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

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Andrew Tao and David Guo decide to not pay attention during ┬áMs. 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?

Problem of the Day #184: The Pizza Buffet September 19, 2011

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Sid went out to eat pizza last weekend. He never leaves without eating at least $1000000$ “pieces” of pizza. However, he’s on a diet this week and he wants to cut a single pie into small bits to trick his stomach. What is the minimum number of straight cuts he must make on a circular pie to produce enough pieces?

Problem of the Day #183: H.E.R.P. D.E.R.P. September 18, 2011

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The robots H.E.R.P. and D.E.R.P. (don’t ask me what the acronyms stand for, because I don’t know either) are in love with each other, but they are $100$ feet away from each other. H.E.R.P. and D.E.R.P. decide to meet each other, and they give each other directions. They send out signals of their location every foot, and expect to arrive within $50$ iterations. However, due to the radio wave noise in their environment, there is a probability of $\displaystyle \frac{2}{7}$ that the signal will be incorrectly interpreted, in which case the two robots will turn $90^{\circ}$ either to the left or to the right, with equal probability. What is the expected value of iterations necessary for H.E.R.P. and D.E.R.P. will meet each other?

Problem of the Day #182: U Can’t Touch This! September 17, 2011

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Sreenath finds himself hovering in an amorphous prison cell. He tries to touch a wall and finds that he cannot get within $1$ meter of the surface that surrounds him. If Sreenath knows that this prison cell has a constant volume of $450$ meters cubed, then what is the maximum volume that Sreenath can occupy?

Problem of the Day #181: I Like Complex Numbers September 16, 2011

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Find the modulus of the sum of all possible $z$ that satisfy $$(z – 1)^{11} = (z + 1)^{11}.$$

Problem of the Day #180: Infinite Power Towers September 15, 2011

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$\def\iddots{{\kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu \raise12mu{.}}}$
What is the maximum value of $a$ for which $a^{a^{a^{\iddots}}}$ converges to a finite value? To what value does it converge?

Problem of the Day #179: Twins! September 14, 2011

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Tandrew Ao has finally found his long-lost twin, Andrew Tao.

Andrew Tao is giving Tandrew Ao some math problems to work on. However, Tandrew Ao’s world has a math that is slightly different from the math of Andrew Tao’s world.

For example, when given the same equation: $7x^2 = -28$, Andrew Tao will say $\pm 2i$ as the answer. Tandrew Ao will say $\pm 2$ as his answer. Both are correct, depending on which math they use.

For another example, Andrew Tao will solve $4x^2 + 4x + 1 = 0$ as $x = \displaystyle -\frac{1}{2}$, while Tandrew Ao will solve it as $x = \displaystyle \frac{i}{2}$.

If so, then What would be Tandrew Ao’s solution for the equation $x^3 – 7x^2 + 14x – 8 = 0$ be?

Problem of the Day #178: Twins! September 13, 2011

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Andrew Tao has finally found his long-lost twin, Tandrew Ao.

Tandrew Ao, who came from the parallel universe, gives Andrew Tao an unusual gift – a cylinder that constantly changes its radius and height but has the same volume ($400 \pi$ cubic inches). Andrew Tao can carry it only in a rectangular box container made of Aonium; otherwise the cylinder will explode into a burst of $\gamma$-rays. Aonium is expensive and therefore Andrew Tao wants to use as little Aonium as possible. If a cubic inch of Aonium is $\$1$, the Aonium container has to be at least half an inch thick, and the Aonium is sold only in cubic inches, what is the least amount of money that Andrew Tao has to spend to bring his gift back home?

Problem of the Day #177: Asking Season September 12, 2011

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It is “asking” season, and Joe the cow would like a date for a moovie. Bessie will only go to the moovie with Joe if he can figure out the perfect square below $100000$ with the most factors. Help Joe out.

Problem of the Day #176: Counting Squares September 11, 2011

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Determine the number of perfect squares between $11111111111000000000$ and $11111111111999999999$, inclusive.