Problem of the Day #413: Shuffling ShenanigansDecember 26, 2012

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Pavan is shuffling cards. Given a deck of $4n$ cards $a_1, a_2, \dots, a_{4n}$, he will shuffle them into the order $$a_4, a_8, \dots, a_{4n}, a_3, a_7, \dots, a_{4n-1}, a_2, a_6, \dots, a_{4n-2}, a_1, a_5, \dots, a_{4n-3}$$ One day, Luke gifts Pavan a deck of cards labelled $1, 2, \dots, 4k$. Pavan applies his shuffle to the deck a finite number of times. He then realizes that he has shuffled the deck into perfectly reversed order: $4k, 4k-1, \dots, 1$.

Determine all possible values of $k$.

Problem of the Day #412: Acute TriangulationDecember 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 #411: Counting DivisorsDecember 25, 2012

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Let $D(x)$ denote the number of positive divisors of $x$. Determine the number of values of $x$ for which $x = D(x)^4$.

Problem of the Day #410: Digital Sums RatiosDecember 25, 2012

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Dilip and Bridget are playing with numbers. Dilip first picks a positive integer $x$ and finds $D$, the sum of its digits. Then, Bridget computes $B$, the sum of the digits of $2x$. What is the maximum possible value of $\frac{D}{B}$?

$\bf{Bonus:}$ Is $\frac{B}{D}$ bounded? If so, what is its maximum possible value?