This website uses cookies to store your personal settings, to personalize advertisements (by hosting webzdarma.cz) and to analyze visit rate. By using this web you agree with it.
101001110000010011
011111011001
101010
011111100000110010
001001100111
111100
 ¤ Contact ¤ 
 ¤ Options ¤ 
 ¤ Actualization ¤ 
 ¤ Stop potratům ¤ 
Rychlá řešení dlouho bolí
Čekáte-li nečekané dítě
 ¤ HEX Counter ¤ 
2 f 0 a 8
 ¤ Certificate ¤ 
Valid HTML 4.01 Valid CSS Valid RSS 2.0

» Gallery » Logical puzzles  011100100011001101 

Tři rovnice - difficulty 4.5 (former difficulty 4)
Máme dánu soustavu tří rovnic o třech neznámých x, y, z:

x + y + z = 1
x2 + y2 + z2 = 2
x3 + y3 + z3 = 4

Čemu je roven součin x * y * z ?
Čemu je roven výraz 1/x + 1/y + 1/z ?
(x+y+z)3 = (x3+y3+z3) + 6xyz + 3xy2 + 3y2z + 3xz2 + 3yz2 + 3x2z + 3x2y
6xyz = (x+y+z)3 - (x3+y3+z3) - 3*(x2*(y+z) + y2*(x+z) + z2*(x+y)) = (x+y+z)3 - (x3+y3+z3) - 3*((x2+y2+z2)(x+y+z) - (x3+y3+z3)) = 1 - 4 - 3*(2*1 - 4) = 3
xyz = 1/2

(x+y+z)2 = (x2+y2+z2) + 2xy + 2xz + 2yz
(xy + xz + yz) = ((x+y+z)2 - (x2+y2+z2))/2 = (1 - 2)/2 = -1/2
1/x + 1/y + 1/z = (xy + xz + yz)/(xyz) = (-1/2)/(1/2) = -1
Difficulty:12345678910
 ¤ TOP ¤ 
 ¤ Searching ¤ 
 ¤ Biblenet ¤ 
Verse:
Back to top
Copyright © 2004-2020 Tomáš Vala
Optimized for Firefox
Website map | Mobile version | A+ A A-