There exist five positive integers in arithmatic progression arranged in increasing order such that

- The sum of the first and the second is the fifth.
- The sum of the squares of the first and the second is the square of the third.
- The sum of the cubes of the first, second and the third is the cube of the fourth.
- The sum of these five numbers is the square of the middle one.

What is the sum of these five integers?

The answer is 25.

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We only need clues (1) and (4):

Say the numbers are $a,a+d,a+2d,a+3d,a+4d$ .

Clue (1): $2a+d=a+4d$ , so we have $a=3d$ and we can rewrite the list as $3d,4d,5d,6d,7d$ .

Clues (2) and (3) follow directly from this; they're the relations $3^2+4^2=5^2$ and $3^3+4^3+5^3=6^3$ respectively.

Clue (4): $25d=25d^2$ ; since the integers are positive, $d=1$ and the sum is $\boxed{25}$ .