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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==
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== 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 , \ldots, 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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==Video solution==
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== Video solution ==
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https://www.youtube.com/watch?v=vUftJHRaNx8 [Video contains solutions to all day 2 problems]
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https://youtu.be/vUftJHRaNx8 [Video contains solutions to all day 2 problems]
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== See also ==
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{{IMO box|year=2021|num-b=5|after=Last Problem}}
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[[Category:Olympiad Algebra Problems]]

Latest revision as of 09:44, 18 June 2023

Problem

Let $m \ge 2$ be an integer, $A$ be a finite set of (not necessarily positive) integers, and $B_1, B_2, B_3 , \ldots, B_m$ 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://youtu.be/vUftJHRaNx8 [Video contains solutions to all day 2 problems]

See also

2021 IMO (Problems) • Resources
Preceded by
Problem 5 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Last Problem
All IMO Problems and Solutions
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