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Problem of the Day #132: Albert’s Lasers July 29, 2011

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Albert has just finished meticulously setting up a row of $100$ lasers for an advanced optics project, all of which are initially pointed upwards. While Albert is taking a break from his many hours of work, Sreenath stumbles upon the lasers and, not knowing why they are there, decides to play with them. For each laser, Sreenath randomly chooses to turn it $60^\circ$ to the right, $30^\circ$ to the right, $30^\circ$ to the left, $60^\circ$ to the left, or not to turn it at all. This causes the beams of some of the lasers to intersect. For example, if Sreenath turns one laser to the right $30^\circ$, it will intersect all other lasers to the right except those already pointing $30^\circ$ or $60^\circ$ to the right. Albert’s Rage Factor when he returns from the break is equal to the number of pairs of lasers that intersect. What is the expected value of Albert’s Rage Factor?

Problem of the Day #123: The Alberts’ Triangle Game July 20, 2011

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Albert Sr. (Senior) and Albert Jr. (Junior) are playing a game. Senior picks a random point inside a triangle with sides $5$, $7$, and $9$. He then draws three lines, one through each of the vertices of the triangle, that also go through the point he picked. This divides the triangle into six smaller triangles. Junior then picks the three smaller triangles with the largest areas. If the sum of the areas of the three triangles is more than $\frac{2}{3}$ of the area of the original triangle, Junior wins the game. If Senior and Junior play $10000$ games, what is the closest integer to the number of games Junior can expect to win?

Problem of the Day #112: Albert’s Sequence of Block July 9, 2011

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Albert has a large collection of blocks of lengths $1$, $2$, $3$, and $4$. Albert wishes to arrange these blocks in a line such that the length of his line of blocks is $15$. However, the blocks that have length equal to a prime number are negatively charged, so they repel one another and cannot be placed adjacent to another block whose length is prime. Also, Albert can only place blocks immediately to the right of other blocks. If Albert chooses a possible configuration of blocks at random whose length is $15$, the probability that the rightmost block has a prime length can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime integers. Compute $a+b$.

Problem of the Day #100: Tangent Circles June 27, 2011

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Circle $A$ has radius $3$ and circle $B$ has radius $2$. Their centers are $13$ units from one another. A common internal tangent $\overline{PQ}$ is drawn such that $P$ lies on $A$ and $Q$ lies on $B$. Next, circles $A’$ and $B’$ are constructed outside of circles $A$ and $B$ such that circle $A’$ is tangent to $\overline{AB}$, $\overline{PQ}$, and circle $A$, and circle $B’$ is tangent to $\overline{AB}$, $\overline{PQ}$, and circle $B$. The distance between the centers of $A’$ and $B’$ can be expressed in the form $\frac{a\sqrt{b}-c\sqrt{d}}{e}$, where $a$, $c$, and $e$ are relatively prime and $b$ and $d$ are not divisible by the square of any prime. Find $a+b+c+d+e$.

Problem of the Day #92: Binomial Coefficients Modulo 5 June 19, 2011

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Find the number of coefficients of $(1+x)^{493}$ that are congruent to $0$ mod $5$.

Problem of the Day #85: A Fibonacci Sum June 12, 2011

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In honor of the number $3$, Albert decides to evaluate $\displaystyle\sum_{k=0}^{n}\frac{1}{3^k}(F_{3k})^3$, where $F_j$ is the $j^{\text{th}}$ Fibonacci number. He finds that it can be expressed in the form $\frac{1}{100\cdot 3^n}(a\cdot F_{9n+9}+b\cdot F_{9n}+c\cdot F_{3n+3}+d\cdot F_{3n}+e)$, where $a$, $b$, $c$, $d$, and $e$ are real numbers that may or may not be in terms of $n$. Find the ordered quintuple $(a,b,c,d,e)$.

Problem of the Day #75: Running Away from a Monster June 2, 2011

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Albert is at the origin of the coordinate plane. Unfortunately, so is a very scary monster. Albert wants to maximize his distance from the monster. However, because he cannot think straight (due to the scariness of the monster), he can only move by randomly picking a direction parallel to the $x$- or $y$-axis and taking a step that direction. After each step Albert’s step length decreases by half. For example. Albert’s first move could be to any of $(1,0)$, $(0,1)$, $(-1,0)$, or $(0,-1)$, and if he decided to move to $(1,0)$, his next move could be to any of $(\frac{3}{2},0)$, $(1,\frac{1}{2})$, $(\frac{1}{2},0)$, or $(1,-\frac{1}{2})$. Albert continues this ad infinitum. If he ends on the point $(x,y)$, what is the probability that $\frac{1}{2}\leq|x|\leq\frac{3}{2}$ and $\frac{1}{2}\leq|y|\leq\frac{3}{2}$?

Problem of the Day #68: Placing Two Points in a Triangle May 26, 2011

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Albert has an isosceles triangle $ABC$ with sides $AB=AC=3$ and $BC=2$. He then places two points $\mathcal{P}$ and $\mathcal{Q}$ somewhere in the interior of $ABC$. For all points $\mathcal{X}$ that lie within $ABC$ or are on the perimeter of $ABC$, Albert defines $f(\mathcal{X}) :=\min(\mathcal{PX}, \mathcal{QX})$.  He also lets $G$ be the maximum value of $f(\mathcal{X})$ for all possible $\mathcal{X}$. If Albert places $\mathcal{P}$ and $\mathcal{Q}$ to minimize $G$, find $G$.

Problem of the Day #62: An Infinite Structure of Cubes May 20, 2011

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Albert has an infinite number of cubes, with side lengths $1$, $\frac{1}{2}$, $\frac{1}{4}$, etc. He aligns the largest cube such that each of its $6$ faces is parallel to either the $xy$, $yz$, or $xz$ plane. Albert constructs a structure in the following way: he picks up the next largest cube that he hasn’t used in his structure yet, chooses any face of the structure so far, and places the cube somewhere on that face. The structure begins with the cube of side length $1$, and Albert continues building until he has used up all the cubes. Let $x_{min}$ be the minimum $x$-coordinate of any of the vertices in his structure, and let $x_{max}$ be the maximum $x$-coordinate of any of the vertices in his structure. Define $y_{min}$, $y_{max}$, $z_{min}$, and $z_{max}$ similarly. What is the maximum possible value of $(x_{max}-x_{min})(y_{max}-y_{min})(z_{max}-z_{min})$?

Moving Across the Coordinate Plane May 13, 2011

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Albert is standing at the origin of the Cartesian plane, desperately in need of cake. Looking around, he spots some delicious chocolate cake at the point $(100,100)$. Albert immediately departs for the cake. He knows that if he goes outside the square with corners $(0,0)$, $(100,0)$, $(100,100)$, and $(0,100)$ the cake will disappear and he will starve. When Albert is at the point $(x,y)$, the maximum speed he can move is given by $v(x,y) = 5+\frac{y}{20}$. What is the minimum time required for Albert to reach the cake?