jump to navigation

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

Posted by Alex in : potd , add a comment

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

Posted by Alex in : potd , add a comment

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

Posted by Alex in : potd , 1 comment so far

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

Posted by Alex in : potd , add a comment

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 #364: Divisibility Pairs March 17, 2012

Posted by Saketh in : potd , add a comment

Suppose that Albert writes down all numbers of the form $p^0+q^0, p^1+q^1, p^2+q^2, \ldots$ up to $p^n+q^n$, where $p$ and $q$ are distinct primes. Determine, in terms of $n$, the number of unordered pairs of distinct members of this sequence such that one divides the other.

Problem of the Day #363: Minimum Distance from Incenter to Centroid March 16, 2012

Posted by Saketh in : potd , add a comment

If $\triangle ABC$ is a right triangle with hypotenuse $\overline{BC}$ of length $1$, determine the minimum possible distance between its incenter and its centroid.

Problem of the Day #362: Digitally Odd Numbers March 15, 2012

Posted by Saketh in : potd , add a comment

Suppose that $n$ is a five digit multiple of $5$, all of whose digits are odd. Suppose further than $\frac{n}{5}$ also has five odd digits. How many possible values of $n$ are there?

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

Posted by Alex in : potd , add a comment

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

Posted by Alex in : potd , add a comment

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

Posted by Alex in : potd , add a comment

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