China National Mathematical Olympiad 2025 - PROBLEMS

Problem 1

Let \( \alpha > 1 \) be an irrational number and \( L \) be an integer such that \( L > \frac{\alpha^2}{\alpha - 1} \). A sequence \( x_1, x_2, \dots \) satisfies:

\[ x_{n+1} = \begin{cases} \left \lfloor \alpha x_n \right \rfloor & \text{if } x_n \leqslant L \\ \left \lfloor \frac{x_n}{\alpha} \right \rfloor & \text{if } x_n > L \end{cases} \]

Prove that:

1. \(\{x_n\}\) is eventually periodic.
2. The eventual fundamental period of \(\{x_n\}\) is an odd integer which doesn't depend on the choice of \(x_1\).

Problem 2

Let \( ABC \) be a triangle with incenter \( I \). Denote the midpoints of \( AI \), \( AC \), and \( CI \) by \( L \), \( M \), and \( N \) respectively...

Prove that \( PQ \), \( LN \), and \( EF \) are concurrent.

Problem 3

Let \( a_1, a_2, \dots, a_n \) be integers such that \( a_1 > a_2 > \cdots > a_n > 1 \). Let \( M = \operatorname{lcm} ( a_1, a_2, \dots, a_n ) \).

For any finite nonempty set \( X \) of positive integers, define:

\[ f(X) = \min_{1 \leqslant i \leqslant n} \sum_{x \in X} \left\{ \frac{x}{a_i} \right\} \]

Suppose \( X \) is minimal and \( f(X) \geqslant \frac{2}{a_n} \). Prove that:

\[ |X| \leqslant f(X) \cdot M. \]

Problem 4

The fractional distance between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is defined as:

\[ \sqrt{ \left\| x_1 - x_2 \right\|^2 + \left\| y_1 - y_2 \right\|^2 } \]

where \( \left\| x \right\| \) denotes the distance between \( x \) and its nearest integer.

Find the largest real \( r \) such that there exist four points on the plane whose pairwise fractional distances are all at least \( r \).

Problem 5

Let \( p \) be a prime number and \( f \) be a bijection from \( \{0,1,\dots,p-1\} \) to itself. Suppose that for integers \( a,b \in \{0,1,\dots,p-1\} \):

\[ |f(a) - f(b)|\leqslant 2024 \quad \text{if } p \mid (a^2 - b) \]

Prove that there exist infinitely many \( p \) such that there exists such an \( f \), and there also exist infinitely many \( p \) such that there doesn’t exist such an \( f \).

Problem 6

Let \( a_1, a_2, \dots, a_n \) be real numbers such that:

\[ \sum_{i=1}^n a_i = n, \quad \sum_{i = 1}^n a_i^2 = 2n, \quad \sum_{i=1}^n a_i^3 = 3n \]

(i) Find the largest constant \( C \), such that for all \( n \geqslant 4 \):

\[ \max \{ a_1, a_2, \dots, a_n \} - \min \{ a_1, a_2, \dots, a_n \} \geqslant C. \]

(ii) Prove that there exists a positive constant \( C_2 \), such that:

\[ \max \{ a_1, a_2, \dots, a_n \} - \min \{ a_1, a_2, \dots, a_n \} \geqslant C + C_2 n^{-\frac 32} \]

where \( C \) is the constant determined in (i).