2024 IMO Problems/Problem 1
Determine all real numbers such that, for every positive integer , the integer
is a multiple of . (Note that denotes the greatest integer less than or equal to . For example, and .)
Contents
[hide]Solution 1
To solve the problem, we need to find all real numbers
is divisible by
Step 1: Break Down
Let
Step 2: Express the Sum in Terms of
Using this, we have:
So, the sum becomes:
Step 3: Modulo
We are interested in
Since
Step 4: Analyze the Fractional Part
Our goal is to find all
Step 5: Test
If
which satisfies the condition for all
Step 6: Consider
For
Thus,
Step 7: Conclude that Only
Since
Step 8: Verify for Integer
Let
This sum is always divisible by
The only real numbers
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)
Video Solution
See Also
2024 IMO (Problems) • Resources | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |