China Mathematical Olympiad 2018 Q6
Given the positive integer $n ,k$ $(n>k)$ and $ a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq  \{1,2,\cdots,n\}$ ,$|\mathbb{I} |=k$,have $\sum_{i\in \mathbb{I}  }x_i\le \sum_{i\in \mathbb{I}  }a_i$ , find the maximum value of $x_1x_2\cdots  x_n.$
To solve the problem, we need to find the maximum value of the product \( x_1 x_2 \cdots x_n \) given the constraints on \( x_i \) and \( a_i \). We will use induction and a lemma involving polynomials to achieve this.

1. **Base Case:**
   - For \( n = 2 \) and \( k = 1 \), the problem is trivial since \( x_1 \leq a_1 \) and \( x_2 \leq a_2 \). Thus, \( x_1 x_2 \leq a_1 a_2 \).

2. **Inductive Hypothesis:**
   - Assume the statement holds for all smaller cases, i.e., for any \( n' < n \) and \( k' < k \).

3. **Define \( t_i \):**
   - Let \( t_i = x_i - a_i \) for each \( 1 \leq i \leq n \). If \( t_i = 0 \) for some \( i \), then \( x_i = a_i \) and we are done. Assume \( t_1 \geq t_2 \geq \cdots \geq t_m > 0 > t_{m+1} \geq \cdots \geq t_n \).

4. **Sum Constraint:**
   - For any set \( \mathbb{I} \subseteq \{1, 2, \cdots, n\} \) with \( |\mathbb{I}| = k \), we have:
     \[
     \sum_{i \in \mathbb{I}} t_i \leq 0
     \]

5. **Lemma:**
   - Let \( P \) be a polynomial of degree \( n \) with one negative root and \( n-1 \) positive roots, say \( y_1 < 0 < y_2 \leq \cdots \leq y_n \). Suppose \( f(0) > 0 \) and \( f'(0) < 0 \). Then \( f'(x) < 0 \) in the interval \( x \in [0, y_2] \).

   **Proof of Lemma:**
   - By Rolle's theorem, there is a root of \( f'(x) \) in each interval \( [y_i, y_{i+1}] \). Since \( f'(x) \) has at most \( n-1 \) roots, it has exactly one root in \( [y_1, y_2] \). Given \( f(0) > 0 \), \( f(x) > 0 \) in \( [y_1, y_2] \). Thus, \( f'(x) < 0 \) in \( [0, y_2] \).

6. **Case I: \( t_m \leq k-1 \):**
   - Define \( b_i = a_i - \frac{t_m}{k-1} \). Normalize \( b_i \) such that \( b_i \in (k-2, k-1) \). Then:
     \[
     \sum_{j \in I} b_i \geq \sum_{j \in I} x_i
     \]
     for \( |I| = k-1 \) and \( I \subset \{1, 2, \cdots, n\} \setminus \{m\} \). By the inductive hypothesis:
     \[
     \prod_{i \neq m} x_i \leq \prod_{i \neq m} b_i = \prod_{i \neq m} \left( a_i - \frac{t_m}{k-1} \right)
     \]
     Denote \( t_m \) by \( t \):
     \[
     \prod_{i=1}^n a_i \leq (a_m + t) \prod_{i \neq m} \left( a_i - \frac{t}{k-1} \right)
     \]
     Define this polynomial in \( t \) by \( f(t) \). Notice:
     \[
     f(0) = a_1 a_2 \cdots a_n > 0
     \]
     \[
     f'(0) = a_1 a_2 \cdots a_n \left( -\frac{1}{k-1} \sum_{i \neq m} \frac{1}{a_i} + \frac{1}{a_1} \right) < a_1 a_2 \cdots a_n \left( -\frac{n-1}{(k-1)k} + \frac{1}{k-1} \right) = 0
     \]
     By the lemma, \( f(t) < f(0) = a_1 a_2 \cdots a_n \).

7. **Case II: \( t_m > k-1 \):**
   - Let \( p = k - m \). Replace \( t_1, \cdots, t_m \) by \( t_1 - p\epsilon, \cdots, t_m - p\epsilon \) and \( t_{m+1}, \cdots, t_n \) by \( t_{m+1} + m\epsilon, \cdots, t_n + m\epsilon \) such that \( t_m - p\epsilon \geq 0 \geq t_n + m\epsilon \). This is called an \( \epsilon \)-operation.

   - Suppose after an \( \epsilon_1 \)-operation, \( t_m \) becomes \( k-1 \) and after an \( \epsilon_2 \)-operation, \( t_{m+1} \) becomes \( 1 \). Denote \( \epsilon = \min\{\epsilon_1, \epsilon_2\} \). After an \( \epsilon \)-operation, \( t_1, t_2, \cdots, t_n \) become \( s_1, s_2, \cdots, s_n \). Define the polynomial \( f \) by:
     \[
     f(t) = (a_1 + s_1 + pt)(a_2 + s_2 + pt) \cdots (a_m + s_m + pt)(a_{m+1} + s_{m+1} - mt) \cdots (a_n + s_n - mt)
     \]
     Notice:
     \[
     f(0) = (a_1 + s_1) \cdots (a_1 + s_n) > 0
     \]
     \[
     f'(0) = (a_1 + s_1)(a_2 + s_2) \cdots (a_n + s_n) \left( p \sum_{i=1}^m \frac{1}{a_i + s_i} - m \sum_{i=m+1}^n \frac{1}{a_i + s_i} \right) < \frac{1}{pm} \left( \frac{1}{a_m + s_m} - \frac{1}{a_{m+1} + s_{m+1}} \right) < 0
     \]
     By the lemma:
     \[
     x_1 x_2 \cdots x_n = f(\epsilon) < f(0)
     \]
     and \( f(0) \leq a_1 a_2 \cdots a_n \) by Case I.

Thus, the maximum value of \( x_1 x_2 \cdots x_n \) is \( a_1 a_2 \cdots a_n \).

The final answer is \( \boxed{ a_1 a_2 \cdots a_n } \).