Find the truth - Humanity at stake

It is unknown to humans if $\frac { e }{ \pi }$ is rational or irrational. But this question can be solved.

Obviously, $\frac { e }{ \pi }$ is either rational or irrational because it is real.

Now let's take 2 cases:

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Case 1: $\frac { e }{ \pi }$ is rational

Now let $\frac { e }{ \pi } =r$ , where $r$ is rational.

So statement 1 is true and 2 is false .

Statement 3 is saying that $\frac { 1 }{ r } +1$ is rational. The reciprocal of a rational number is rational (except if it is zero). So $\frac { 1 }{ r }$ is rational. Since rationals are closed under addition, $r+1$ becomes rational, so statement 3 is true .

So, for Case 1 , the mean is $\frac { 1+3 }{ 2 }$ or $\boxed {2}$ .

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Case 2: $\frac { e }{ \pi }$ is irrational

Now let $\frac { e }{ \pi } =r$ , where $r$ is irrational.

So statement 1 is false and 2 is true .

Statement 3 is saying that $\frac { 1 }{ r } +1$ is rational. So $r$ is also a rational because it is the reciprocal of $1$ subtracted from a rational number. But this is not satisfied by the case, so 3 is false

So the mean for Case 2 is $\frac { 2 }{ 1 }$ or $\boxed {2}$ .

So the answer to this question is $\boxed {2}$ regardless of the rationality of the ratio $e$ is to $\pi$ .

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