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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 #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 #166: Cubic Properties September 1, 2011

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$9$ has an interesting property that its cube can be expressed as a sum of three different, nonzero cubes. ($9^3 = 8^3 + 6^3 + 1^3$, to be more specific.) What is the next smallest integer that has this property?

Problem of the Day #164: Late Night Problems August 30, 2011

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Find the smallest factor of $164^8$ that is bigger than $164^4$.

Problem of the Day #162: 162 August 28, 2011

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Let $x$ be the smallest positive integer for which $162 \cdot x$ is divisible by $x + 162$. Find $$x + \frac{162 \cdot x}{x + 162}$$

Problem of the Day #161: Potato Chip Geometry August 27, 2011

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SeungIn has an interesting potato chip, the shape of which can be described as a union of a circle with radius $.00002$ miles and a circle with radius $.12$ feet, with $10$ square inches of overlap. If an inch is $2.54$ centimeters, then the area of the chip can be expressed as $x$ meters squared. Find $x$.

Problem of the Day #158: SeungIn’s Program August 24, 2011

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Because the slackers have neglected their duty of posting problems for MPOTD, SeungIn decides to take action and make a program that generates problems everyday at 4 p.m. However, since SeungIn’s programming is not perfect (and because SeungIn prefers some subjects to others), the distribution of the types of problems are uneven, and, once in a while, it will print out two problems instead of one. When Billy inspected the code, he found the distribution of the types of problems:

Algebra – $\displaystyle \frac{5}{18}$

Geometry – $\displaystyle \frac{1}{9}$

Number Theory – $\displaystyle \frac{7}{18}$

Combinatorics – $\displaystyle \frac{2}{9}$

Billy also finds that the program generates two problems with $\displaystyle \frac{1}{7}$ chance. However, Billy was not able to fix the code until SeungIn had to implement it to the website in order to make it make problems everyday at 4 p.m. Given the information above, what is the probability that, within a given three-week period, there will be $6$ or fewer Number Theory problems generated by SeungIn’s program?

Problem of the Day #157: EARTHQUAKE!!! August 23, 2011

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A magnitude $6.0$ earthquake struck Virginia today (Aug. 23rd, 2011). (It was actually magnitude $5.9$, but assuming that it’s $6.0$ will make your math easier.) The relative magnitude is the magnitude felt by someone within a certain distance from the epicenter (the starting point of the earthquake), and is calculated with the formula

$$\large rm(x) = am \cdot (1 – \log_{10} (1+x^{\frac{3}{19}}))$$

where $rm(x)$ is the relative magnitude, $am$ is the absolute magnitude, and $x$ is the distance from the epicenter to the impact point in miles.

Find the area of the zone within which $rm(x)$ was at least $3.0$, and divide the answer by $\pi mile^2$. The final result will be your answer.