Problem of the Day #216: Red Lines Blue LinesOctober 21, 2011

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Alex takes a set of $N$ points and draws either a red line or a blue line between each pair. What is the smallest $N$ such that he will always create a closed path consisting of four segments, all of the same color?

Problem of the Day #215: Sharing Some ApplesOctober 20, 2011

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Sam recently bought some apples, $4$ for each of his friends. He is trying to divide them as evenly as possible. If none of his apples weighs more than $k$ times any other, determine the smallest value $r$ for which he can guarantee that no group weighs more than $r$ times any other. Express $r$ in terms of $k$.

Problem of the Day #214: Albert’s Number LineOctober 19, 2011

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Albert has arranged the numbers from $1$ to $N$ in a line such that the sum of any two adjacent numbers is a perfect square. What is the greatest $N$ for which this is possible?

Problem of the Day #213: Length of an Angle BisectorOctober 18, 2011

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Triangle $ABC$ has sides of length $39$, $48$, and $71$. The bisector of angle $A$ cuts $BC$ at point $D$. Find the length of $AD$ in terms of $cos(\frac{A}{2})$.

Problem of the Day #212: Number of Distinct Qualifying SetsOctober 17, 2011

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Alex is picking certain subsets of the set $\{1,2,3,…,50\}$. To qualify, a set must have a sum of at least $777$. Determine the number of distinct sets from which Alex can choose.

Problem of the Day #211: A Pig Stuck inside Its PenOctober 16, 2011

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Albert the Pig is chained by a stiff leash, of length $x$, to one of the corners of a pen, in the shape of a regular hexagon with side length $1$. Albert must stay within the pen. The leash does not bend but Albert’s neck allows him to reach areas up to $0.1$ units beyond his leash. Find $x$ such that the area of land that Albert can reach is maximized.

Problem of the Day #210: Albert’s SequenceOctober 15, 2011

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Albert has a sequence defined by \begin{align*}a_0 &= a_1 = a_2 = 0 \\ a_3 &= 1 \\ a_n &= 3 a_{n-2} - a_{n-4}.\end{align*} Find $a_{31}$.

Problem of the Day #209: Maximum Possible InradiusOctober 14, 2011

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Find the largest possible inradius of a quadrilateral with side lengths $4$, $5$, $6$, and $7$.

Problem of the Day #208: Binary CoinsOctober 13, 2011

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Saketh wants to purchase some food worth a random value of cents between $0$ and $1,000,000$, inclusive. He has coins worth $1$ cent, $2$ cents, $4$ cents, and so on to infinite powers of $2$, with an infinite number of coins for each value. What is the expected number of coins that Saketh will use to purchase his food, given that he always uses the fewest possible number of coins?

Problem of the Day #207: Points on a PlaneOctober 12, 2011

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Four points are arranged on a plane such that the distance between every pair is either $a$ or $b$. If $b>a$, find all possible values of $\frac{b}{a}$.