## End of Third Two-Week PeriodAugust 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 $135$.

## Problem of the Day #117: Las cajas del CidJuly 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 FunJuly 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 rectangleJuly 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 CitiesJune 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 cubeJune 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 ThoughtsJune 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 $k(10)$.