Question

Fool's cutest problem of the day:

In a certain exam, there are exactly \( n\) questions.

It is also known that that \(2^{n-k} \) students gave wrong answers to at least \( k \) questions where \( k \in [1,n] \). If the total number of wrong answers given is \( 65535\), then how many questions are there in exam?

Answer 1

Just to notifying few great minds: @satellite73 @Zarkon @JamesJ @myininaya @amistre64 @Mr.Math @asnaseer

Answer 2

\[\mbox{n wrong answers}\quad\cdots\quad 1\mbox{ student}\]\[\mbox{n-1 wrong answers}\quad\cdots\quad 2^1-2^0=1\mbox{ student}\]\[\mbox{n-2 wrong answers}\quad\cdots\quad 2^2-2^1=2\mbox{ students}\]\[\vdots\]\[\mbox{k wrong answers}\quad\cdots\quad 2^{n-k}-2^{n-k-1}=2^{n-k-1}\mbox{ students}\]\[\vdots\]\[\mbox{1 wrong answer}\quad\cdots\quad 2^{n-1}-2^{n-2}=2^{n-2}\mbox{ students}\]\[\mbox{total wrong answers: }\quad n+\sum\limits_{k=1}^{n-1} k\cdot 2^{n-k-1}=n+2^{n-1}\sum\limits_{k=1}^{n-1} k\cdot 2^{-k}=\]\[=n+2^{n-1}\cdot 2^{1-n}(-n+2^n-1)=2^n-1\]\[2^n-1=65535\quad\Rightarrow\quad n=16\]

Answer 3

From what I can gather, there is one one answer on this exam, and im most likely to get it wrong :)

Answer 4

n?