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Problem of the Day #373: Palindromey Strings March 26, 2012

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A palindromey string is a string of lowercase letters that is either a palindrome or the concatenation of two palindromey strings. Find the number of palindromey strings of length $12$.

Problem of the Day #372: Binary Tree Path Lengths March 25, 2012

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Inspired by a problem from the 2012 TJ IOI.

The nodes in a complete binary tree of infinite height are referred to by ordered pairs $(x, y)$, where $x$ and $y$ are 0-indexed values denoting the row and the column of the nodes. The root is $(0, 0)$. Find the length of the shortest path between node $(1000, 3)$ and $(1234, 123456)$.

Problem of the Day #371: Connecting Components March 24, 2012

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Suppose we have a list of $100$ nodes. Every second, two nodes are randomly chosen and an edge is drawn in between the two nodes. What is the expected number of seconds that must elapse before the number of connected components of the graph is less than or equal to $5$?

Problem of the Day #368: Paths through a Grid, Part II March 21, 2012

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Albert starts at the top left corner of a $16$ by $16$ grid and is allowed to move up, down, or right for each step. How many ways are there for him to reach the bottom right corner, given that no cell is traversed more than twice?

Problem of the Day #367: Non-Repetitive Paths through a Grid March 20, 2012

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Albert starts at the top left corner of a $16$ by $16$ grid and is allowed to move either down one cell or right one cell for each step. He is not allowed to move in the same direction more than $3$ times in a row. How many ways are there for Albert to reach the bottom right corner?

Problem of the Day #366: Canyon Numbers March 19, 2012

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Today is MPotD’s one year anniversary! A canyon number is a number with at least three digits that satisfies the following properties:

  1. The first and the last digit are the same.
  2. There is a unique minimum digit, $k$.
  3. The digits up to and including $k$ form a strictly decreasing sequence.
  4. The digits including $k$ and beyond form a strictly increasing sequence.

Find the number of $10$-digit canyon numbers.

Problem of the Day #365: Circles and Triangles March 18, 2012

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Circles $A$, $B$, and $C$ have radii $12$, $16$, and $20$ respectively and are externally tangent to each other. Find the area of the region inside the incircle of $\triangle ABC$ but outside circles $A$, $B$, and $C$.

Problem of the Day #361: PI March 14, 2012

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Happy $\pi$ day! Find the maximum possible value $h$ such that it is possible to fit $12$ right circular cones, with base radius $\frac{1}{\sqrt{\pi}}$ and height $h$, in a cube with side length $2$.

Problem of the Day #360: 360 Albedegrees March 13, 2012

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Albert has invented a new way to measure angles: Albedegrees. An angle that measures $360$ degrees also measures $360$ Albedegrees, but Albedegrees have the property that an angle with degree measure $x$ has three times the Albedegree measurement as an angle with degree measure $\frac{x}{2}$. In terms of $n$, find the degree of each angle in a regular $n$-gon.

Problem of the Day #359: A Knight on an Infinite Chessboard March 12, 2012

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Given any constant starting location, how many positions on an infinite-size chessboard can be reached after exactly $100,000$ knight moves?