There is a triangle with side lengths \(a\), \(b\) and \(c\). Can you form a triangle with side lengths \(\frac{a}{b}\), \(\frac{b}{c}\) and \(\frac{c}{a}\)? Does it depend on what \(a\), \(b\) and \(c\) are? Give a proof if it is always possible or never possible. Otherwise, construct examples to show the dependence on \(a\), \(b\) and \(c\).
Recall that a triangle can be drawn with side lengths \(x\), \(y\) and \(z\) if and only if \(x+y>z\), \(y+z>x\) and \(z+x>y\).
There is a triangle with side lengths \(a\), \(b\) and \(c\). Does there exist a triangle with side lengths \(|a-b|\), \(|b-c|\) and \(|c-a|\)? Does it depend on what \(a\), \(b\) and \(c\) are?
Recall that a triangle can be formed with side lengths \(x\), \(y\) and \(z\) if and only if all the inequalities \(x+y>z\), \(y+z>x\) and \(z+x>y\) hold.
There is a triangle with side lenghts \(a\), \(b\) and \(c\). Does there exist a triangle with sides of lengths \(a^2+bc\), \(b^2+ca\) and \(c^2+ab\)? Does it depend on the values of \(a\), \(b\) and \(c\)?
Could you meet a person inhabiting this planet who asks you “Am I a Goop?"
On this planet you meet a couple called Tom and Betty. You hear Tom ask someone: “Are Betty and I both Goops?"
What kind is Betty?
You learn that one of the aliens living on this planet is a wizard. You learnt that by overhearing a certain question being asked on the planet. What question could that have been?
Suppose you meet a person inhabiting this planet and they ask you “Am I a Crick?" What would you conclude?
You meet two friends, Katja and Anja. Katja once asked Anja “Is at least one of us a Goop?"
What kinds are Katja and Anja?
You later learn that there is exactly one wizard on this planet of Cricks and Goops. You would like to find out who that is.
You meet an alien called Andrew. He asks you “Am I the kind that could ask whether I am not the wizard?"
Do you have enough information to tell for sure who the wizard is by now?
You meet another alien, whose name is Teddy. He asks you “Am I the kind who could ask whether I am a Goop?"
Can anything be deduced about Teddy?