The number of isosceles triangles with integer sides if no side exeeds 2016 is

Options

A. `(1008)^(2)` if equal sides not exceed 1008
B. `2(1008)^(2)` if equal sides exceed 1008
C. `3(1008)^(2)` if equal sides have any length not exceeding 2016
D. `(2016)^(2)` if equal sides have any length not exceeding 2016

Explanation

To find the number of isosceles triangles with integer sides where no side exceeds 2016, we can follow these steps:

### Step 1: Understand the properties of isosceles triangles
An isosceles triangle has two sides of equal length. Let's denote the lengths of the two equal sides as \( n \) and the base (the third side) as \( a \).

### Step 2: Apply the triangle inequality
For any triangle with sides \( a, n, n \), the triangle inequality must hold:
1. \( n + n > a \) → \( 2n > a \) → \( a < 2n \)
2. \( n + a > n \) → \( a > 0 \)
3. \( n + a > n \) → \( a > 0 \)

From the first inequality, we have \( a < 2n \). The second and third inequalities simply state that \( a \) must be positive.

### Step 3: Determine the range of \( n \)
Since no side can exceed 2016, we have:
- \( n \leq 2016 \)

### Step 4: Determine the range of \( a \)
From the triangle inequality, we have:
- \( 1 \leq a < 2n \)

### Step 5: Count the possible values of \( a \)
For each integer value of \( n \), the possible values of \( a \) range from 1 to \( 2n - 1 \). Therefore, the number of possible values for \( a \) is:
- \( 2n - 1 \)

### Step 6: Sum the possible values of \( a \) for all valid \( n \)
Now we need to sum \( 2n - 1 \) for all integer values of \( n \) from 1 to 2016:
\[ \text{Total} = \sum_{n=1}^{2016} (2n - 1) \]
This can be simplified as:
\[ \text{Total} = \sum_{n=1}^{2016} 2n - \sum_{n=1}^{2016} 1 \]
\[ = 2 \sum_{n=1}^{2016} n - 2016 \]

### Step 7: Calculate the sum of the first 2016 integers
The sum of the first \( m \) integers is given by the formula:
\[ \sum_{n=1}^{m} n = \frac{m(m + 1)}{2} \]
Thus, for \( m = 2016 \):
\[ \sum_{n=1}^{2016} n = \frac{2016 \times 2017}{2} = 2033136 \]

### Step 8: Substitute back into the total
Now substituting back:
\[ \text{Total} = 2 \times 2033136 - 2016 = 4066272 - 2016 = 4064256 \]

### Conclusion
The total number of isosceles triangles with integer sides where no side exceeds 2016 is **4064256**.

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