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Problem of the Day #165: Redecorating the Barn August 31, 2011

Posted by Saketh in : potd , add a comment

Alex is firing paintballs at the barn’s side wall. While each shot he fires is of a different color, he also wishes to produce “hybrid” colors by overlapping some of these splotches of paint. Unfortunately, he doesn’t have very good aim and cannot make any guarantees when it comes to producing overlaps.

Since Alex is in possession of a very large paint shooter, he knows that each shot will cover at least half of the barn wall. If he wishes to guarantee that at least one of the hybrid colors he produces will take up at least one fifth of the barn wall, how many paintballs must he fire?

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 #163: Random Hand Grabbing August 29, 2011

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$16$ people are playing a game called “Human Knot.” Each person reaches for another’s hand randomly with each hand such that each pair of hands is equally likely to grab each other. What is the probability that, once untangled, the people form a single closed loop?

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

Posted by Seungln in : potd , add a comment

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

Posted by Seungln in : potd , add a comment

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 #160: Ladder Climbing August 26, 2011

Posted by Alex in : potd , add a comment

There are $160$ kids, each next to an infinitely tall ladder. Each kid starts at height $0$, and rungs on the ladder differ by height $1$. Every second, each kid flips a fair coin. If the coin lands heads, the kid climbs up one step. Else, the kid stops climbing forever. Find the expected value of the average height of the kids after each one of them as finished climbing.

Problem of the Day #159: Top Ranked Muffins August 25, 2011

Posted by Saketh in : potd , add a comment

Mitchell has purchased $1024$ muffins at the local bakery. Mitchell knows that he can determine the heavier of any pair of muffins in one use of his extremely accurate Weight Comparison Device. What is the least number of comparisons he must make to determine the three heaviest muffins?

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

Posted by Seungln in : potd , add a comment

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?

Summer Contest Results August 23, 2011

Posted by Sreenath in : announcement , add a comment
  1. Andrew Tao
  2. Eugene Chen
  3. Lewis Chen
  4. Brian Shimanuki
  5. Steven Hao
  6. Nick Haliday
  7. Kun Liu
  8. 한웅 (Andrew Kim)
  9. Rick Huang
  10. Andy Jiang

Full rankings: (more…)

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

Posted by Seungln in : potd , add a comment

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.