2023 CMO Problems/Problem 1

Find the smallest real number $\lambda$ such that any positive integer $n$ can be expressed as the product of 2023 positive integers $n=x_1 x_2 \cdots x_{2023}$ , where for each $i \in$ $\{1,2, \ldots, 2023\}$ , either $x_i$ is a prime number or $x_i \leq n^\lambda$ .

Solution 1

1. Let $n=2^{2024}$ . Then there exist $x_i \geq 4$ and $\lambda \geq \frac{1}{1012}$ .

2. Assume $n=x_1 x_2 \cdots x_{1012} p_1 p_2 \cdots p_k$ where $x_1 \sim x_{1012} \in \mathbb{N}^{+}$ and $x_i \leq n^{\frac{1}{1012}}$ with $x_i p_j>n^{\frac{1}{1012}}$ . Also, let $p_1 \leq p_2 \leq$ $\cdots \leq p_k$ be primes and $x_1 \leq x_2 \leq \cdots \leq x_{1012}$

We will show that $k \leq 1011$ . Suppose otherwise, that $k \geq 1012$ . Then $n \geq x_1^{1012} \cdot p_1^{1012} \Rightarrow x_1 p_1 \leq n^{\frac{1}{1012}}$ which leads to a contradiction. Therefore, the minimum $\lambda$ is: $\lambda_{\min }=\frac{1}{1012}$ ~moving|szm

2023 CMO(CHINA) (Problems • Resources)
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2023 CMO Problems/Problem 1

Solution 1

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