A Nice Olympiads Trick | No Calculator Allowed

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[@DarkGir1]

Same Idea as my approach, but yours is too complicated written down: Let x = sqrt(9*10*11*12+1), then x^2 = 9*10*11*12+1 => x^2 - 1 = 9*10*11*12 (applying 3. binomial formula) => (x-1)(x+1) = 9*10*11*12 (rearranging right side) => (x-1)(x+1) = (9*12)(10*11) (multiply brackets) => (x-1)(x+1) = 108*110 (Since the difference of x-1 and x+1 is 2 and the difference of 108 and 110 is as well 2, we bring both sides to the same form) => (x-1)(x+1) = (109-1)(109+1) Now, it is eas to see that x=109 is the solution