Olympiad problems

1958 February Putnam, A3

Real numbers are chosen at random from the interval $[0,1].$ If after choosing the $n$-th number the sum of the numbers so chosen first exceeds $1$, show that the expected value for $n$ is $e$.

2012 NIMO Problems, 8

Tags: geometry , geometric transformation , reflection , integration , trigonometry , probability , expected value

Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$. [i]Proposed by Lewis Chen[/i]

2007 Princeton University Math Competition, 2

Tags: probability , expected value

Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs?