Difference between revisions of "2024 IMO Problems/Problem 1"
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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>.) | 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>.) | ||
| + | |||
| + | ==Video Solution(In Chinese)== | ||
| + | https://youtu.be/LW54i7rWkpI | ||
==Video Solution== | ==Video Solution== | ||
https://www.youtube.com/watch?v=50W_ntnPX0k | https://www.youtube.com/watch?v=50W_ntnPX0k | ||
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| + | ==Video Solution== | ||
| + | Part 1 (analysis of why there is no irrational solution) | ||
| + | |||
| + | https://youtu.be/QPdHrNUDC2A | ||
| + | |||
| + | Part 2 (analysis of even integer solutions and why there is no non-integer rational solution) | ||
| + | |||
| + | https://youtu.be/4rNh4sbsSms | ||
| + | |||
| + | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
Latest revision as of 00:11, 27 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 
.)
Video Solution(In Chinese)
Video Solution
https://www.youtube.com/watch?v=50W_ntnPX0k
Video Solution
Part 1 (analysis of why there is no irrational solution)
Part 2 (analysis of even integer solutions and why there is no non-integer rational solution)
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
