Given three positive integers a, b and c such that a² + b² - c² = 1. Let the number of unique triangles formed with sides a, b and c with perimeter less or equal than 50 million represent the surface area of an ellipsoid of axes lengths n, 2n and 3n.

What is the square of the ceiled positive solution of n.

Where did you come across this problem? What is your maths background? This site is aimed towards helping you to solve the problem rather than just giving you an answer. But this is quite an intriguing question!

Initially I think you need to find how many positive integer triples a,b,c exist such that:-

a² + b² - c² = 1

a+b+c ≤ 50,000,000

You don't actually need to know the values of a,b,c that form the set of the solutions. Just work out how many solutions there are. However, I personally think that there's no shortcut, so you probably do need to work out what all the a,b,c triples are. Unless someone else on the forum has a clever method?

To start off, there are 24,999,999 solutions of the form b=1 and c=a.

And that's not all. Here are the additional solutions up to perimeter 50 with b≤a (since if a≠b then a and b can be trivially swapped to give another solution)

Step-by-step explanation:

a. the are of the rectangle is 18.25

b. 18.25 times two is 36.50

rae me : d

∠lno ≅ ∠lnm

∠oln ≅ ∠mln

step-by-step explanation:

given   that triangle lno is congruent to triangle lnm.

we have to fill up the blanks in the given question.

according to above order of congruency   kept in mind we can say that

∠lno ≅ ∠lnm

∠oln ≅ ∠mln

ln ≅ ln due to common in both.

hence, by sas congruency,

the side angle side rule   states that if two sides and the included angle of one triangle are congruent to another triangle, then these two triangles are congruent.