## Problem of the Day #409: Sohail’s Function
*May 1, 2012*

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

Sohail is tinkering around with a function $f$. Suddenly, he realizes that his function satisfies the equation $f(x+y) + f(xy) = f(x)f(y) + 1$ for all pairs of integer values $x$ and $y$. Find all possible values of $f(2012)$.

## Problem of the Day #408: Rolling Dice
*April 30, 2012*

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

Find the expected number of rolls of a fair six-sided die before the sequence of rolls contains $1, 2, 3, 4, 5, 6$ (in that order) as a subsequence.

## Problem of the Day #407: Cow Tipping
*April 29, 2012*

*Posted by Saketh in : potd , 1 comment so far*

Alex is going cow-tipping! Each cow he encounters is shaped like a cube, and has each edge and face colored either black or white. As an exceptionally skilled cow-tipper, Alex knows that many seemingly different cows are not rotationally distinct. How many different cow colorings are there?

## Problem of the Day #406: True or False
*April 28, 2012*

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

Albert is taking a test where each of the $1000$ questions has one of two answers: *true* or *false*. Having not prepared for the test, he does not know the answer to a single problem and resorts to intelligent guessing. He knows that the number of questions on the test with answer *true* is a power of two, with each power of two having an equal probability and each possible distribution being equally likely for a certain *true* count. Assuming Albert guesses optimally, how many questions is he expected to answer correctly?

## Problem of the Day #405: Funny Functions
*April 27, 2012*

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

Function $f$ is defined such that $f(x^2 – y^2) = (x-y)(f(x)+f(y))$ for all real $x$ and $y$. If $f(1) = 2012$, find $f(2012)$.

## Problem of the Day #404: Biking Paths
*April 26, 2012*

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

The city planning committee is laying down a set of bike trails between $n$ major attractions. They plan to submit a proposal asking for $kn$ of the $\binom{n}{2}$ possible trails to be built, where $k$ is some constant.

To accommodate athletes who wish to perform laps, the committee wants to guarantee that a cycle of length $4$ will be built. However, they don’t know which $kn$ trails will be chosen. What is the least value of $k$ for which a cycle of length $4$ will definitely exist?

## Problem of the Day #403: The Lone Prime Factor
*April 25, 2012*

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

Albert added two perfect cubes together, resulting in a number with exactly one prime factor. Find all possible values for that factor.

## Problem of the Day #402: Equal Subset Products
*April 24, 2012*

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

Alex takes a set of $k$ consecutive positive integers and partitions them into two disjoint sets such that the product of their elements is equal. Find all values of $k$ such that this is possible.

## Problem of the Day #401: The Duck and The Fox
*April 23, 2012*

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

A number of different versions of the following classic puzzle exist:

A duck, pursued by a fox, escapes to the center of a perfectly circular pond. The fox cannot swim, and the duck cannot take flight from the water. The fox is $k$ times faster than the duck. Assuming the fox pursues an optimal strategy, how can the duck reach the edge of the pond and fly away without being eaten?

First, find a strategy by which the duck can escape if $k=4$. Then, determine the smallest value of $k$ such that it is impossible for the duck to escape.

## Problem of the Day #400: Squirrel City
*April 22, 2012*

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

In Squirrel City there is a $100$ by $100$ grid on which every treehouse is located, one at each lattice point. The administration is doing some construction and needs to chop down trees for wood.

They know that no citizen will complain unless there exists a treehouse from which at least two different stumps can be observed. Furthermore, there must be at least one treehouse left for the squirrels to use.

Determine the greatest number of trees which can be chopped down without inciting any complaints. Assume that the radius of each tree is negligible when compared to the gaps between the trees.