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...+ a | m \in M\} </math>. Kobayashi's theorem states that the set of prime numbers that divide at least one element of ''M + a'' is infinite.</blockquote> ...p>3</sup> = a''. Since the left-hand side is irreducible over the rational numbers, by [[Thue equation|Thue's theorem]], each equation has a finite number of ...

For positive real numbers <math>a, b, c:</math> '''Example 2.''' [[International Mathematical Olympiad]] (IMO) 1995. ...

...ts+x_n^2}{n}}.</math><ref>{{Cite book |last=Djukić |first=Dušan |title=The IMO compendium: a collection of problems suggested for the International Mathem ...chy–Schwarz_inequality#real_number_proof|Cauchy–Schwarz inequality on real numbers]], setting one vector to {{math|(1, 1, ...)}}: ...

...tical Olympiad |url=http://www.imo-official.org/problems.aspx |website=www.imo-official.org |access-date=29 September 2020}}</ref> ...t in this time. The problem committee submitted it to the jury of the XXIX IMO marked with a double asterisk, which meant a superhard problem, possibly to ...

[[File:Ship diagram-numbers.svg|thumb|right|Net tonnage is calculated by measuring a ship's internal vo ...nt-of-ships.aspx |date=2013-12-02 }}. International Maritime Organization (IMO). {{retrieved | access-date=2011-10-17}}</ref> ...

...ion|compass and straight edge]], and the values are called [[constructible numbers]]. ...], whereas if the codomain is taken to be the [[projectively extended real numbers]], these entries take the value <math>\infty</math> (see [[division by zero ...

...ment. Emerging regulations by the [[International Maritime Organization]] (IMO) have all but ceased application of biocides, causing a rush to research en ...copper, zinc, and all other heavy metal coatings have been outlawed by the IMO.<ref name=Heavy_Metal_Paints>{{cite journal|last=Almeida|first=Elisabete|au ...

...who receive joint prostheses.<ref name="medical devices"/> To reduce these numbers, the surfaces of the devices used in these procedures have been altered in ...rg/en/OurWork/Environment/Anti-foulingSystems/Pages/Default.aspx "Focus on IMO - Anti-fouling systems"]. International Maritime Organisation.</ref> ...

...eventing Collisions at Sea |publisher=International Maritime Organization (IMO), United Nations |date=October 20, 1972 |at=Part B, Section I, Rule 6}}</re ...anization (IMO), United Nations |access-date=3 July 2018 |quote=Members of IMO are governed by [[International Regulations for Preventing Collisions at Se ...