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Problem of the Day #277: The Holiday Present December 21, 2011

Posted by Saketh in : potd , add a comment

Albert is preparing holiday gifts for his friends. He knows that Sreenath has trolled a lot this past year, so he will be giving him $1$x$1$x$2$ bricks of coal. The coal will be packed in a $2$x$2$x$n$ box.

Suppose that $a_n$ is the number of distinct ways the coal can be packed for a given value of $n$. For which values of $n$ is $a_n$ a perfect square?

Problem of the Day #276: Christmas Cookies December 20, 2011

Posted by Seungln in : potd , add a comment

Albert, Alex, Arjun, Billy, Mitchell, Saketh, SeungIn and Sreenath have a huge pile of Christmas cookies. They do an epic sleepover at Saketh’s house, where the cookies are. In the middle of the night, Albert wakes up and eats $1$ more than $\frac{1}{3}$ of the cookies, and goes back to bed. Later that night, Alex eats $2$ more thanĀ $\frac{1}{3}$ of the remaining cookies, and goes back to bed. Arjun gets up later, eats $3$ more than $\frac{1}{3}$ of the remaining cookies, and goes back to bed, and so on. The $n$th person eats $n$ more than $\frac{1}{3}$ of the cookies. If there are $238$ cookies remaining after each person has eaten cookies, how many cookies were there to start with?

Problem of the Day #275: Graphs with Special Cycles December 19, 2011

Posted by Alex in : potd , add a comment

An undirected, unweighted graph of $8$ nodes and each edge of the graph is part of exactly one cycle. Find the number of possible graphs.

Problem of the Day #274: Isosceles Triangle and a Circle December 18, 2011

Posted by Alex in : potd , add a comment

The base of an isosceles triangle, with length $8$, is tangent to a circle of radius $4$. Given that the isosceles triangle has an area of $32$, find the perimeter of the triangle such that area contained in the triangle but not in the circle is maximized.

Problem of the Day #273: Count the Factors of Two December 17, 2011

Posted by Saketh in : potd , add a comment

One day, Alex erroneously simplifies $\frac{(2n)!}{n!}$ to $2!$, concluding that the fraction has exactly $1$ factor of $2$. In terms of $n$, how many does it really have? You may not use the floor function (or anything equivalent) in your answer.

Problem of the Day #272: Quartic Sum of Cubes December 16, 2011

Posted by Saketh in : potd , add a comment

Find all pairs of positive integers $x$ and $y$ such that $x^3 + y^3 = 1073^4$.

Problem of the Day #271: Mutual Strangers at a Party December 15, 2011

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$100$ people are at a party, and each is friends with at most $10$ of the other guests. Determine the greatest $n$ for which there must exist a set of $n$ guests such that no two of them are friends.

Problem of the Day #270: Meme Generator December 14, 2011

Posted by Seungln in : potd , 2 comments

Albert, Billy, SeungIn and Sreenath are engaging in a meme war, in which every person makes memes out of everyone else. All of them are lazy, however, and they each make computer programs that will make memes for them. None of the programs are perfect, however, and have only a certain chance to be successful. If the sum of the probability that one person will succeed twice in a row is $1$ and the sum of the probability that one person will succeed, fail and then succeed again, in that order, is also $1$, and assuming that Billy is the best programmer and that Sreenath is the worst, what is the difference between Billy’s probability of success and Sreenath’s probability of success?

Problem of the Day #269: Bessie’s Barns December 13, 2011

Posted by Saketh in : potd , add a comment

Bessie and the girls have rented $4$ barns on the ocean front for Bovine Beach Week. The landlord is rather eccentric: he has required that the first and last barns must contain an odd number of cows, and the two in between must contain an even number of cows. Additionally, Bessie wants to make sure that none of the barns is unused. If there are $10$ cows in total, and they are all distinct (very unique, in fact), how different possible housing arrangements can Bessie choose from?

Problem of the Day #268: An Infinite Series December 12, 2011

Posted by Saketh in : potd , add a comment

Let $a_1 = \frac{1}{2011}$ and let $a_{n+1} = a_n \cdot (a_n+1)$. Evaluate

$$\frac{1}{a_1+1}+\frac{1}{a_2+1}+\frac{1}{a_3+1}+\cdots$$