2023 CMO Problems

Day 1

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

Problem 2

Find the largest real number $C$ such that for any positive integer $n$ and any real numbers $x_1, x_2, \ldots, x_n$ , the following inequality holds: $\sum_{i=1}^n \sum_{j=1}^n(n-|j-i|) x_i x_j \geq C \sum_{i=1}^n x_i^2$

Solution

Problem 3

Given a prime number $p \geq 5$ , let $\Omega=\{1,2, \ldots, p\}$ . For any $x, y \in \Omega$ , define: $r(x, y)= \begin{cases}y-x, & \text { if } y \geq x \\ y-x+p, & \text { if } y<x\end{cases}$

For a non-empty subset $A$ of $\Omega$ , define: $f(A)=\sum_{x \in A} \sum_{y \in A}(r(x, y))^2$

A subset $A$ of $\Omega$ is called a "good subset" if $0<|A|<p$ and for any subset $B$ of $\Omega$ with $|B|=|A|$ , it holds that $f(B) \geq f(A)$ .

Find the largest positive integer $L$ such that there exist $L$ pairwise distinct good subsets $A_1, A_2, \ldots, A_L$ of $\Omega$ satisfying $A_1 \subseteq A_2 \subseteq \cdots \subseteq A_L$ .

Solution

Day 2

Problem 4

Let non-negative real numbers $a_1, a_2, \ldots, a_{2023}$ satisfy $a_1+a_2+\cdots+a_{2023}=100$

Define $N$ as the number of elements in the set $\left\{(i, j) \mid 1 \leq i \leq j \leq 2023, a_i a_j \geq 1\right\}$

Prove that $N \leq 5050$ and provide necessary and sufficient conditions for the equality to hold.

Solution

Problem 5

In an acute triangle $\triangle A B C, K$ is a point on the extension of $B C$ . Through $K$ , draw lines parallel to $A B$ and $A C$ , denoted as $K P$ and $K Q$ respectively, such that $B K=$ $B P$ and $C K=C Q$ . Let the circumcircle of $\triangle K P Q$ intersect $A K$ at point $T$ . Prove: (1) $\angle B T C+\angle A P B=\angle C Q A$ ; (2) $A P \cdot B T \cdot C Q=A Q \cdot C T \cdot B P$ .

Solution

Problem 6

The numbers $1,2, \ldots, 99$ are placed on the vertices of a given regular 99 -gon, with each number appearing exactly once. This arrangement is called a "state." Two states are considered "equivalent" if one can be obtained from the other by rotating the 99 -gon in the plane.

Define an "operation" as selecting two adjacent vertices of the 99-gon and swapping the numbers at these vertices. Find the smallest positive integer $N$ such that for any two states $\alpha$ and $\beta$ , it is possible to transform $\alpha$ into a state equivalent to $\beta$ with at most $N$ operations.

Solution

2023 CMO(CHINA) (Problems • Resources)
Preceded by 2022 CMO Problems	Followed by 2024 CMO Problems
1 • 2 • 3 • 4 • 5 • 6
All CMO(CHINA) Problems and Solutions

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2023 CMO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6

See Also