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Problem of the Day #358: Albert March 11, 2012

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Inspired by a problem from the  2012 VCU Programming Contest

Albert likes to draw circles. He first draws a circle of radius $100$. Then he draws a circle of radius $15$ whose center is $d$ units away from the center of the bigger circle. Find $d$ such that $\frac{3}{4}$ of the area of the smaller circle is within the bigger circle.

Problem of the Day #331: Albert February 13, 2012

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Albert and Billy are playing a game:

At the starting of each round, both players write down a number. They then play rock-paper-scissors $8$ times.

If a player wins at least as many times as the number he wrote, his score increases by the cube of the number written. Else, the score is cut in half.

Albert is exceptionally good at rock-paper-scissors, and wins $\frac{3}{4}$ of the time.

Assuming both players play optimally, compute the probability that Albert has the higher score after $5$ such rounds.

Problem of the Day #289: Escape! January 2, 2012

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Albert and Alex are playing tag. Since Albert is too fast for Alex to catch, Alex decides to make a robot to tag him. The robot moves twice as fast as Albert. The robot moves directly towards Albert at all times.

Albert begins at the origin, and the robot begins at $(0,1)$. If the robot tags Albert at the point $(x,0)$, find $x$.

Problem of the Day #173: Factorials September 8, 2011

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Find all ordered pairs $(x,y)$ such that $$1!\,2!\,3!\cdots x!=y!$$

Summer Contest Results August 23, 2011

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  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 #156: Dr. Kim’s Solar Cells, Part 2 August 22, 2011

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Dr. Kim has recently ordered a shipment of ten very expensive solar cells. As the cells arrive one at a time, Dr. Kim places them in a stack on his desk (with the last one received on the top). Izzy occasionally walks in, trips, and smashes the solar cell on the top of the stack. Given that she has already smashed the fifth solar cell Dr. Kim received, in how many ways can Izzy smash the remaining cells?

End of Summer Contest August 16, 2011

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Problem #149 was the last contest problem. The contest will end in one week (Tuesday, August 23 at 4PM EST).

End of Fourth Two-Week Period August 16, 2011

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Please join us in congratulating Brian Shimanuki, the winner of the third two-week period. This period spanned problems $136$ through $149$.

As the top-scoring solver over the past two weeks that hasn’t already won a two-week prize, Brian wins a $\$10$ gift card of his choice. You can view the full rankings for these two weeks after the jump. (more…)

Problem of the Day #147: ▲ August 13, 2011

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In triangle ABC, $\angle ABC=95^\circ$ and $\angle ACB=50^\circ$. Point P is constructed inside the triangle such that $BP=CP$ and $\frac{[BPC]}{[ABC]}=\frac{1}{5}$. Find the integer closest to $\angle BPC$.

Problem of the Day #145: Albert August 11, 2011

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Let S be a sequence of random variables, all 0 or 1, such that the probability of $S_n$ being 0 is $\frac{1}{n^2}$. Let $p$ be the expected value of the probability that the product of the elements of a randomly chosen subsequence of S is nonzero. Find the integer closest to $10000\cdot p$.

You may use any computational aid.