11-sinf Matematika olimpiada №4

1. 1!+2!+3!+.....+10!  sonining oxirgi raqamini toping.

2.  ni  hisoblang.

3. ifodani  soddalshtiring

4. Funksiyaning eng kichik qiymatini toping. ?(?) = (2? + 6)⁵ − 4(2? + 6)⁴, bunda |? + 3| ≤ 2. 

5. Agar a = √5 + 1 va b = √5 − 1 bo‘lsa, 2a² − ab − a² ifodaning qiymatini toping.

6. y = x² − 2x + 3 parabolaga y = 3 to‘g‘ri chiziqqa nisbatan simmetrik bo‘lgan parabola tenglamasini tuzing.

7. Soddalashtiring:

8. ABCD - kvadrat, AC-diagonal. Agar AE=BE+CE ( a + b = c ) bo‘lsa, ∠AEB burchakni toping(chizmaga qarang!).

9. Raqamlari yig‘indisi 10 dan kam bo‘lmagan, raqamlari ko‘paytmasi esa 10 dan katta bo‘lmagan uch xonali sonlar nechta?

10. Doskaga ?₁, ?₂, ?₃, . . . , ?₂₀₀ sonlari yozilgan. Ma’lumki, ?₁ = 3, ?₂ = 9. Agar ixtiyoriy n natural son uchun ?_n+2 = ?_n+1 − ?_n tenglik o‘rinli bo‘lsa, ?₂₀₀ ni toping.

11. BC = BE = CD ,∠DAE = 20°, ∠BCD = 60° ,∠ADE = ? (chizmaga qarang)

12. Tenglamani yeching:

13. bo’lsa, P(−1) ni toping.

14. Soddalashtiring: 

15. (3 − cos²x− 2sinx)(lg²y+ 2lgy+ 4) ≤ 3 bo‘lsa, sin²x+ 20y+ 1 ni toping.

16. Nargizada 2 ta olma va 3 ta nok bor. U 5 kun ketma-ket har kuni singliga bittadan meva beradi. Bu ishni necha usul bilan amalga oshirish mumkin?

17. ABC uchburchakda ctgA+ctgB= 3 va AB= 12 bo‘lsa, ABC uchburchak yuzini toping.

18. u_n ketma-ketlik quyidagicha berilgan: u₁ = 1, u_n+1 = u_n + 8n. U holda, u₅₀ −u₃₀ ni toping.

19. Uchlari A(2; −3), B(6; −1), C(6; 4) va D(2; 2) nuqtalarda bo‘lgan ABCD to‘rtburchak yuzini toping.

20. Natural son roppa-rosa 2 ta tub bo‘luvchiga, bu sonning kvadrati esa 45 ta turli natural bo‘luvchiga ega. Berilgan sonning kubi eng ko‘pi bilan nechta natural bo‘luvchiga ega bo‘lishi mumkin?

21. Soddalashtiring: bunda, |?| < 1.

22. Agar    bo‘lsa sin²α ni toping.

23. a²b⁵ = 6²⁰ tenglikni qanoatlantiruvchi nechta (?; ?) butun sonlar juftligi mavjud?

24. Muntazam uchburchak 36 ta yuzi 1 ga teng bo‘lgan kichkina muntazam uchburchaklardan iborat (chizmaga qarang!). ABC uchburchak yuzini toping.

25. Soatning minut mili 144° ga burilganda, soatning soat mili qanday burchakka burilada?

26. ?(3) ∙ (? − 2) + ?(? − 1) = 3? bo‘lsa, ?(1) ni toping.

27. Tengsizliklar sistemasinining butun yechimlari sonini toping.

28. ∠CAD = 2∠ACD, AC =13a, BD = 5a va AB = CD bo’lsa, 20⋅cos ∠ACD ni toping.(chizmaga qarang!)

29. Agar ifodaning qiymatini toping.

30. Arifmetik progressiyada ?₂ = 12, ?₇ − ?₄ = 9 bo‘lsa, ?₁₉ ni toping.