Problem of the Day #61: Evaluation of a certain two-variable polynomialMay 19, 2011

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The polynomial $P(x, y)$ satisfies the equality $P(x, y) = \begin{cases} 1 & x=y \\0 & \text{else} \end{cases}$ for $x, y \in \{0, 1, 2, \cdots, 60\}$. Additionally, the degree of $P$ in $x$ and the degree of $P$ in $y$ are both at most $60$. Find $P(61, 61)$.

Problem of the Day #51: Recurrence mod 53May 9, 2011

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Suppose that the sequence $a_0, a_1, a_2, \cdots$ of elements of $\mathbb{Z}^{107}$ satisfies the following properties:

1. $a_0 = (1, 0, 0, \cdots, 0)$.
2. For $n \geq 0$, if $a_n = (x_0, x_1, \cdots, x_{106})$ and $a_{n+1} = (y_0, y_1, \cdots, y_{106})$, where $x_0, x_1, \cdots, x_{106}, y_0, y_1, \cdots, y_{106}$ are integers, then $y_k = x_{k-1} + 2 x_{k-2} + 3 x_{k-3} + \cdots + 106 x_{k+1}$ for all $k$ (where indices are considered cyclically, so $x_{k} = x_{k+107}$ for all $k$).

Find the remainder when the second entry of $a_{53}$ is divided by $53$.

Problem of the Day #42: Subtracting divisorsApril 30, 2011

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A number $n$ is written on a blackboard. In one turn, Albert may choose a divisor $d$ of the number $k$ written on the board and replace $k$ by $k – d$. He wishes to change the number to $1$ in the least possible number of moves; call this number $f(n)$.

For how many integers $n < 1000$ is $2^{f(n) – 1} \leq n$?

Problem of the Day #33: Stacking coinsApril 21, 2011

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In a certain restaurant, there are $100$ distinguishable tables. Each table can support one stack of zero or more coins, but no more than $100$ coins can be placed on a single table. How many ways are there to stack $250$ distinguishable coins on the tables? (Two arrangements are considered the same if each table has the same coins in the same order in both arrangements.)

Problem of the Day #27: Solution to a nonlinear recurrenceApril 15, 2011

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Given that the sequence $\{a_{n}\}_{n=0}^\infty$ satisfies $a_0 = 1$ and $a_{n+1} – a_n =2^n a_{n}^2$ for all $n \geq 0$, find a closed-form expression for $a_n$.

Problem of the Day #24: Product of areas of triangles in regular n-gonApril 12, 2011

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Given a regular $n$-gon $\mathcal{P}$ with side length $1$, find the product of the areas of all the triangles whose vertices are vertices of $\mathcal{P}$.

Problem of the Day #14: Sum and difference of areas in hexagonApril 2, 2011

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Let $ABCDEF$ be a convex hexagon. $AD$ intersects $BE$ at $G$, $BE$ intersects $CF$ at $H$, and $AD$ intersects $CF$ at $I$. Additionally, $GF$ intersects $IE$ at $J$, $HD$ intersects $GC$ at $K$, and $IB$ intersects $HA$ at $L$. Let $[\mathcal{P}]$ denote the area of polygon $\mathcal{P}$. Given that \begin{align*} [GHI] &= 4 \\ [GDE] &= 5 \\ [HBC] &= 6 \\ [IFA] &= 7 \end{align*} find the minimum possible value of $[JFE] + [KDC] + [LBA] – [GKHLIJ]$.

Problem of the Day #5March 24, 2011

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Let $a_n$, for $n$ a positive integer, be the $n$th smallest positive integer whose decimal (base-$10$) representation contains no occurrence of the same digit twice in a row. Find $a_{ 82^6}$.