Given that X1 times x2 times X3 until x2006 equals 1, and x1, X2... X2006 are all positive numbers, then (1 + x1) (1 + x2) ）What is the minimum value of (1 + x3)... (1 + x2006)

Given that X1 times x2 times X3 until x2006 equals 1, and x1, X2... X2006 are all positive numbers, then (1 + x1) (1 + x2) ）What is the minimum value of (1 + x3)... (1 + x2006)

(1 + x1) (1 + x2)... (1 + x2006) ≥ 2 radical (x1) * 2 radical (x2) *... 2 radical (x2006)
=(2 ^ 2006) * radical (x1 * x2 *... X6)
=2^2006
So it's 2 ^ 2006

Given X1 + x2 + X3 +... + x2006 = X1 ^ 2 + x2 ^ 2 + X3 ^ 2 +... + x2006 ^ 2 = 2006, find the value of X1 + x2 ^ 2 + X3 ^ 2 +... + x2006 ^ 2006

2006 ( x1=x2=x3=…… =The following is the derivation process: X1 + x2 + X3 +... + x2006 = X1 ^ 2 + x2 ^ 2 + X3 ^ 2 +... + x2006 ^ 2. By subtracting these two formulas left and right, we get: X1 (x1-1) + X2 (x2-1) + +X2006 (x2006-1) = 0 if we want this formula to hold, then we have X1 = x2 = X3 = =X2006 = 0 or X1 = x2 = X3 = =X2006 = 1, if X1 = x2 = X3 = =If x2006 = 0, there will be no X1 + x2 + X3 +... + x2006 = X1 ^ 2 + x2 ^ 2 + X3 ^ 2 +... + x2006 ^ 2 = 2006, so there will be X1 + x2 + X3 +... + x2006 = 1

Known: | x1-1 | + | x2-2 | + | x3-3 | +. + | x2002-2002 | + | x2003-2003 | = 0
Find the value of: 2 ^ X-2 ^ x2-2 ^ X3 -. - 2 ^ x2002 + 2 ^ x2003. (Note: similar to x2003, the following 2003 should be the subscript of x)

Using the properties of absolute value
|a|>=0
therefore
If | a | + | B | = 0
There must be a = b = 0
therefore
x1-1=0
x2=2=0
……
x2003-2003=0
therefore
x1=1
x2=2
……
x2003=2003
2^X1-2^X2-2^X3-.-2^X2002+2^X2003
=2-2^2-2^3…… -2^2002+2^2003
2^2003-2^2002
=2^2002(2-1)=2^2002
So start from the back
Every two are combined
The results are as follows
2+2^2=6