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.
100101000100110111
010001011111
100000
101101011111101101
100100011011
110011
 ¤ Contact ¤ 
 ¤ Options ¤ 
 ¤ Actualization ¤ 
 ¤ Stop potratům ¤ 
Rychlá řešení dlouho bolí
Čekáte-li nečekané dítě
 ¤ HEX Counter ¤ 
3 2 d 7 c
 ¤ Certificate ¤ 
Valid HTML 4.01 Valid CSS Valid RSS 2.0

» Gallery » Logical puzzles  001111100101111010 

Tři rovnice - difficulty 4.2 (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 © 2004-2022 Tomáš Vala
Optimized for Firefox
Website map | Mobile version | A+ A A-