MathDB

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Part of 2010 Bundeswettbewerb Mathematik

Problems(2)

t (a, b, c)= min{b/a,c/b} where a <= b <= c , sidelengths

Source: Germany Federal - Bundeswettbewerb Mathematik 2010, round 2, p1

4/14/2020
Let a,b,ca, b, c be the side lengths of an non-degenerate triangle with a≤b≤ca \le b \le c. With t(a,b,c)t (a, b, c) denote the minimum of the quotients ba\frac{b}{a} and cb\frac{c}{b} . Find all values that t(a,b,c)t (a, b, c) can take.
inequalitiesalgebrasidelengthsgeometric inequality
1...1 2 1...1 is a prime

Source: Germany Federal - Bundeswettbewerb Mathematik 2010, round 1, p1

4/9/2020
Exists a positive integer nn such that the number 1...1⏟n ones21...1⏟n ones\underbrace{1...1}_{n \,ones} 2 \underbrace{1...1}_{n \, ones} is a prime number?
number theoryprime numbersDigits