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Problem of the Day #428: The Jar of Numbers January 3, 2013

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Owl has a jar of numbers. Any two numbers in the jar have a common divisor greater than one, yet the greatest common divisor of all the numbers is one. Furthermore, there does not exist any pair of numbers in the jar such that one divides the other.

Can you establish an upper bound on the number of numbers in the jar?

Problem of the Day #419: Zero Zeroes December 29, 2012

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Determine, in terms of $n$, the length of the longest base-$n$ arithmetic progression that does not contain the digit 0.

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 #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 #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 #403: The Lone Prime Factor April 25, 2012

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Albert added two perfect cubes together, resulting in a number with exactly one prime factor. Find all possible values for that factor.

Problem of the Day #402: Equal Subset Products April 24, 2012

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Alex takes a set of $k$ consecutive positive integers and partitions them into two disjoint sets such that the product of their elements is equal. Find all values of $k$ such that this is possible.