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End of Third Two-Week Period August 2, 2011

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Please join us in congratulating Eugene Chen, the winner of the third two-week period. This period spanned problems $122$ through $135$.

As the top-scoring solver over the past two weeks that hasn’t already won a two-week prize, Eugene wins a $\$10$ gift card of his choice. You can view the full rankings for these two weeks after the jump. (more…)

Problem of the Day #135: Spider! August 1, 2011

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Seungln and a spider are at opposite corners of a 335x415x581 box composed of unit cubes. The spider begins tunneling directly towards Seungln. How many cubes does the spider pass through?

Problem of the Day #125: Dr. Kim’s Solar Cells July 22, 2011

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Izzy is standing at one corner of a 20 ft by 20 ft classroom. She begins walking towards the midpoint of one of the opposite walls. When she reaches a wall, she will bounce off in the opposite direction at the same angle at which she hit the wall. If she is ever within 3 feet of one or more solar cells, she will trip towards it, destroying the solar cell(s). Dr. Kim would like to place his solar cells so they will not be destroyed during Izzy’s rampage. The area of the region in which Dr. Kim can safely deploy his solar cells can be expressed in the form $\frac{a-b\sqrt{c}}{d}$, where $a$, $b$, and $d$ are relatively prime positive integers and $c$ is not divisible by the square of any prime. Compute $a+b+c+d$.

End of Second Two-Week Period July 19, 2011

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Please join us in congratulating Lewis Chen, the winner of the second two-week period. This period spanned problems $108$ through $121$.

As the top-scoring solver over the past two weeks that hasn’t already won a two-week prize, Lewis wins a $\$10$ gift card of his choice. You can view the full rankings for these two weeks after the jump. (more…)

Problem of the Day #117: Las cajas del Cid July 14, 2011

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The legendary Indian national hero Sid has $3$ boxes. He also regularly receives gifts from his many fans. When he receives a gift, he randomly places it in one of his $3$ boxes. Sid has received $10$ gifts so far. The expected value of the number of empty boxes can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $a+b$.

Problem of the Day #113: Fibonacci Fun July 10, 2011

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The Fibonacci sequence is defined as $F_1=1,F_2=1$ and $F_n=F_{n-1}+F_{n-2}$. Compute the real part of $$\sum\limits_{k=1}^{999}\; ki^{\left(F_k\right)^2}$$ You may use a four-function calculator.

Problem of the Day #111: Albert’s rectangle July 8, 2011

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Two points are chosen at random from the region bounded by the lines $x=0$, $y=0$, $x=1000$, and $y=1000$. These points are opposite corners of a rectangle with sides parallel to the axes. The expected value of the area of the rectangle can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $a+b$.

Problem of the Day #102: Billy’s Cities June 29, 2011

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Billy is in charge of creating plans for roads between a number of cities. To create a plan for $n$ cities, he flips a coin for each of the $\binom{n}{2}$ possible roads. If it lands heads, he builds the road, and if it lands tails, he doesn’t. A plan is considered acceptable if for every choice of two cities, there exists a path between them (can go through other cities). If Billy creates $4620$ plans for $3$ cities using this method, compute the expected value of the number of acceptable plans.

Problem of the Day #95: A cube in a sphere in a cube June 22, 2011

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A sphere is inscribed within a cube. A cube is then inscribed within the sphere such that the faces of the inner cube are all parallel to the corresponding faces on the outer cube. Let $S$ be the set of the distances from the center of one face of the larger cube to the vertices of the smaller cube. If the side length of the smaller cube is $1$, compute the sum of the squares of the elements in $S$.

Problem of the Day #87: Sinusoidal Thoughts June 14, 2011

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The expression $(\sin{x}+\cos{x})^n$ can be written as $k(n)\cdot\sin^n\left(f(x)\right)$, where $k(n)$ is a function of $n$ and $f(x)$ is a polynomial in $x$. Compute $k(10)$.