Problem 094

[Oliver1978]

Thanks I've solved it in the meantime.

[Animesh111]

I need clarification in the problem statement of 'almost equilaterals' in problem 94. It says that "Find the sum of the perimeters of all almost equilateral triangles with integral side lengths and area and whose perimeters do not exceed one billion (1,000,000,000)." My doubt is whether the sum of perimeters should be less than a billion or value of perimeter of any triangle should be less than 1 billion.
Kindly tell me

[Georg]

Consider only triangles whose perimeters do not exceed one billion (1,000,000,000).

[mpiotte]

I need clarification in the problem statement of 'almost equilaterals' in problem 94. It says that "Find the sum of the perimeters of all almost equilateral triangles with integral side lengths and area and whose perimeters do not exceed one billion (1,000,000,000)." My doubt is whether the sum of perimeters should be less than a billion or value of perimeter of any triangle should be less than 1 billion.
Kindly tell me

Please, do not create a new topic if one already exist.
You will notice the same exact question was asked and answered before.
Merged threads.

[Animesh111]

there is a number 235000835, if we form triangles as sides 235000835,235000835,235000836, we get an almost equilateral triangle satisfying the constraints of the problem. But this number is not included in the final perimeter sum as adding this number increases the sum above 1 billion but the perimeter is less than 1 billion. so what is the question asking for??

[Georg]

What is the area of your triangle?

[Animesh111]

it was an overflow error, this triplet is not a valid solution, i checked now.

[enigmaticcam]

Got the help I needed. Thanks

[TinyTavi]

For problem 94 it states:
Find the sum of the perimeters of all almost equilateral triangles with integral side lengths and area

Is it supposed to be integer? Or is there something major I am missing?

[DJohn]

Find the sum of the perimeters of all almost equilateral triangles with integral side lengths and area

Is it supposed to be integer? Or is there something major I am missing?

"Integral side lengths" means the side lengths are integers - "integral" is the adjective corresponding the noun "integer". Since the perimeter is the sum of the side lengths, it's going to be an integer too.

[skoczian]

Shouldn't this thread be added to the existing thread "Problem 094"?

[hk]

Done.

[rbagdazian]

In the article on Heronian triangles (https://en.wikipedia.org/wiki/Heronian_triangle) the table of almost equilateral Heronian triangles does not include the following: 227, 227, 228. I get the following integral area for this case: 22378. Why isn't this in the table?

[bloebje]

In the article on Heronian triangles (https://en.wikipedia.org/wiki/Heronian_triangle) the table of almost equilateral Heronian triangles does not include the following: 227, 227, 228. I get the following integral area for this case: 22378. Why isn't this in the table?

The area is close to an integer, but it is slightly smaller than 22378.

[rbagdazian]

Ah, that's good to know. It's probably why my algo is failing to obtain the correct solution. I guess I have to rethink my approach to testing for integral area.

[Vinny360]

Is 5479171588 as the final answer at least close?

Edit: Nevermind, I just found out where is my error. I worked only with those triangles that differed by 1, not with those that were in fact equilateral...

Edit 2: Wait, there are no equilateral triangles with integral area. Soooo ... What the hell am I missing?

You are missing some triangles. (I'll give you a hint - you are missing more than one triangle)