You can just ask GPT-4 to rewrite it using unicode characters:
Rewrite this post using unicode characters:
> The IMO are a bit more involved...
Answer:
> The IMO are a bit more involved...
e.g. IMO 2023:
> Problem 1. Determine all composite integers n > 1 that satisfy the following property: if d₁, d₂, ..., dₖ are all the positive divisors of n with 1 = d₁ Although ChatGPT 4 makes a pretty decent start on this already:
> To determine all composite integers 𝑛 > 1 that satisfy the given property, we need to closely examine the condition set forth: for a composite number 𝑛 with its divisors d₁, d₂, ..., dₖ where 1 = d₁ Let's break down the problem:
1. Composite Number Requirement: We are looking for composite numbers, which are integers greater than 1 that are not prime. These numbers have more than two distinct positive divisors.
2. Divisor Property: The property dᵢ | (dᵢ₊₁ + dᵢ₊₂) means that dᵢ₊₁ + dᵢ₊₂ is a multiple of dᵢ.
3. Systematic Examination: We will analyze this property systematically for composite numbers starting from the smallest composite number (4) and going upwards.
Let's start the analysis:
- For 𝑛 = 4, the divisors are 1, 2, 4. Here, 1 | (2 + 4), so 4 satisfies the condition.
- For 𝑛 = 6, the divisors are 1, 2, 3, 6. Here, 1 | (2 + 3) and 2 | (3 + 6), so 6 satisfies the condition.
- For 𝑛 = 8, the divisors are 1, 2, 4, 8. Here, 1 | (2 + 4), but 2 ∤ (4 + 8), so 8 does not satisfy the condition.
Let's implement a systematic method to identify such composite numbers up to a reasonable limit (let's say up to 100 for a start) and check if they satisfy the given condition.
The composite integers greater than 1 and up to 100 that satisfy the given property are: 4, 8, 9, 16, 25, 27, 32, 49, 64, and 81.
Each of these numbers has the characteristic that for all their positive divisors d₁, d₂, ..., dₖ (where 1 = d₁ < d₂ < ... < dₖ = 𝑛), the condition dᵢ divides dᵢ₊₁ + dᵢ₊₂ is true for every 1 ≤ i ≤ k − 2.