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.
100110101010111000
011011111111
000101
110111111001101001
000110100110
110100
 ¤ Contact ¤ 
 ¤ Options ¤ 
 ¤ Actualization ¤ 
 ¤ Stop potratům ¤ 
Rychlá řešení dlouho bolí
Čekáte-li nečekané dítě
 ¤ HEX Counter ¤ 
2 f 0 f e
 ¤ Certificate ¤ 
Valid HTML 4.01 Valid CSS Valid RSS 2.0

» Gallery » Logical puzzles  011110011000101010 

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-