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Problem of the Day #265: Delivering a Textbook December 9, 2011

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Albert is situated on the coordinate plane at the point $(2,0)$. He wishes to give a textbook to his friend, who lives at $(0,1)$. Every step Albert takes, he moves one unit in either the positive $x$, positive $y$, negative $x$, or negative $y$ direction. How many paths of length $10$ can Albert walk such that he starts and ends at his home and he visits his friend’s house exactly once?

Problem of the Day #245: Distance between Two Centers November 19, 2011

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Albert draws a random isosceles triangle with perimeter $6$. What is the expected distance between the circumcenter and the incenter of the triangle?

Problem of the Day #239: Connecting Points on a Circle November 13, 2011

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Albert has $20$ distinct points evenly spaced on the circumference of a circle. He draws $10$ line segments that connect each point to exactly one other point. What is the expected value of the number of intersections there will be? Note that this is not the same as asking how many intersection points there will be.

Problem of the Day #226: Halloween Madness October 31, 2011

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It’s Halloween, and after Albert finishes his college apps he wants to go trick-or-treating. Albert’s neighborhood consists of 6 houses equally spaced around a circle of radius $1$ kilometer. He starts from his own house, visits the rest of the houses exactly once in a random order, and returns back to his house. What is the expected length of Albert’s trick-or-treating path?

Problem of the Day #225: Projecting Circles onto Planes October 30, 2011

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Albert has a plane and a disk of radius $15$ in $3$-dimensional space, both randomly oriented. The circle does not intersect the plane. If the circle is projected onto the plane, what is the expected value of the area of the resulting planar figure?

Problem of the Day #223: A Not-So-Random Walk October 28, 2011

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Albert is currently located at the number $0$ on the number line. Every turn, Albert has a $\frac{1}{x}$ chance of moving one unit in the positive direction and a $1-\frac{1}{x}$ of standing still, where $x$ is the number Albert is currently on. What is Albert’s expected distance from the origin after $20$ moves?

Problem of the Day #210: Albert’s Sequence October 15, 2011

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Albert has a sequence defined by \[\begin{align*}a_0 &= a_1 = a_2 = 0 \\ a_3 &= 1 \\ a_n &= 3 a_{n-2} - a_{n-4}.\end{align*}\] Find $a_{31}$.

Problem of the Day #206: Climbing Stairs October 11, 2011

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Albert is climbing a staircase with $n$ stairs. He is currently at the bottom of the stairs. For any step, let $m$ be the number of steps above him until the top. Each turn, Albert has a $\frac{1}{2^k}$ chance of going up $k$ stairs and a $\frac{1}{2^m}$ chance of staying on the same step. What is the expected number of turns it will take Albert to reach the top of the stairs?

Problem of the Day #202: Binary Strings October 7, 2011

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Albert creates a random binary string of length $20$. He searches from left to right through the string and replaces any instance of $101$ with $0$. He does this same procedure for $010$ and $1$. Albert continues applying these steps until there are no valid moves left. What is the expected value of the number Albert’s final binary string represents?

Problem of the Day #141: Delivering a Birthday Cake August 7, 2011

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Sreenath is on the surface of a sphere of radius $1$. He wants to hand-deliver a (belated) birthday cake to Bitcable as quickly as possible, which is located at a random point on the sphere. Unfortunately, Sreenath is also very hungry. After each second of travelling, Sreenath eats half of the remaining cake. Every time Sreenath consumes cake, his walking speed increases by $\frac{\pi}{32}$ units per second. Sreenath starts out with a walking speed of $\frac{\pi}{32}$ units per second. Let $p$ be the fraction of cake Bitcable can expect to receive. Compute the integer closest to $10000\cdot p$.