Consider the system of equations `x_1+x_(2)^(2)+x_(3)^(3)+x_(4)^(4)+x_(5)^(5)=5` and `x_1+2x_2+3x_3+4x_4+5x_5=15` where `x_1,x_2,x_3,x_4,x_5` are positive real numbers. Then numbers of `(x_1,x_2,x_3,x_4,x_5)` is ___________.

Explanation

To solve the system of equations given by:

1. \( x_1 + x_2^2 + x_3^3 + x_4^4 + x_5^5 = 5 \)
2. \( x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5 = 15 \)

where \( x_1, x_2, x_3, x_4, x_5 \) are positive real numbers, we can use the method of inequalities, specifically the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step 1: Apply AM-GM Inequality

We will apply the AM-GM inequality to each term in the first equation:

- For \( x_2^2 \):
\[ x_2^2 + 1 \geq 2x_2 \quad \text{(AM-GM)} \]

- For \( x_3^3 \):
\[ x_3^3 + 1 + 1 \geq 3x_3 \quad \text{(AM-GM)} \]

- For \( x_4^4 \):
\[ x_4^4 + 1 + 1 + 1 \geq 4x_4 \quad \text{(AM-GM)} \]

- For \( x_5^5 \):
\[ x_5^5 + 1 + 1 + 1 + 1 \geq 5x_5 \quad \text{(AM-GM)} \]

### Step 2: Summing the Inequalities

Now we sum all these inequalities:

\[ (x_2^2 + 1) + (x_3^3 + 1 + 1) + (x_4^4 + 1 + 1 + 1) + (x_5^5 + 1 + 1 + 1 + 1) \geq 2x_2 + 3x_3 + 4x_4 + 5x_5 \]

This simplifies to:

\[ x_2^2 + x_3^3 + x_4^4 + x_5^5 + 10 \geq 2x_2 + 3x_3 + 4x_4 + 5x_5 \]

### Step 3: Relating to the Second Equation

From the second equation, we know:

\[ x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5 = 15 \]

Adding \( x_1 \) to both sides of our inequality gives:

\[ x_1 + x_2^2 + x_3^3 + x_4^4 + x_5^5 + 10 \geq 15 \]

### Step 4: Simplifying the Inequality

This leads to:

\[ x_1 + x_2^2 + x_3^3 + x_4^4 + x_5^5 \geq 5 \]

### Step 5: Equality Condition

Since the first equation states that \( x_1 + x_2^2 + x_3^3 + x_4^4 + x_5^5 = 5 \), we have equality in the AM-GM inequalities. The equality condition for AM-GM holds when all terms are equal:

\[ x_2^2 = 1, \quad x_3^3 = 1, \quad x_4^4 = 1, \quad x_5^5 = 1 \]

This gives:

\[ x_2 = 1, \quad x_3 = 1, \quad x_4 = 1, \quad x_5 = 1 \]

### Step 6: Finding \( x_1 \)

Substituting these values into the first equation:

\[ x_1 + 1 + 1 + 1 + 1 = 5 \]

Thus,

\[ x_1 = 1 \]

### Conclusion

The only solution set is:

\[ (x_1, x_2, x_3, x_4, x_5) = (1, 1, 1, 1, 1) \]

Therefore, the number of sets \( (x_1, x_2, x_3, x_4, x_5) \) is **1**.

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