Russian | Can you solve this? | Math Olympiad

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  • Опубликовано: 30 окт 2024

Комментарии • 6

  • @fireblast3740
    @fireblast3740 День назад +2

    Well, from another video method: Vieta theorem
    We have:
    sqrt(x)+sqrt(y)=5 (-b)
    sqrt(xy)=sqrt(x)*sqrt(y)=5 (c)
    t^2-5t+5=0
    t(1,2)=(5+-sqrt(25-20))/2
    t1=(5+sqrt(5))/2;t2=(5-sqrt(5))/2
    (t1)^2=x1=y2=(5+sqrt(5))^2/4=(15+5sqrt(5))/2
    (t2)^2=x2=y1=(5-sqrt(5))^2/4=(15-5sqrt(5))/2
    Answer:((15+5sqrt(5))/2;(15-5sqrt(5))/2);((15-5sqrt(5))/2;(15+5sqrt(5))/2)

  • @عبدالواسع-س8م
    @عبدالواسع-س8م 2 дня назад +1

    Well done!
    Thank you so much!

  • @seith7212
    @seith7212 День назад +3

    7:30, how 15 disappeared

    • @Prem-K007
      @Prem-K007 День назад +1

      15 can be written as (30/2) and (30/2)-(15/2) = 15/2
      . another small example :- 3 -(3/2) = 3/2.
      So visually ..it may appear disappeared but it is not actually.
      And also not just 15 is disappeared along with a minus sign is missing which should be noticed
      I'm afraid that you asked this question.

    • @user-mv4ix7jd8o
      @user-mv4ix7jd8o 16 часов назад +1

      you had 15 minus half of it. that's just the other half :)

  • @adribber
    @adribber 2 дня назад +1

    √x + √y = 5 --- ①
    √xy = 5 --- ②
    From ①
    (√x + √y)² = 5²
    x + y + 2√xy = 25 ②
    √x(15 - x) = 5
    [√x(15 - x)]² = 5²
    x(15 - x) = 25
    - x² + 15x = 25
    x² - 15x + 25 = 0
    D = (- 15)² - 4*1*25 = 225 - 100 =125 = 5³ > 0
    x = [- (- 15) ± √D ] / 2*1
    x = ( 15 ± 5√5 ) / 2
    y = 15 - ( 15 ± 5√5 ) / 2
    y = (15 -+ 5√5 ) / 2
    ∴ (x , y) = {(15 + 5√5) / 2 , (15 - 5√5) / 2} , {(15 - 5√5) / 2 , (15 + 5√5) / 2}
    Verifying
    √xy = √[(15 + 5√5) / 2] * [(15 - 5√5) / 2]
    = √[(15 + 5√5)(15 - 5√5) / 4]
    = √[(225 - 125) / 4]
    = √100/4
    = √25
    = 5 Pass!