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Problem of the Day #256: Lost in the Woods November 30, 2011

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Albert is lost in the woods! He is somewhere in a rectangular strip of forest that is $1$ mile wide and arbitrarily long. What is the shortest path he can follow such that he is guaranteed to find his way out of the woods?

Problem of the Day #255: Return of the Tricky Exponent November 29, 2011

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Evaluate

$$ \displaystyle\sum_{i=0}^{2010} \frac{1}{1+(\tan{\frac{i\pi}{4022}})^{\sqrt{2}}} $$

Problem of the Day #254: Reactive Vials November 28, 2011

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A mischievous joker places $6$ vials of uranium and $8$ vials of heavy water randomly in a line. He knows that if any three vials of uranium are next to each other, they will react dangerously. What is the chance that the system reacts?

Problem of the Day #253: A Game of Lexes November 27, 2011

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Alex, Elex, Ilex, and Olex are playing a game with some function $f(x)$. First, Alex selects a real number $x$. Elex then finds the value of $f(x)$, and Ilex computes the value of $f(\frac{1}{1-x})$. Finally, Olex calls out the sum of Elex’s and Ilex’s numbers.

Ulex observes that for any value of $x$ other than $0$ and $1$, Olex will simply state the initial value of $x$. Given this information, find $f(2011)$.

Bonus: Find a closed form expression for $f(x)$.

Problem of the Day #252: Lower Bound on Alex Numbers November 26, 2011

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We define the Alex number $A(a,b)$ to be to greatest real value $R$ for which $x^y + y^x \gt R$ for all $x,y$ such that $0 \lt x \lt a$ and $0 \lt y \lt b$. Find, with proof, an expression for $A(a,b)$ in terms of $a$ and $b$.

Problem of the Day #251: Triangle Cutting November 25, 2011

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Instead of shopping (today is Black Friday!), Sreenath wants to go cut triangles. Points $A$, $B$, and $C$ are located at $(1, 1)$, $(5, 5)$, and $(3, 1)$ respectively. Sreenath chooses point $E$ on $\overline{AC}$ such that $BE$ cuts $\triangle ABC$ into two triangles with equal perimeters. Sreenath then chooses point $Q$ on $\overline{AC}$ such that $BQ$ cuts $\triangle ABC$ into two triangles with equal areas. Find $QE$.

Problem of the Day #250: Billy’s Phillys November 24, 2011

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Billy goes to Philadelphia (obviously to get a very good Philly cheese steak). To lower his risk of bladder cancer, however, Billy must limit his Philly’s cheese limit to $52g$. Assume that Billy can order his Phillys in three different sizes, as follows:

Assume that the density of cheese is $1 \frac{g}{cm^3}$, and that the cheese on each of Billy’s possible Phillys have a $.01$ cm thickness of cheese that covers each of the two sides of the cut bread. Assuming that Billy has an unlimited amount of money for his Phillys, and that Billy will not leave a single bit of Philly to go to waste, what is the maximum possible amount of Billy’s Phillys, and how much will he have to pay for them?

Problem of the Day #249: Prime Squares mod 24 November 23, 2011

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Arthur is thinking of a prime such that its square is congruent to $1$ mod $24$. Find all primes he is definitely not thinking of.

Problem of the Day #248: Triangle Perimeter Range November 23, 2011

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Alex has drawn a triangle with sides of length $a$, $b$, and $c$ such that $ab+bc+ac=247$. Find the range of possible values for the triangle’s perimeter.

Problem of the Day #247: Factor Square Sum November 21, 2011

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There is only one integer equal to the sum of the squares of its $4$ smallest positive integer divisors. Find this integer.