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 divisible by
Because
But
The only real numbers
Solution 2 (logic)
If we assume that a is a whole number, you could say that the last term is automatically a multiple of n, and from there you can add the first term and the penultimate term,
The number cannot have a fraction or every few sequences it will add a number, but, if the number n is the number directly after the fraction adds one, like if the fraction was 1/3, and n was three, it would add one to the sequence, which would already be divisible by n, but one plus a multiple of n/2 (if
Solution 3 (Casework)
Let's determine all such that
is divisible by
for every positive integer
.
First, even integers work: if where
is an integer, then
which is divisible by
.
Conversely, let where
and
. Then
Case 1: is even. Then
is divisible by
, so
must be divisible by
.
By induction, we can show for all
, which implies
. Thus
is an even integer.
Case 2: is odd. Using similar reasoning, we can derive that
for all
, which implies
for all
. This is impossible.
Therefore, only even integers satisfy the condition.
~brandonyee
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 |