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Difference between revisions of "2021 IMO Problems/Problem 6"

(Created page with "==Problem== Let <math>m>2</math> be an integer, <math>A</math> be a finite set of (not necessarily positive) integers, and <math>B_1,B_2,B_3,...,B_m</math> be subsets of <math...")
 
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==Problem==
 
==Problem==
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Let <math>m>2</math> be an integer, <math>A</math> be a finite set of (not necessarily positive) integers, and <math>B_1,B_2,B_3,...,B_m</math> be subsets of <math>A</math>. Assume that for each <math>k = 1, 2,...,m</math> the sum of the elements of <math>B_k</math> is <math>m^k</math>. Prove that <math>A</math> contains at least <math>m/2</math> elements.
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Let <math>m\ge 2</math> be an integer, <math>A</math> be a finite set of (not necessarily positive) integers, and <math>B_1,B_2,B_3,...,B_m</math> be subsets of <math>A</math>. Assume that for each <math>k = 1, 2,...,m</math> the sum of the elements of <math>B_k</math> is <math>m^k</math>. Prove that <math>A</math> contains at least <math>m/2</math> elements.
  
 
==Video solution==
 
==Video solution==
 
https://www.youtube.com/watch?v=vUftJHRaNx8 [Video contains solutions to all day 2 problems]
 
https://www.youtube.com/watch?v=vUftJHRaNx8 [Video contains solutions to all day 2 problems]

Revision as of 23:03, 30 July 2021

Problem

Let $m\ge 2$ be an integer, $A$ be a finite set of (not necessarily positive) integers, and $B_1,B_2,B_3,...,B_m$ be subsets of $A$. Assume that for each $k = 1, 2,...,m$ the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $m/2$ elements.

Video solution

https://www.youtube.com/watch?v=vUftJHRaNx8 [Video contains solutions to all day 2 problems]

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