## Problem of the Day #378: Filling a Grid, Part I
*March 31, 2012*

*Posted by Alex in : potd , add a comment*

Given a grid with $R$ rows and $C$ columns, what is the maximum number of cells that can be colored such that no colored cell is adjacent to more than two other colored cells? Express your answer in terms of $R$ and $C$. Two cells are adjacent if they share an edge.

## Problem of the Day #377: Happy $n_b^\text{th}$ Birthday!
*March 30, 2012*

*Posted by Albert in : potd , add a comment*

Sreenath has foolishly forgotten how old Jon is. He knows Jon is $57$, $73$, or $91$, but can’t remember which. In an attempt to figure it out, Sreenath poses a clever math problem for Jon on his birthday (today). “Happy $n_b^{\text{th}}$ Birthday, Jon!” Sreenath says (where $b$, the base of $n$, is an integer greater than 2). Solve for $n$, given that it is three digits long.

## Problem of the Day #376: Log Entry 01
*March 29, 2012*

*Posted by Albert in : potd , add a comment*

Alex has decided to provide you with the following data:

$$

\begin{eqnarray}

\ln(2) & \approx & 0.693147181 \\

\ln(3) & \approx & 1.09861229 \\

\ln(5) & \approx & 1.60943791 \\

\ln(7) & \approx & 1.94591015

\end{eqnarray}

$$

Your mission, should you choose to accept it, is to find the number of digits in the decimal expansion of $35^{12000}$.

## Problem of the Day #375: Log Entry 00
*March 28, 2012*

*Posted by Albert in : potd , add a comment*

Alex has decided to provide you with the following data:

$$

\begin{eqnarray}

\ln(2) & \approx & 0.693147181 \\

\ln(3) & \approx & 1.09861229 \\

\ln(5) & \approx & 1.60943791 \\

\ln(7) & \approx & 1.94591015

\end{eqnarray}

$$

Your mission, should you choose to accept it, is to find the smallest positive integer $x$ such that $2^{10 x}$ has more than $3x+1$ digits.

## Problem of the Day #374: Increasing Sequence Numbers
*March 27, 2012*

*Posted by Alex in : potd , add a comment*

Find the number of integers satisfying the property that the concatenation of all the odd-indexed digits (the leftmost digit is index 1), starting from the leftmost digit, followed by all the even-indexed digits, starting from the rightmost even-indexed digit, forms a strictly increasing sequence.

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

*Posted by Alex in : potd , add a comment*

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*

*Posted by Alex in : potd , add a comment*

*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*

*Posted by Alex in : potd , add a comment*

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 #370: Rows of Counters
*March 23, 2012*

*Posted by Saketh in : potd , add a comment*

Alex is playing a game in which he manipulates a row of black and white counters. To change the row, he is allowed to remove any black counter and replace it with either a white counter and a black counter (in that order), or two black counters. He is not allowed to perform any other manipulations.

If he starts with a single black counter, how many distinct rows of $10$ counters can he arrange?

## Problem of the Day #369: Circumradius of a Pyramid
*March 22, 2012*

*Posted by Saketh in : potd , add a comment*

If a right pyramid with a square base and angles of $120^{\circ}$ between its faces has an inradius of length $1$, find its circumradius.