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Problem of the Day #423: Cookie Queries December 31, 2012

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Albert is hungry. He visits a local djinn and wishes for a bottomless jar of cookies.

As always, there’s a catch. The crafty djinn marks $n$ of the cookies from $1$ to $n$ using some icing. He then arranges them in some fixed order $\pi([n])$ hidden away from Albert’s view.

Now, if Albert can name a subset $S$ of $[n]$ such that some permutation of $S$ is a contiguous subsequence of $\pi([n])$, he gets to eat $|S|$ cookies from the jar.

All queries (Albert’s choices for $S$) must be written down before any of them are processed. What is the least number of queries Albert can make to guarantee he will receive at least one cookie?

$\textbf{Bonus:}$ Determine, for fixed $n$ and $q$, the set of $q$ queries that maximizes the expected number of cookies Albert will get to eat.

Problem of the Day #422: Divergent Geometric Mean December 31, 2012

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Can you find a convergent sequence of reals with a divergent geometric mean? Alternatively, give a proof that no such sequence exists.

Problem of the Day #421: Dim Sum Night December 30, 2012

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Anand is eating at a Chinese restaurant with his family. He is going to order for all of them; he will submit one ordered tuple of non-negative integers $(x,y,z)$ representing the number of dumplings, spring rolls, and steamed buns he wants.

Now, well distributed orders (like $(10, 10, 10)$) are typically more satisfactory than unbalanced orders (such as $(29, 1, 0)$). After all, we don’t want everyone fighting over the lone spring roll! Let the quality $Q$ of a given order be modeled by the function $Q(x,y,z) = xyz$.

Anand knows how much his family will eat, so he wants $x+y+z = 30$. He will select a tuple $(x,y,z)$ at random from all those that satisfy this constraint. Determine the expected value of $Q(x,y,z)$, the quality of the order.

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 #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 #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.