LKLogic - math olympiad solutions [video]
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sqrt( (1000)(1001)(1002)(1003) +1 )
Like high-school math problems, they are manufactured puzzles (i.e. with a neat solution), which narrows the search space and tells you if you are on the right track. But unlike those, these have 2-3 tricks, so you get no feedback after the first step. It seems a bit like NP-complexity: hard to find the method; easy to check.
I'm sure there are many math olympians here on HN (and winners), who would find these unchallenging. Speaking for myself, I can do most in my head, and most of the remainder on paper - when I need their answer is when I learn the most.
A maybe-criticism: the answers are very clear, but they don't say why those steps were chosen. To me, the channel is more like a slightly-cleverer peer, rather than a super-genius master. Sometimes I find a better method, and it becomes a springboard for related ideas. And all that might well be better for learning.
Because it's a puzzle, it will be in the form sqrt(x^2), so the answer will be x.
The only perfect square I know for this level that could match is (a+b)^2 = a^2 + 2ab + b. With b=1, the first term needs to be in the form a^2 + 2a or a(a+2).
I had no intutition for what this could be, then noticed it needs to be close to a perfect square itself, and this closest to this would be the two inner factors times the two outer factors (for it to work out neatly). Writing as binomils for ease of calculation:
((1000+0)(1000+3)) ((1000+1)(1000+2))
(1,000,000 + 3000) (1,000,000 + 3000 + 2)
sqrt( (1,000,000 + 3000) + 1)^2 ) = 1,000,000+3000+1 = 1,003,001