How many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter that does not exceed $10^8$?
This problem is a combinatorics problem and is quite complex, so let’s break it down.
We have to find the number of triangles with integral sides and at least one integral angle, and their perimeter should not exceed 10^8.
We also know that for any triangle with integers sides a, b, c (a ≤ b ≤ c), if one of its angles is an integral number of degrees, then (a^2 + b^2 – c^2) / (2ab) must be a rational number. This follows from the cosine rule in trigonometry, where cos(C) = (a^2 + b^2 – c^2) / (2ab) for a triangle with sides a,b,c and angle C opposite to side c.
By substituting p = a+b+c, c = p-a-b, we have f(a,b,p) = a^2 + b^2 – (p-a-b)^2 = 2ab – p^2. If it is a rational number, then p^2 <= 2ab < 2p^2 for p > 2.
Iterating through all possible values of p and for each fixed p, iterating through all possible pairs (a,b), one can solve this problem. You will need to run 2 loops – the outer one for p from 3 to 10^8, and the inner one for a going from p/3 to p/2. For each p and a, find the number of possible b such that 2ab-p^2 > 0 and b > a and b < (p-a).
The pseudocode would look like:
```
count = 0
for p = 3 to 10^8
for a = p/3 to p/2
b = (p^2)/(4a)
if floor(b) > a and 2*a * floor(b) > p^2
count++
“`
This solution has a time complexity of O(n^2) but would require a huge computational resource because 10^8 is a large number. To further optimize the code, we can make use of binary search and prefix sum to reduce the time complexity.
However, note that running this for 10^8 is computationally intensive and may not be feasible on a regular computer.
This problem may require analytical methods to solve efficiently rather than a purely brute force approach, due to the high limit on the perimeter. It’s likely a problem that was meant to be simplified using various mathematical concepts and then solved, instead of solved directly.
More Answers:
Primitive TrianglesA Modified Collatz Sequence
Linear Combinations of Semiprimes