Problem of the Day #256: Lost in the WoodsNovember 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 ExponentNovember 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 VialsNovember 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 LexesNovember 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 NumbersNovember 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 CuttingNovember 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 PhillysNovember 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:

• Small Philly: costs $\$ 3.25$, comes on a rectangular bread of$8$cm by$24$cm • Medium Philly: costs$\$4.25$, comes on a rectangular bread of $11$ cm by $27$ cm