## 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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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 #276: Christmas CookiesDecember 20, 2011 Posted by Seungln in : potd , add a comment Albert, Alex, Arjun, Billy, Mitchell, Saketh, SeungIn and Sreenath have a huge pile of Christmas cookies. They do an epic sleepover at Saketh’s house, where the cookies are. In the middle of the night, Albert wakes up and eats$1$more than$\frac{1}{3}$of the cookies, and goes back to bed. Later that night, Alex eats$2$more than$\frac{1}{3}$of the remaining cookies, and goes back to bed. Arjun gets up later, eats$3$more than$\frac{1}{3}$of the remaining cookies, and goes back to bed, and so on. The$n$th person eats$n$more than$\frac{1}{3}$of the cookies. If there are$238$cookies remaining after each person has eaten cookies, how many cookies were there to start with? ## Problem of the Day #270: Meme GeneratorDecember 14, 2011 Posted by Seungln in : potd , 2 comments Albert, Billy, SeungIn and Sreenath are engaging in a meme war, in which every person makes memes out of everyone else. All of them are lazy, however, and they each make computer programs that will make memes for them. None of the programs are perfect, however, and have only a certain chance to be successful. If the sum of the probability that one person will succeed twice in a row is$1$and the sum of the probability that one person will succeed, fail and then succeed again, in that order, is also$1$, and assuming that Billy is the best programmer and that Sreenath is the worst, what is the difference between Billy’s probability of success and Sreenath’s probability of success? ## Problem of the Day #257: 4December 1, 2011 Posted by Seungln in : potd , add a comment$4$is an interesting number. What is the most concise way of expressing$256$using only the number$4$, excluding$4^4$and the mathematically equivalent$\displaystyle \frac{1}{\frac{1}{4^4}}$,$4 \cdot 4 \cdot 4 \cdot 4$or$\displaystyle \frac{1}{\frac{1}{4 \cdot 4 \cdot 4 \cdot 4}}$? You are limited to addition, subtraction, multiplication, division and exponents. ## Problem of the Day #250: Billy’s PhillysNovember 24, 2011 Posted by Seungln in : potd , add a comment Billy goes to Philadelphia (obviously to get a very good Philly cheese steak). To lower his risk of bladder cancer, however, Billy must limit his Philly’s cheese limit to$52g$. Assume that Billy can order his Phillys in three different sizes, as follows: • Small Philly: costs$\$3.25$, comes on a rectangular bread of $8$ cm by $24$ cm
• Medium Philly: costs $\$ 4.25$, comes on a rectangular bread of$11$cm by$27$cm • Large Philly: costs$\$5.25$, comes on a rectangular bread of $14$ cm by $30$ cm
Assume that the density of cheese is $1 \frac{g}{cm^3}$, and that the cheese on each of Billy’s possible Phillys have a $.01$ cm thickness of cheese that covers each of the two sides of the cut bread. Assuming that Billy has an unlimited amount of money for his Phillys, and that Billy will not leave a single bit of Philly to go to waste, what is the maximum possible amount of Billy’s Phillys, and how much will he have to pay for them?

## Problem of the Day #224: Curved TrianglesOctober 29, 2011

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Saketh has a triangle with area of $84$ that has its corners curved into arcs of circles with radius $2$. What is the maximum possible area of this curved triangle?

## Problem of the Day #205: R.I.P. Steve JobsOctober 10, 2011

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This is kind of old news now, but MPOTD decided to pay its own tribute to Steve Jobs with this problem.

The iPhone 4S (still) has the $3.5$ in. screen (i.e. a screen that is measured as $3.5$ in. from one corner to its opposite corner). Assume that the ratio of the short side of the touchscreen to the long side of the touchscreen does not matter, as long as the area of the screen is at least $4 in.^2$. If, before he passed away, Steve Jobs was planning on making the iPhone 5 have a $5$ in. screen, keeping its shortside-to-longside proportion the same, what is the difference between the maximum and the minimum areas possible for the iPhone 5′s touchscreen?

## Problem of the Day #186: SquaresSeptember 21, 2011

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$365$ is an interesting number because it can be expressed as a sum of three consecutive squares ($10^2 + 11^2 + 12^2$) AND as a sum of two consecutive squares ($13^2 + 14^2$). What is the next biggest integer for which this property can happen?

## Problem of the Day #185: Not Paying Attention in AP Chinese ClassSeptember 20, 2011

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Andrew Tao and David Guo decide to not pay attention during  Ms. X’s AP Chinese class. Because they are sitting directly next to each other, however, they decide that at least one of the two have to be paying attention at any given time in order to avoid grabbing the attention of Ms. X, who would undoubtedly give both of them a bad grade. Andrew, however, has a short attention span and can only pay attention for $n$ minutes at a time, where $n$ is a positive integer between $5$ and $20$, before he has to take a $3$ minute break from paying attention. If the class is $90$ minutes and David decides to go in a cycle of $14$ minutes of attention and $4$ minutes of not paying attention, what is the sum of the values of $n$ that will allow the two of them to get away with their lack of attention, regardless of where in their cycle each of them start?