There is a two-digit number whose digits are the same, and has got the following property : when squared, it produces a four-digit number, whose first two digits are the same and equal to the original's minus one, and whose last two digits are the same and equal to half of the original's. Find that number.

Answer

To solve the problem step by step, let's break it down clearly.

### Step 1: Define the two-digit number
Let the two-digit number be represented as \( 11x \), where \( x \) is the digit (since both digits are the same).

### Step 2: Square the number
When we square the number, we get:
\[ (11x)^2 = 121x^2 \]
This is the four-digit number we will analyze.

### Step 3: Analyze the properties of the squared number
According to the problem:
- The first two digits of the squared number are the same and equal to \( x - 1 \).
- The last two digits are the same and equal to \( \frac{x}{2} \).

### Step 4: Formulate the four-digit number
The four-digit number can be expressed as:
\[ 1000(x - 1) + 100(x - 1) + 10\left(\frac{x}{2}\right) + \left(\frac{x}{2}\right) \]
This simplifies to:
\[ 1100(x - 1) + 5x \]

### Step 5: Set up the equation
Equating the squared number to the four-digit number gives us:
\[ 121x^2 = 1100(x - 1) + 5x \]
Expanding the right-hand side:
\[ 121x^2 = 1100x - 1100 + 5x \]
This simplifies to:
\[ 121x^2 = 1105x - 1100 \]

### Step 6: Rearrange to form a quadratic equation
Rearranging gives us:
\[ 121x^2 - 1105x + 1100 = 0 \]

### Step 7: Solve the quadratic equation
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 121 \), \( b = -1105 \), and \( c = 1100 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-1105)^2 - 4 \cdot 121 \cdot 1100 \]
Calculating:
\[ = 1221025 - 532400 = 688625 \]

Now, taking the square root:
\[ \sqrt{688625} = 829 \]

Now substituting back into the quadratic formula:
\[ x = \frac{1105 \pm 829}{2 \cdot 121} \]

Calculating the two possible values for \( x \):
1. \( x = \frac{1105 + 829}{242} = \frac{1934}{242} \approx 8 \)
2. \( x = \frac{1105 - 829}{242} = \frac{276}{242} \approx 1.14 \)

Since \( x \) must be a digit (0-9), we take \( x = 8 \).

### Step 8: Find the original number
The original two-digit number is:
\[ 11x = 11 \cdot 8 = 88 \]

### Final Answer
The two-digit number is \( 88 \).

---