I did it a different way and ended up with 2 simpler equations: n^2 = a + b - 1 2n = a - b Adding one to the other you can get rid of b and solve for n in terms of a. a needs to be of the form: a = 2*k^2, k an integer, k>=0 to have an integer solution. You end up with a simpler solution (no division by 8 nor mod 4 condition): a = 2k^2 b = 2k^2 - 4k + 2 for all integers k >= 0. With solutions being all pairs (a,b) and (b,a).
Wish we could go back to the good old days when maths was just numbers😂 why are we adding LETTERS🤔 This is MATHS not ENGLISH. next they will make imaginary numbers and pretend they make sense😂 what a bunch of silly billies. I am very bad at English so sadly I could not do this question and even worse I could not share with my friends like you told me to because I have no friends. I sent it to my brother though but he is stupid so he couldn’t do it. Thank you for the elegant solution and smart kangaroo, when I learn English and what a skwaire rewt is, I will try question again🙌
If k = 0 (mod 4) then k = 4n, thus
1/8 k^2 = 1/8(4n)^2 = 16/8 n^2 = 2n^2
1/8(k+4)^2 = 1/8(4n + 4)^2 = 16/8(n + 1)^2 = 2(n + 1)^2
Take two sequential non-negative integers, square then double them.
Great way to simplify it!
I did it a different way and ended up with 2 simpler equations:
n^2 = a + b - 1
2n = a - b
Adding one to the other you can get rid of b and solve for n in terms of a.
a needs to be of the form:
a = 2*k^2, k an integer, k>=0
to have an integer solution.
You end up with a simpler solution (no division by 8 nor mod 4 condition):
a = 2k^2
b = 2k^2 - 4k + 2
for all integers k >= 0.
With solutions being all pairs (a,b) and (b,a).
Very creative, great solution!
Another less complicated solution: (a.b)= (2z^2, 2(z-1)^2) for all z being an integer
Great solution!
This also ties to the increasing number of atoms in each second row of the periodic table. i.e. the series 0, 2, 8, 18, 32, 50, ...
Yesss the goofy thumbnails we've all been waiting forrrr. Please could you do an Indian. Olympiad question next? Much love!
I’ll give it a try!
a=2
b=8
a=k^2, b=(k-sqrt(2))^2
Can we use AM GM inequality and relations
Nice video
Thank you! ❤️
That kangaroo...
Wish we could go back to the good old days when maths was just numbers😂 why are we adding LETTERS🤔 This is MATHS not ENGLISH. next they will make imaginary numbers and pretend they make sense😂
what a bunch of silly billies.
I am very bad at English so sadly I could not do this question and even worse I could not share with my friends like you told me to because I have no friends. I sent it to my brother though but he is stupid so he couldn’t do it. Thank you for the elegant solution and smart kangaroo, when I learn English and what a skwaire rewt is, I will try question again🙌
Bro is 8 year old
👍🇧🇷