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

*Posted by Saketh in : potd , add a comment*

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*

*Posted by Saketh in : potd , add a comment*

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*

*Posted by Albert in : potd , add a comment*

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*

*Posted by Saketh in : potd , add a comment*

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*

*Posted by Saketh in : potd , add a comment*

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*

*Posted by Alex in : potd , add a comment*

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*

*Posted by Seungln in : potd , add a comment*

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
- Large Philly: costs $\$ 5.25$, comes on a rectangular bread of $14$ cm by $30$ cm

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

*Posted by Saketh in : potd , 1 comment so far*

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*

*Posted by Saketh in : potd , add a comment*

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*

*Posted by Saketh in : potd , add a comment*

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