Competition problem — Combinatorics

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Combinatorics Difficulty 9.1 IMO level Find the answer

Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1 . In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds:

- $(1)$ one of the numbers on the blackboard is larger than the sum of all other numbers;

- $(2)$ there are only zeros on the blackboard.

Player $B$ must then give as many cookies to player $A$ as there are numbers on the blackboard. Player $A$ wants to get as many cookies as possible, whereas player $B$ wants to give as few as possible. Determine the number of cookies that $A$ receives if both players play optimally.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

To solve this problem, we need to carefully analyze the game dynamics and the optimal strategies for both players, $A$ and $B$ .

Initially, the blackboard contains 2020 copies of the number 1. The players' moves involve manipulating these numbers under certain rules:

1. Player $A$ erases two numbers, $x$ and $y$ .
2. Player $B$ then writes either $x+y$ or $|x-y|$ back on the blackboard.

The game ends under two conditions:
- One number becomes larger than the sum of all others.
- All numbers become zero.

To find the optimal strategies for both players:

### Analyzing the Strategies:

- **Player $A$ Strategy:**
- Player $A$ aims to keep the game going for as long as possible to maximize the number of cookies received. $A$ will look to choose $x$ and $y$ such that the choices available to $B$ either do not terminate the game quickly or lead to suboptimal reductions.

- **Player $B$ Strategy:**
- Player $B$ wants to end the game as soon as possible to minimize the number of numbers on the blackboard, thus minimizing the cookies given away. Therefore, $B$ tries to quickly reach one of the stopping conditions, preferably making all numbers zero.

### Evaluating the Outcome:

Given the 2020 starting numbers:

1. Game Dynamics:
- Each operation by the two players reduces the total number of numbers on the board by exactly one each round.
- If $m$ is the total number of operations performed, then the number of entries on the blackboard at the end of the game is $2020 - m$ .

2. Equalizing Strategy:
- Player $B$ can minimize the number of cookies by trying to ensure that the numbers gradually reduce to an equal or nearly equal state, especially driving towards zeroes by cleverly using the $|x-y|$ operation whenever possible.

3. Optimal Play:
- Player $B$ can achieve a situation where only 7 numbers remain on the blackboard through optimal reduction strategies despite the initial large number of 1s because:
- Using the $x+y$ or $|x-y|$ operation strategically can quickly consolidate numbers or eliminate non-zero numbers.

Given the optimal plays from both sides, the minimum number of numbers left on the board—and hence cookies to be given—is 7 after the conclusion of their strategies.

Thus, the number of cookies player $A$ receives is:
$\boxed{7}.$

This solution reflects the balance between both players playing optimally to achieve their conflicting goals.

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Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.