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2021 IMO: Difference between revisions

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For the statement to be true, there must be at least three pairs whose sum is each a perfect square.
'''2021 [[IMO]]''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.
There must be p,q,r such that p+q = x^2 and q+r = y^2, p+r = z^2.
(In Russia)


WLOG n<= p<= q<= r <= 2n ... Equation 1
*[[2021 IMO Problems|Entire Test]]
**[[2021 IMO Problems/Problem 1 | Problem 1]], proposed by Australia
**[[2021 IMO Problems/Problem 2 | Problem 2]], proposed by Calvin Deng, Canada
**[[2021 IMO Problems/Problem 3 | Problem 3]], proposed by Mykhailo Shtandenko, Ukraine
**[[2021 IMO Problems/Problem 4 | Problem 4]], proposed by Dominik Burek and Tomasz Ciesla, Poland
**[[2021 IMO Problems/Problem 5 | Problem 5]], proposed by Spain
**[[2021 IMO Problems/Problem 6 | Problem 6]], proposed by Austria


p = x^2 + z^2 - y^2
== See Also ==
* [[IMO Problems and Solutions, with authors]]
* [[Mathematics competition resources]]


q = x^2 + y^2 – z^2
{{IMO box|year=2021|before=[[2020 IMO]]|after=[[2022 IMO]]}}
 
r = y^2 + z^2 – x^2
 
by equation 1
 
2n <= x^2 + z^2 – y^2 <= 4n
 
2n <= x^2 + y^2 – z^2 <= 4n
 
2n <= y^2 + z^2 – z^2 <= 4n
 
 
6n <= x^2 + y^2 + z^2 <= 12n
 
6n <= 3x^2 <= 12n
 
2n <= x^2 <= 4n
 
√(2n) <= x <= 2√n
 
At this time n >= 100, so
 
10 * √2 <= x,y,z <= 20
 
15 <= x,y,z <= 20
 
where
2|x^2 + y^2 – z^2
2|x^2 + z^2 – y^2
2|y^2 + z^2 – z^2
x = 16, y = 18, z = 20 fits perfectly
 
therefore the proposition is true

Latest revision as of 09:42, 18 June 2023

2021 IMO problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. (In Russia)

See Also

2021 IMO (Problems) • Resources
Preceded by
2020 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
2022 IMO
All IMO Problems and Solutions
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