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Problem of the Day #196: Lighting up a hexagon October 1, 2011

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Albert and Arjun are situated on a large hexagon. This hexagon has a candle at each of the vertices. Albert wishes to light all of these candles at the same time; however, each candle is special, and takes a different amount of time to light up. The first candle takes $1$ second to light up, the second takes $2$ seconds, and so on (these candles are consecutively placed). Albert and Arjun decide to split up the work of turning each of these on, and can move from one vertex to an adjacent one in $1$ second. To make sure they light the candles in the right order, they call out to each other to notify of the lighting of a candle. However, the speed of sound is slow in this dimension, and also takes $1$ second to propagate along an edge. Albert begins at vertex $1$ and Arjun begins at vertex $4$. What is the minimum time for the hexagon to be fully lit?

Problem of the Day #194: Running Fast September 29, 2011

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Alex is running very fast on a circular track. The track has an inner radius of $4$ meters and an outer radius of $8$ meters, and allows running on painted lanes at $1$ meter intervals. Alex runs at $3 \pi$ m/s. Alex wants to run around the track as fast as possible, and decides to run as far inside as possible. However, some slow runners begin running at the same time as Alex at $\pi$ m/s, and Alex (being a very considerate fellow) does not want to end up within $\pi$ meters of them on the same lane, and will slow down to $\pi$ m/s running speed if he is at that distance. Alex can switch lanes at any time. What is the fastest possible time that Alex can complete $10$ laps around the track?

Problem of the Day #144: Arjun’s Problems August 10, 2011

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Since Albert has been bugging him to write a problem, Arjun decides to pick one out of a set of hats. However, Aziz (also needing a problem) decides to pick one out the same set of hats. There are three hats: the first has $5$ acceptable problems, the second has $6$, and the third has $8$. If Arjun and Aziz each pick 3 problems (each problem has equal chance of being chosen) from the hats (with replacement), the probability that none of Arjun’s problems were from the same hat as any of Aziz’s problems can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Problem of the Day #114: Digits in Triangles July 11, 2011

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$x$ and $y$ are positive four-digit integers whose digits can be taken from the set of positive integers less than $7$. If $z$ is the maximum possible integer side length of a triangle with sides $x$, $y$, and $z$, find the sum of all $z$ for all $x$ and $y$.

Problem of the Day #103: Triples with Log June 30, 2011

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Find the number of ordered triples $(a,b,c)$ of positive integers such that $ab + c \le 87$ and $\log_a b$ is a positive integer.

Problem of the Day #76: Posts from the Past June 3, 2011

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Recently there has been an invention that allows you to send messages to the future! Hi future! In order to transport these messages, there are many different paths you can choose. These paths are arranged on a $4$ dimensional grid, where the past is at $(0,0,0,0)$ and the future is at $(8,8,8,8)$. However, the path cannot simply be traversed; it must form the shape of a tetrahedron! How many possible paths exist such that the vertices of this tetrahedron are on integer coordinates?

Problem of the Day #70: The Ping is too Large May 28, 2011

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Arjun has been playing video games until very late in the night, sometimes forgetting to post his MPOTD problem. When playing those games, he decides to host a server so his friends can connect faster, but he needs to find the optimal placement to minimize the overall ping from all his friends. Conveniently, everyone’s houses are arranged on a grid. TJHSST is at (10, 10), Paul is at (8, 5), Minh is at (4, 9), Arjun is at (2, 10), Fareez is at (7, 2), and Albert is at (1, 3). There are a string of empty warehouses on the function $$ f(x) = \frac{x^4 + 3x^3 + x + 3}{100} $$ along which the server can be placed. What is the minimum possible sum of the ping times between each house and the server, if ping is defined as the round trip time and packets move at a constant rate of $\frac{1}{5} \frac{units}{ms}$?

Problem of the Day #54: Lots of Circles May 12, 2011

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Isosceles trapezoid $ABCD$ has parallel sides of length $6$ and $8$.  We wish to place circles of radius $1$ and $2$ inside this trapezoid. If the number of circles of radius $1$ can only differ from the number of circles of radius $2$ by at most $2$, what is the maximum number of circles than can be placed inside the trapezoid with no overlap?

Problem of the Day #45: Pre-Arranged Absences May 3, 2011

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Albert has decided that he needs to cram for AP exams. For that purpose he has decided to take pre-arranged absences on strategic days. In this hypothetical situation he is taking AP Chemistry and Psychology on Monday, Calculus BC on Wednesday,  Biology and Physics B the next Monday, and World History and Human Geography on the next Thursday and Friday, respectively. A maximum of $3$ half days is allowed, and they can only be taken:

What is the number of combinations of absences Albert can take?

Problem of the Day #36: Fingers April 24, 2011

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Albert has been watching a particularly scary horror anime. In this anime, fingernails are ripped off of fingers. However, there are a few limitations on what can show up on screen (due to censorship laws). Firstly, there is a limited screentime of $23$ minutes, and different fingers have different times taken for the nail to be ripped off, as well as a different amount of horror points associated with them. Thumbs take $50$ seconds, and are worth $10$ horror points. Index fingers take $40$ seconds, and are worth $8$ horror points. Middle fingers take $50$ seconds, and are worth $10$ horror points. The remaining $2$ fingers have already been ripped off, so viewing them on screen takes $10$ seconds and provides a measly $1$ horror point. Due to weird censorship laws, at least $3$ fingers from each hand must be shown on screen. What is the maximum amount of horror points possible?