨᠢ ᠨᠡᠢᠲᠡ 460 0 ᠲᠥᠭᠥᠷᠢᠭ ᠬᠡᠷᠡᠭᠰᠡ ᠨ ᠡ᠃ 1 ᠵᠠᠬ ᠠ A ᠬᠡᠯᠪᠡᠷᠢ ᠶᠢᠨ ᠬᠤᠪᠴᠠᠰᠤ ᠪᠠ 2 ᠵᠠᠬ ᠠ B ᠬᠡᠯᠪᠡᠷ ᠢ ᠶᠢᠨ ᠬᠤᠪ ᠴᠠᠰᠤ ᠨᠢ ᠨᠡᠢᠲᠡ 28 00 ᠲᠥᠭᠥᠷᠢ ᠭ ᠬᠡᠷᠡᠭᠰᠡᠨ ᠡ᠃ (1) A , B ᠬᠤᠶᠠᠷ ᠬᠡᠯ ᠪᠡᠷᠢ ᠶᠢᠨ ᠬᠤᠪ ᠴᠠᠰᠤ ᠨ ᠤ ᠨᠢᠭᠡᠴᠢ ᠦᠨ ᠡ ᠶᠢ ᠣᠯ᠃ (2) ᠲᠣᠰ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠳᠡᠯᠭᠡᠭᠦᠷ A , B ᠬᠤᠶᠠᠷ ᠬᠡᠯᠪᠡᠷᠢ ᠶᠢᠨ ᠬᠤᠪᠴᠠ ᠰᠤ ᠶᠢ ᠨᠡᠢᠲᠡ 60 ᠵᠠᠬ ᠠ ᠤᠷᠤᠭᠤ ᠯᠬᠤ ᠪ ᠠᠷ ᠪᠠᠢᠨ ᠠ᠂ ᠡᠭᠦᠨ ᠳᠦ A ᠬᠡᠯᠪᠡᠷᠢ ᠶᠢᠨ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠤ ᠲᠣᠭ ᠠ ᠨᠢ B ᠬᠡᠯᠪᠡᠷᠢ ᠶᠢᠨ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠤ ᠲᠣᠭ ᠠᠨ ᠤ 2 ᠲᠠᠬᠢ ᠨ ᠠᠴᠠ ᠪᠠᠭ ᠠ ᠪᠤᠰᠤ ᠪᠠᠢ ᠯᠭᠠᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ B ᠬᠡᠯᠪᠡᠷᠢ ᠶᠢᠨ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠤ ᠦᠨ ᠡ ᠶᠢ 7.5 ᠨᠤᠭᠤᠯ ᠵᠤ ᠥᠭᠬᠦ ᠪᠡᠷ ᠪᠤᠳᠤᠪᠠᠯ᠂ ᠲᠤᠰ ᠳᠡᠯᠭᠡᠭᠦᠷ ᠪᠠᠷᠠᠭ ᠠ ᠤᠷᠤᠭᠤ ᠯᠬᠤ ᠵᠣᠭᠣ ᠰ ᠢ ᠠᠳᠠᠭ ᠲᠤ ᠪᠠᠨ ᠬᠡᠳᠦᠢ ᠪ ᠡᠯᠡᠳᠬᠡᠬᠦ ᠴᠢᠬᠤᠯ ᠠ ᠲᠠᠢ ᠪᠤ ᠢ︖ 25. (9 ᠬᠤᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠴᠡᠭ O ᠢ ᠶᠡᠷ ᠲᠥᠪ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠵᠥᠪ ᠵᠢᠷᠭᠤ ᠭᠠᠨ ᠦᠨᠴᠦᠭᠲᠦ ABC DEF ᠶᠢᠨ ᠷᠠᠳᠢᠦᠰ ᠨᠢ 6cm ᠂ ᠴᠡᠭ P, Q ᠵᠡᠷᠭᠡ ᠪᠡᠷ ᠲᠣᠰ ᠲᠣᠰ A,D ᠬᠤᠶᠠᠷ ᠴᠡᠭ ᠡᠴᠡ ᠭᠠᠷᠴᠦ ᠂ ᠴᠦᠮ 1cm / s ᠦᠨ ᠬᠤᠷᠳᠤ ᠴᠠ ᠪ ᠠᠷ AF ,DC ᠳᠠᠭ ᠠᠤ ᠲᠡᠭᠦᠰᠬ ᠡᠯ ᠦᠨ ᠴᠡᠭ F, C ᠲᠡᠢᠰᠢ ᠬᠦᠳᠡᠯᠭᠡᠭᠡᠨ ᠬᠢᠨ ᠡ᠂ PB ,PE ,QB ,QE ᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠂ ᠬᠦᠳ ᠡᠯᠭᠡᠭᠡᠨ ᠬᠢᠬᠦ ᠬᠤᠭᠤᠴᠠᠭ ᠠ ᠶᠢ t( s) ᠭᠡᠵᠦ ᠵᠢᠱᠢᠨ ᠡ᠃ (1) ᠪᠠᠲᠤ ᠯᠠ᠄ ᠳᠦᠷ ᠪᠡᠯᠵᠢᠨ PB QE ᠪᠣᠯ ᠵᠢᠰ ᠢᠭᠦᠯᠵᠢᠨ; (2) ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦᠭᠲᠦ PB QE ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠪᠠ ᠵᠥᠪ ᠵᠢᠷᠭᠤ ᠭᠠᠨ ᠦᠨᠴᠦ ᠭᠲᠦ ABC DEF ᠶᠢᠨ ᠲᠠᠯᠠᠪᠠᠢ ᠶᠢᠨ ᠬᠠᠷᠢᠴᠠᠯ ᠢ ᠣᠯ᠃ 26. (1 2ᠬᠤᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠦ᠂ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠦᠨᠴᠦᠭᠲᠦ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ ᠧᠮ ᠳᠦ᠂ ᠬᠠᠶᠠᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ 𝑦 = −𝑥 2+ 𝑏𝑥 + 𝑐 ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ A , B ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠵᠤ᠂ 𝑦 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ C ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠬᠤ ᠪᠦᠭᠡᠳ ᠰᠢᠯᠤ ᠭᠤᠨ 𝑦= 𝑥− 6 ᠨᠢ ᠴᠡᠭ B ᠢ ᠳᠠᠭ ᠠᠷᠢᠵᠤ᠂ 𝑦 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠲᠠᠢ ᠴᠡᠭ D ᠲᠦ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ C ᠪᠠ D ᠨᠢ 𝑥 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠦᠨ ᠬᠤᠪᠢ ᠳ ᠤ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠲ ᠡᠢ᠂ P ᠪᠣᠯ ᠬᠡᠷᠴᠢᠮ OB ᠳᠡᠭᠡᠷᠡ ᠬᠢ ᠨᠢᠭᠡᠨ ᠬᠦᠳ ᠡᠯᠬᠦ ᠴᠡᠭ᠂ ᠴᠡᠭ P ᠢ ᠳᠠᠭ ᠠᠷᠢᠭᠤᠯᠤ ᠨ 𝑥 ᠲᠡᠩᠬᠡᠯᠢᠭ ᠲᠦ ᠡᠭᠴᠡ ᠰᠢᠯᠤᠭᠤ ᠨ ᠪᠠᠢᠭᠤᠯ ᠪᠠᠯ ᠬᠠᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠲᠠᠢ ᠴᠡᠭ M ᠳᠦ ᠤᠭᠲᠤ ᠯᠤᠯᠴᠠᠵᠤ᠂ ᠰᠢᠯᠤᠭᠤᠨ BD ᠲᠠᠢ ᠴᠡᠭ N ᠲᠦ ᠤ ᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ᠃ (1) ᠲᠣᠰ ᠬᠠᠶᠠᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠤᠨ ᠠᠨᠠ ᠯᠢᠰ ᠢ ᠯᠡᠷᠬᠡᠢᠯᠡ ᠯ ᠢ ᠣᠯ᠃ (2) ∆MDB ᠦᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠬᠡ ᠦ ᠶᠡᠰ ᠴᠡᠭ P ᠦ ᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ (3)(2) ᠦᠨ ᠨᠦᠬᠦᠴᠡᠯ ᠳᠣᠣ ᠷ ᠠ᠂ 𝑦 ᠲᠡᠩᠬᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ᠂ Q , M , N ᠭᠤᠷᠪᠠ ᠨ ᠴᠡᠭ ᠢᠶᠡᠷ ᠤᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠭᠤᠷᠪᠠᠯ ᠵᠢᠨ ᠢ ᠲᠡᠭᠰᠢ ᠦ ᠨᠴᠦᠭᠲᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠪᠣᠯᠭᠠᠬᠤ Q ᠴᠡᠭ ᠤᠷᠤᠰᠢᠨ ᠠ ᠤᠤ︖ ᠬᠡᠷᠪᠡ ᠤᠷᠤᠰᠢᠪᠠ ᠯ ᠴᠡᠭ Q ᠦᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠰᠢᠭᠤᠳ ᠪᠢᠴᠢᠵᠦ ᠭᠠᠷᠭ ᠠ᠃ ᠬᠡᠷᠪᠡ ᠤᠷᠤᠰᠢᠬᠤ ᠦᠭᠡᠢ ᠪᠣᠯ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠨ ᠢ ᠲᠣ ᠳᠤᠷᠬᠠᠢᠯᠠ᠃