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Problem of the Day #226: Halloween Madness October 31, 2011

Posted by Billy in : potd , 2 comments

It’s Halloween, and after Albert finishes his college apps he wants to go trick-or-treating. Albert’s neighborhood consists of 6 houses equally spaced around a circle of radius $1$ kilometer. He starts from his own house, visits the rest of the houses exactly once in a random order, and returns back to his house. What is the expected length of Albert’s trick-or-treating path?

Problem of the Day #225: Projecting Circles onto Planes October 30, 2011

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Albert has a plane and a disk of radius $15$ in $3$-dimensional space, both randomly oriented. The circle does not intersect the plane. If the circle is projected onto the plane, what is the expected value of the area of the resulting planar figure?

Problem of the Day #224: Curved Triangles October 29, 2011

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Saketh has a triangle with area of $84$ that has its corners curved into arcs of circles with radius $2$. What is the maximum possible area of this curved triangle?

Problem of the Day #223: A Not-So-Random Walk October 28, 2011

Posted by Billy in : potd , 2 comments

Albert is currently located at the number $0$ on the number line. Every turn, Albert has a $\frac{1}{x}$ chance of moving one unit in the positive direction and a $1-\frac{1}{x}$ of standing still, where $x$ is the number Albert is currently on. What is Albert’s expected distance from the origin after $20$ moves?

Problem of the Day #222: INVersIoNs, iNveRsIoNS, InVeRsIoNs October 27, 2011

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Bessie the Cow is given a sequence of numbers $1 \ldots 10$ and is told she can do a swap of a pair of numbers in that sequence $5$ times. Dessie, Bessie’s friend, has to guess the sequence after Bessie has applied the inversions. Find Dessie’s chance of being correct.

Problem of the Day #221: Trees and Leaves October 26, 2011

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A certain tree grows in an unusual way. At each level, starting at the trunk, it either spawns $2$ more branches, or becomes a leaf. How many possible tree configurations have $10$ leaves?

Problem of the Day #220: Uniform Spheres are Cool October 25, 2011

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A sphere of radius $R$ has its center at the origin. There is a point $p$ outside the sphere at $(x,0,0)$ with a magical property value $v$ which affects the surrounding space as $f(v, d) = \frac{v}{d}$, where $v$ is the magical property value and $d$ is the distance from that point. Amazingly, you are informed that you can balance out the affect of point $p$ on the surface of the sphere with a point inside the sphere. Find that point’s position and magical property value.

Problem of the Day #219: Sum of the Inradii October 24, 2011

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Triangle $ABC$ has a right triangle at $B$. We draw the altitude from $B$ to the hypotenuse, and we label its foot $H$. If $AH = 16\sqrt{\pi}$, find the sum of the inradii of the triangles $ABC$, $BAH$, and $BCH$.

Problem of the Day #218: Area of a Convex Hull October 23, 2011

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For some set of points, a convex hull is a subset of points that forms the vertices of a convex polygon completely covering all the points in the set. Find the area of the convex hull of all the lattice points within the circle $$(x-4)^2 + (y-5)^2 = 16.$$

Problem of the Day #217: Number of Integral Solutions to an Equation October 22, 2011

Posted by Saketh in : potd , add a comment

Determine the number of ordered tuples $(w,x,y,z)$ such that $w^3 + x^3 + y^3 + z^3 = 2011$.