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Billy’s Pizza May 31, 2011

Posted by Saketh in : bonus , 2 comments

Billy obtained a triangular slice of pizza $ABC$ to go with his soda. If $m\angle A = m\angle B = 72 ^\circ$ and $AB = 1$, compute the exact value of $BC$.

Problem of the Day #73: Billy’s Soda May 31, 2011

Posted by Sreenath in : potd , add a comment

Billy is deciding between two different cups of soda. They each have a base of diameter $4$ and mouth of diameter $5$. The smaller size has a height of $6$ units, while the larger has a height of $8$ units.

Albert suggests that the larger one has $\frac{64}{27}$ the volume of the smaller one. Billy suspects he is mistaken. Help Albert compute the correct ratio of the volumes.

Problem of the Day #72: Cake Explorer May 30, 2011

Posted by Saketh in : potd , add a comment

Mitchell is trapped in a tetrahedral cake and wishes to scout for an exit by launching four probes, one aimed at each vertex of the cake.

Determine the point inside the cake where he can position himself such that the sum of the distances traveled by the probes is minimized.

Problem of the Day #71: 7th Time’s the Charm May 29, 2011

Posted by Aziz in : potd , add a comment

Given that side $a$ has a measure of $30$ meters, determine the difference between the area of the circumcircle of triangle $AB’C$ and the area of the circumcircle of triangle $BB’C$.

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

Posted by Arjun in : potd , add a comment

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 #69: Avoid that RickRoll! May 27, 2011

Posted by Seungln in : potd , add a comment

Albert is being chased down a number line by Rick Astley, who is belting out Never Gonna Give You Up live as he goes. Albert knows that Rick will start at $0$ and can only take steps of length $23$ or $37$ to the right. He also knows that the number line starts at $0$ and extends infinitely off to the right.

Wishing to avoid being rickrolled, Albert decides to occupy an integer location such that it is impossible for Rick to be within hearing distance. If Rick sings loudly enough that he can be heard at any point within $5$ units of his location, determine the rightmost location where Albert will be safe.

Playing with Cyclic Quadrilaterals May 27, 2011

Posted by Saketh in : bonus , add a comment

Alex has created a model of a cyclic quadrilateral using some modelling clay. He observes that it has the following properties:

SeungIn sees a spider crawling on the model and attempts to swat it. Unfortunately, he ends up squishing the quadrilateral instead. Both of its diagonals shrink by exactly $1$ unit, but they remain perpendicular to each other. Determine the area of the new shape.

Problem of the Day #68: Placing Two Points in a Triangle May 26, 2011

Posted by Billy in : potd , add a comment

Albert has an isosceles triangle $ABC$ with sides $AB=AC=3$ and $BC=2$. He then places two points $\mathcal{P}$ and $\mathcal{Q}$ somewhere in the interior of $ABC$. For all points $\mathcal{X}$ that lie within $ABC$ or are on the perimeter of $ABC$, Albert defines $f(\mathcal{X}) :=\min(\mathcal{PX}, \mathcal{QX})$.  He also lets $G$ be the maximum value of $f(\mathcal{X})$ for all possible $\mathcal{X}$. If Albert places $\mathcal{P}$ and $\mathcal{Q}$ to minimize $G$, find $G$.

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

Posted by Mitchell in : potd , add a comment

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 $P(0)$.

Problem of the Day #66: Banner Across an Arch May 24, 2011

Posted by Albert in : potd , add a comment

After constructing a giant arch in the shape of one sinusoidal hump, Billy wants to place a banner across it to advertise mpotd.com. But the giant arch is… GIANT! (built to the standard $1$km high and $\frac{12\sqrt{2}}{5}$km wide), so Billy needs to figure out how he minimize cost while still making the banner visible from a certain distance. To minimize costs, Billy decides to hang the rectangular banner from two spots at equal hight on opposite sides of the arch, and have the banner touch the ground. He also remembers reading that a nominal area, $A$, for the banner is one such that $\frac{A}{d^2} = \frac{1}{64}$, where $d$ is the maximum viewing distance from the banner. Billy remembers that the arch was purposefully built a distance of $64(\sqrt{3}-1)$km from a mountain range (for whatever reason), so he figures this is the maximum distance needed. What length of banner paper should Billy buy?