## Task B-4.3.

How many isosceles trapezoids, with bases of different lengths, have integer side lengths and a perimeter of 2010?
Solution. Let $a$ and $c$ ($a > c$) be the bases of the trapezoid, and $b$ be the leg of the trapezoid. Then, from the triangle inequality, we conclude that

$$
a < 2b + c
$$

If we add $a$ to both sides, we get

$$
2a < a + 2b + c = 2010 \quad \text{or} \quad a < \frac{2010}{2} = 1005
$$

Thus, $a \in \{1, 2, 3, \ldots, 1004\}$.

If $a = 1004$ and $c < a$, then

$$
b = \frac{2010 - a - c}{2} = \frac{1006 - c}{2}
$$

We conclude that $c$ is an even number less than 1003, i.e., $c \in \{2, 4, 6, \ldots, 1002\}$, and there are 501 such numbers.

If $a = 1003$ and $c < a$, then

$$
b = \frac{2010 - a - c}{2} = \frac{1007 - c}{2}
$$

We conclude that $c$ is an odd number less than 1004, i.e., $c \in \{1, 3, 5, \ldots, 1001\}$, and there are 501 such numbers.

If $a = 1002$ and $c < a$, then

$$
b = \frac{2010 - a - c}{2} = \frac{1008 - c}{2}
$$

We conclude that $c$ is an even number less than 1002, i.e., $c \in \{2, 4, 6, \ldots, 1000\}$, and there are 500 such numbers.

Similarly, for $a = 1001$, we get 500 new trapezoids, and we continue this process all the way down to $a = 4$ and $a = 3$, for which there is only one desired trapezoid each. $a$ cannot be 1 or 2.

The total number of trapezoids with the given properties is

$$
2(1 + 2 + 3 + \cdots + 501) = 2 \cdot \frac{501 \cdot (1 + 501)}{2} = 251502
$$