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

Posted by Saketh in : potd , add a comment

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 Puzzle December 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 Numbers December 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 Squares December 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 Tongues December 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 Fanclub December 26, 2011

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The Albert T. Gural Fanclub currently has $7$ members: Saketh, Sreenath, Albert, Billy, SeungIn, Arjun, Aziz. Because this club only has $7$ members so far, each member (except Albert) decides to recruit more people. Saketh, Sreenath, Billy, SeungIn, Arjun and Aziz have the same recruiting abilities, and are going to be able to recruit $1$, $2$, $3$, $4$, $5$ or $6$ new members, not necessarily in that order, with equal probability. The six members decide to split the pile of $\$ 21,000$ proportional to the amount of members each person recruited. If SeungIn suddenly receives temporary foresight that lasts long enough to show him that Arjun will only recruit $2$ people, how much money can he expect to receive?

Problem of the Day #281: Sreenath’s Gift December 25, 2011

Posted by Alex in : potd , 2 comments

For those of you who celebrate it, Merry Christmas!

Sreenath wanted dinosaurs for Christmas. However, he’s been naughty this year, so Santa decided to leave this math problem:

(Help Sreenath evaluateĀ $\displaystyle\prod_{i=2}^{280} \displaystyle\sum_{j=0}^{\infty} i^{-j}$)

Problem of the Day #280: Gift Labeling December 24, 2011

Posted by Alex in : potd , 1 comment so far

Santa Saketh has hired monkeys to label the many gifts he has. Each second, Saketh’s monkeys randomly write a number in the range $[1, 100]$ and Saketh labels that gift if it has not already been labeled. After how many seconds will the expected number of labeled gifts exceed $50$?

Bonus: the probability a number is written is inversely proportional to its number of digits. For example, $1$ has twice the probability of being written as $12$. Solve the problem with this new information.

Problem of the Day #279: Elves, Elves, Everywhere! December 23, 2011

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Sohail and Kevin are playing with some elves. They take turns presenting one coin to any elf that is still awake. Each elf will accept a total of $5$ coins before dispensing a present and going to sleep.

If there are $2011$ elves present, and Sohail gets to go first, how many presents is he guaranteed to win? Both Sohail and Kevin play optimally.

Problem of the Day #278: The Perfect Parasol December 22, 2011

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Sreenath received a $2$x$2$x$10$ box of coal bricks from Albert. He wants to use it as a parasol. If the sun is directly overhead, and Sreenath can orient the box any way he wishes, what is the greatest possible area of its shadow?