Problems

Given an odd integer $n\ge 3$ and a regular $n$-gon with vertex set $V$, let $\mathcal{P}$ denote the set of all regular polygons with vertices in $V$.

For example, if $n=15$, then $\mathcal{P}$ includes:

• 1 regular 15-gon
• 3 regular pentagons
• 5 equilateral triangles

Two players, Alice and Bob, play the following game:

• Initially, all points in V are uncolored.
• Alice moves first.
• They take turns coloring an uncolored point:

• Alice colors red.
• Bob colors blue.
  • The game ends when all points are colored.

A polygon in $\mathcal{P}$ is called red-dominated if it has more red vertices than blue ones.

Find the largest integer $k$ such that, no matter how Bob plays, Alice can guarantee at least $k$ red-dominated polygons in $\mathcal{P}$.

Prove that for any odd prime $p$, the number of positive integers $n$ satisfying

is at most

where $c$ is a constant independent of $p$.

Let $n_1,n_2,\dots,n_{26}$ be pairwise distinct positive integers satisfying the following conditions:

• In the decimal representation of each $n_i$, every digit belongs to the set $\{1,2\}$.

• For any $i,j$, $n_j$ cannot be obtained from $n_i$ by adding some digits on the right.

Find the least possible value of

where $S(m)$ denotes the sum of all digits of $m$ in decimal representation.

Let

be the set of all non-negative integers.

For each $n\in\mathbb{N}$, define the Catalan number

Prove that for any positive integer $m$, we have

Given a positive integer $n$, find the smallest positive integer $k$ with the following property:

After coloring any $k$ unit squares black and the remaining squares white in a $2n\times 2n$ grid, it is always possible to make all squares black using a finite number of the following two operations:

1. Select a $2\times2$ square that contains exactly $3$ black squares and $1$ white square, and change the color of the white square to black.

2. Select a $2\times2$ square that contains exactly $2$ black squares and $2$ white squares, and change the color of all squares within the square: black squares become white, and white squares become black.

Note: The same $2\times2$ square can be operated on multiple times, and each square can change color multiple times.

Every positive integer is colored either blue or red. Prove that there exists an infinite sequence $\{a_n\}$ of positive integers with

such that the sequence

consists entirely of positive integers of the same color.

Let $n\ge 2$ be an integer. Two players, Alice and Bob, play the following game.

Initially, all $\binom n2$ edges of the complete graph $K_n$ are uncolored.

They take turns coloring edges red, with Alice starting first.

In each move, a player selects one or two uncolored edges to color red, with the constraint that after each move, no three vertices can have all three edges between them colored red.

The game ends when no more edges can be colored.

Prove that there exists a real number $\varepsilon>0$ such that for any integer $n\ge 2$, no matter how Bob plays, Alice can ensure that at the end of the game, the number of red edges does not exceed