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Problem of the Day #77: Triangles and Angles June 4, 2011

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If $\angle B’EC$ and $\angle ABA’$ are right angles, find the measure of $\angle CDB$.

Problem of the Day #71: 7th Time’s the Charm May 29, 2011

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Given that side $a$ has a measure of $30$ meters, determine the difference between the area of the circumcircle of triangle $AB’C$ and the area of the circumcircle of triangle $BB’C$.

Problem of the Day #56: Spring Cleaning May 14, 2011

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Five people, Sreenath, Mitchell, Alex, Albert and Sam, live in the same house and need to clean up before their parents come visit.

Each person gets his own room. A person will clean only their own room. Each person’s own room has a certain mess level, greater values requiring more cleaning. A cycle consists of everyone cleaning a certain amount of their own room and dumping the rest of their mess into another person’s room. Each person is required to clean between $5$ and $20$ of their mess each cycle. Whatever is left uncleaned is dumped into a randomly chosen other person’s room. Whenever someone dumps their remaining mess, the person receiving it will gain an additional 5 mess units as a cost for dumping.

Mess Factor of:
Sreenath — 150
Mitchell — 80
Alex — 50
Albert — 10
Sam — 70

The goal is to minimize the number of cycles it takes to clean the house.

Determine the expected number of cycles it would take if everyone acted independently and if they came up with and followed an optimal plan before beginning the cleanup.

Problem of the Day #46: Happy Alberts from Outer Space May 4, 2011

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In another world within the 4th dimension, Albert has been chosen to be cloned for the intergalactic space wars.

In their world, every day is an Albert day. Albert loves the idea and would like to travel there. To do so, Albert will use the 4th dimension, much like one would use a line to get from one point to another point.

Albert will jump off his own world onto a space station, The Albert, located in the Albertverse.

If we define Albert’s world by the points: (1,0,2,0), (-20,4,15,0), (17,13,20,0), and (-10,6,12,0) and the location of The Albert as: (15,4,3,-2), then find the spot in Albert’s world which minimizes the distance he has to jump in order to reach The Albert.

Problem of the Day #37: Winner April 25, 2011

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Albert and Arjun are competing in a running tournament with $128$ total participants. They are the best of buddies, but also very competitive, so they’d like to race against each other in the tourney.

The tourney follows a double elimination format, where the winners of the first round move into the Winner Bracket while the losers go into the Loser Bracket.

Every match result thereafter follows the Best of Three format, where a runner needs to win two races against his opponent before moving on in the Bracket.

Compute the number of ways that when Arjun competes against Albert, he has less losses than Albert.

Problem of the Day #28: The 7′s of Modularity April 16, 2011

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Find the last 3 digits of $$7777^{777^{77^7}}$$

Problem of the Day #19: Race of Complexity April 7, 2011

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Several capitalists have developed a sickly condition known as capital overflow. To cure themselves, they decide to construct a race track in the shape of a regular pentagon and race against each other.

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Problem of the Day #16: ‘Sid Nils’ April 4, 2011

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The legendary Indian national hero Sid is a master of Spades. His patented strategy of “N-I-L” is world renowned. However, due to a recent streak of consistently failed nils, he has hired you to help him figure out why he is losing.

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Problem of the Day #11: Difference of Squares March 30, 2011

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If $$z=\sqrt{\sqrt{6\pi^2-5-4i^2}+i}$$
Find $$Re(z)^2 – Im(z)^2$$

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