Problem: Find the perimeter of pentagon PQRST, given that PQ=3, QR=4, ST=x, TP=x+1 and all side lengths are integers.

Solution:
To find the perimeter of pentagon PQRST, we need to add up the lengths of all five sides. Let's label the sides and their lengths:

- Side PQ = 3
- Side QR = 4
- Side RS = x (unknown)
- Side ST = x (unknown)
- Side TP = x+1 (unknown)

Now, we can use the fact that the sum of the interior angles of a pentagon is equal to 540 degrees to find the value of x. The sum of the angles of a pentagon can be found using the formula:

sum of angles = (n-2) x 180 degrees, where n is the number of sides

So, for a pentagon, we have:

sum of angles = (5-2) x 180 degrees = 540 degrees

We also know that the angles of a regular pentagon are all equal, so each angle is 540 degrees / 5 = 108 degrees. We can use this information to find the value of x as follows:

- The angle at vertex R is shared by two sides, RS and QR, so it must be 108 degrees / 2 = 54 degrees.
- Similarly, the angle at vertex T is shared by two sides, TP and PQ, so it must also be 54 degrees.
- The angle at vertex S is shared by three sides, RS, ST, and QR, so it must be 540 degrees - 2 x 54 degrees - 2 x 72 degrees = 180 degrees - 144 degrees = 36 degrees.
- Finally, we can use the fact that the sum of the angles in a triangle is 180 degrees to find the angle at vertex P. The angle at vertex P is shared by two sides, PQ and TP, so the third angle must be 180 degrees - 54 degrees - 36 degrees = 90 degrees. This means that triangle PQR is a right triangle with PQ as the hypotenuse.

Now that we know the value of x, we can find the length of the remaining sides:

- RS = ST = x = 72 / (180 / π) = 4π
- TP = x+1 = 4π+1

Finally, we can add up the lengths of all five sides to find the perimeter:

Perimeter = PQ + QR + RS + ST + TP
Perimeter = 3 + 4 + 4π + 4π + 1
Perimeter = 8 + 8π

Therefore, the perimeter of pentagon PQRST is 8 + 8π.