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.
111000011011100110
101010100110
001110
011010110000000000
100111010111
010110
 ¤ Contact ¤ 
 ¤ Options ¤ 
 ¤ Actualization ¤ 
 ¤ Stop potratům ¤ 
Rychlá řešení dlouho bolí
Čekáte-li nečekané dítě
 ¤ HEX Counter ¤ 
3 1 9 4 9
 ¤ Certificate ¤ 
Valid HTML 4.01 Valid CSS Valid RSS 2.0

» Gallery » Logical puzzles  101100000101001011 

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 © 2004-2021 Tomáš Vala
Optimized for Firefox
Website map | Mobile version | A+ A A-