Floors

This is a seemingly simple problem that I found very interesting, with a surprising answer. Find the sum of all positive solutions to 2x2xx=52x^2-x\lfloor x\rfloor=5 (HMNT 2011 G5).

#FloorFunction

Note by Cody Johnson
7 years ago

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Comments

Is G guts? My first reaction was "why is this a geometry problem!?" :P

Michael Tang - 7 years ago

Step One: Let x=q+rx=q+r where qZq\in\mathbb{Z} and 0r<10\le r<1.

Cody Johnson - 7 years ago

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It is true that either q=1q=1 or q=2q=2. We can quickly find the answer afterwards.

Daniel Liu - 7 years ago

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Yeah we can also bound it like this: 2x25=xxx22x^2-5=x\lfloor x\rfloor\le x^2

Xuming Liang - 7 years ago

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@Xuming Liang Can this be applied to negative numbers too? Why is 52-\frac52 a solution, yet (52)2=6.25>5\left(-\frac52\right)^2=6.25>5?

Cody Johnson - 7 years ago

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@Cody Johnson For negatives, xxx2x\lfloor x\rfloor \geq x^2.

Michael Lee - 7 years ago
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