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Problem of the Day #418: Not Cupcake Flour December 29, 2012

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Snickers sells pastry flour. She fills orders by using a balance and a set of $9$ rocks to weigh the bags. Each rock weighs an integer number of pounds, and all of the rocks have distinct weights.

By placing some subset of the rocks on one side of the balance, Snickers can measure any integer weight between $1$ and $500$ pounds, inclusive. How many distinct sets of rocks could she have?

Problem of the Day #417: Another Diophantine Equation December 28, 2012

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Determine the number of ordered pairs $(x,y)$ satisfying $$x^2 – xy + y^2 = 417$$

Problem of the Day #416: Seven-Sums December 28, 2012

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A set of positive integers has the property that no seven distinct elements of it have a sum which is divisible by $7$. At most how many elements can this set contain?

Problem of the Day #415: Penny Flipping December 27, 2012

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Yuqing is playing with a triangle of pennies, similar to those shown below.

pennytriangle

The triangle is of unspecified size, but we know that all but one of the pennies is initially heads up (with the last being tails up). He can manipulate the pennies by flipping any entire row, or by rotating the whole triangle $120^{\circ}$.

Yuqing wants to turn all the coins tails up. Determine, for all triangle sizes, all positions for the coin that is initially tails up such that it is possible for him to do so.

Problem of the Day #414: Square Sum Factors December 27, 2012

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Dasith the number wizard can manipulate numbers very quickly. For his greatest trick, he asks his audience for any positive integer $p$. He then pulls out of his hat 3 integers $x$, $y$, and $z$, with $0 < x^2 + y^2 + z^2 < p^2$, such that $p | (x^2 + y^2 + z^2)$.

This trick is possible for some choices of $p$, and impossible for others. Find any infinite subset of the positive integers such that this trick is possible for any choice of $p$ in that set.

Problem of the Day #413: Shuffling Shenanigans December 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 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 #411: Counting Divisors December 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 Ratios December 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?

Problem of the Day #409: Sohail’s Function May 1, 2012

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Sohail is tinkering around with a function $f$. Suddenly, he realizes that his function satisfies the equation $f(x+y) + f(xy) = f(x)f(y) + 1$ for all pairs of integer values $x$ and $y$. Find all possible values of $f(2012)$.