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Problem of the Day #310: The volume of a simply defined region January 23, 2012

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Let $\theta$ be a real number with $0 < \theta < \pi$. Let $A$ and $O$ be distinct points in three-dimensional space. Let $S$ be the set of all points $P$ in space with both $OP < 1$ and $\angle AOP <\theta$. Find the volume of $S$ in terms of $\theta$.

(This problem is possible to solve with calculus, but there’s also a very nice solution that avoids it.)

Problem of the Day #146: Product of cosines August 12, 2011

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Find the value of \[- \log_2\left( \prod_{k=0}^{1023} \cos\left(\frac{k \pi}{2048}\right)\right).\]

Problem of the Day #139: Number of tricky sets August 5, 2011

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A subset $S$ of $\mathbb{Z}^3$ is called tricky if the following two conditions hold:

Find the number of tricky subsets of $\mathbb{Z}^3$.

Note: $\mathbb{Z}^3$ is the set of all ordered triples of integers. Additionally, if $x = (x_1, x_2, x_3)$ and $y = (y_1, y_2, y_3)$, then $x + y$ is defined to be $(x_1 + y_1, x_2 + y_2, x_3 + y_3)$.

Problem of the Day #130: Minimizing acute angles July 27, 2011

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If $A_1, A_2, \cdots, A_{100}$ are points in the plane, no three collinear, what is the minimum possible number of triples $(i, j, k)$ of distinct integers such that $1 \leq i, j, k \leq 100$, $i < k$, and $\angle A_i A_j A_k$ is acute?

Problem of the Day #122: I like pentagons July 19, 2011

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Let ABCDE be a pentagon with $AB \parallel CE$, $BC \parallel AD$, $AC \parallel DE$, $\angle ABC=120^\circ$, $AB=6$, $BC=10$, and $DE = 30$. The ratio of the area of $\triangle ABC$ to the area of $\triangle EBD$ 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 #116: Binomial coefficient sum July 13, 2011

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Given that \[\sum_{n = 1003}^{2006} \frac{1}{n} \binom{n}{2006 - n} = \frac{p}{q},\] where $p$ and $q$ are relatively prime integers with $q > 0$, find the smallest positive integer $m$ such that $qm – p$ is divisible by $2011$.

Problem of the Day #108: Cubic mod 2011 July 5, 2011

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Positive integers $x, a \leq 2011$ satisfy the property that $x^3 – x^2 – a$ is divisible by $2011$. Find the number of possible values of $a$.

Problem of the Day #84: Number of bounded sequences with a certain sum June 11, 2011

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How many sequences $(a_1, a_2, \cdots, a_k)$, where $a_1, a_2, \cdots, a_k$ are positive integers less than or equal to $84$, satisfy $a_1 + a_2 + \cdots + a_k = 168$?

Problem of the Day #79: Number of times crossing a line June 6, 2011

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A random path from $(0, 0)$ to $(79, 79)$ composed of steps $(0, 1)$ and $(1, 0)$ is selected uniformly from the set of all such paths. Find the expected value of the square of the number of points on the path which are also on the line $y = x$.

Problem of the Day #67: Evaluation of a polynomial May 25, 2011

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Let $n$ be a positive integer. Given that the polynomial $P(x)$ satisfies the relation $2 P(x+1) – P(x) = x(x-1)(x-2)\cdots(x-n)$ for all $x$, find the value of $P(0)$.