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Problem of the Day #420: Billy the Baker December 30, 2012

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Billy the baker is testing out a new secret ingredient. He makes a huge square brownie to be divided amongst three of his friends.

Billy wants feedback on small, medium, and large sizes. Taking out his knife, he cuts the brownie into $5$ pieces, and rearranges them to form $3$ smaller squares of distinct sizes. Find a way this can be done.

Problem of the Day #412: Acute Triangulation December 26, 2012

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Peter is creating a painting for his art class. He starts by painting a number of non-overlapping acute triangles, and he observes that their union is a square. Further, he observes that any two triangles that meet do so either at a shared vertex, or at a shared side. What is the least number of triangles he could have painted?

Problem of the Day #398: Sine Sum April 20, 2012

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Suppose that $M$ is the centroid of triangle $ABC$. Suppose further that the circumcircle of triangle $AMC$ is tangent to line $AB$. Determine the maximum possible value of $$\sin\angle CAM + \sin\angle CBM$$

Problem of the Day #369: Circumradius of a Pyramid March 22, 2012

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If a right pyramid with a square base and angles of $120^{\circ}$ between its faces has an inradius of length $1$, find its circumradius.

Problem of the Day #365: Circles and Triangles March 18, 2012

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Circles $A$, $B$, and $C$ have radii $12$, $16$, and $20$ respectively and are externally tangent to each other. Find the area of the region inside the incircle of $\triangle ABC$ but outside circles $A$, $B$, and $C$.

Problem of the Day #363: Minimum Distance from Incenter to Centroid March 16, 2012

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If $\triangle ABC$ is a right triangle with hypotenuse $\overline{BC}$ of length $1$, determine the minimum possible distance between its incenter and its centroid.

Problem of the Day #361: PI March 14, 2012

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Happy $\pi$ day! Find the maximum possible value $h$ such that it is possible to fit $12$ right circular cones, with base radius $\frac{1}{\sqrt{\pi}}$ and height $h$, in a cube with side length $2$.

Problem of the Day #360: 360 Albedegrees March 13, 2012

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Albert has invented a new way to measure angles: Albedegrees. An angle that measures $360$ degrees also measures $360$ Albedegrees, but Albedegrees have the property that an angle with degree measure $x$ has three times the Albedegree measurement as an angle with degree measure $\frac{x}{2}$. In terms of $n$, find the degree of each angle in a regular $n$-gon.

Problem of the Day #356: Circling Circles March 9, 2012

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Two smaller circles $\omega_1$ and $\omega_2$ are drawn internally tangent to a circle $\omega$ of radius $2012$, with distinct points of tangency $A$ and $B$. If $\omega_1$ and $\omega_2$ meet in two distinct points, and the line $AB$ passes through one of these points, find the sum of the radii of $\omega_1$ and $\omega_2$.

Now, prove the implication that the sum of the radii is invariant for all possible $\omega_1$ and $\omega_2$.

Problem of the Day #355: The Fourth Side March 8, 2012

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Quadrilateral $ABCD$ has $AB = BC = 5$ and $AD = 4$. If $m\angle ABD = 2 m\angle CDB$ and $m\angle CBD = 2 m\angle ADB$, find all possible values for the length of side $\overline{CD}$.