## Problem of the Day #287: Taxes in AlbertlandDecember 31, 2011

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Alex decides to spend a year ($365$ days) in Albertland, a country with some very unusual fiscal policies. He pays his taxes in the form of one coin each day, with coins in Albertland having values of $1$, $2$, $3$, $4$, or $5$ alberts.

The government asks for a certain denomination of coin each day. However, once Alex pays a coin of value $n$, he is exempt from paying another coin of the same value for the next $n$ days, even if the government asks for one. How many distinct totals could Alex pay for that year?

## Problem of the Day #286: 3D Slider PuzzleDecember 30, 2011

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Sid has decided to take the classic “slider puzzle,” pictured below, to the third dimension.

He divides a $3$x$3$x$3$ frame into $27$ $1$x$1$x$1$ cells, $26$ of which contain a unit cube. These cubes are labeled from $1$ to $26$. Given any initial configuration, the goal is to slide the cubes around so that cube $n$ and cube $27-n$ swap positions. For how many distinct initial configurations is this possible?

## Problem of the Day #285: Three-Digit NumbersDecember 29, 2011

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Alex is going to give Albert a set of $n$ distinct three-digit numbers. Albert wants to pick $5$ disjoint subsets such that each has the same sum, but he doesn’t know which numbers he’ll receive. Find the smallest value of $n$ for which Albert will always be able to find his subsets.

## Problem of the Day #284: Infinitely Many Perfect SquaresDecember 28, 2011

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Find all ordered pairs of integers $(a,b)$ such that $a\cdot 2^n+b$ is a perfect square for every non-negative integer $n$.

## Problem of the Day #283: Bunny TonguesDecember 27, 2011

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Saketh has three different bunnies, each with long tongues, just like this guy.

Saketh later discovers that these bunnies’ tongues grow or shrink every night with these following rules.

The length of Bunny 1′s tongue increases by 4 times the length of Bunny 2′s tongue, but decreases by 3 times the length of Bunny 3′s tongue.

The length of Bunny 2′s tongue increases by twice the length of Bunny 1′s tongue, but decreases by the length of Bunny 3′s tongue.

The length of Bunny 3′s tongue doubles, but decreases by twice the length of Bunny 1′s tongue.

However, Saketh notices that there seems to be no difference in the lengths of the tongues of each bunny between each day. What is the ratio of the lengths of the tongues of the bunnies? Express the ratio in the format (length of the tongue of Bunny 1):(length of the tongue of Bunny 2):(length of the tongue of Bunny 3).

## Problem of the Day #282: The Albert T. Gural FanclubDecember 26, 2011

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