Difference between revisions of "2024 IMO Problems/Problem 1"
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| − | + | Determine all real numbers <math>\alpha</math> such that, for every positive integer <math>n</math>, the integer | |
<cmath>\lfloor \alpha \rfloor + \lfloor 2\alpha \rfloor + \dots +\lfloor n\alpha \rfloor</cmath> | <cmath>\lfloor \alpha \rfloor + \lfloor 2\alpha \rfloor + \dots +\lfloor n\alpha \rfloor</cmath> | ||
| − | is | + | is a multiple of <math>n</math>. (Note that <math>\lfloor z \rfloor</math> denotes the greatest integer less than or equal to <math>z</math>. For example, <math>\lfloor -\pi \rfloor = -4</math> and <math>\lfloor 2 \rfloor = \lfloor 2.9 \rfloor = 2</math>.) |
Revision as of 10:40, 16 July 2024
Determine all real numbers 
such that, for every positive integer 
, the integer
![\[\lfloor \alpha \rfloor + \lfloor 2\alpha \rfloor + \dots +\lfloor n\alpha \rfloor\]](http://latex.artofproblemsolving.com/d/e/8/de8ec4b01ecf552960e95a201881ef7e35e7a2ce.png)
is a multiple of . (Note that 
denotes the greatest integer less than or equal to 
. For example, 
and 
.)
