The image below is made up of 57 matchsticks:
Define a move to be removing 3 matchsticks in any of the following shapes:
What is the largest number of successive moves that can be made?
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All points on a Cartesian plane are colored either \(\color{red}{\text{red}}\) or \(\color{blue}{\text{blue}}\).
For all positive numbers \(r,\) there exists a non-degenerate isosceles triangle with legs of length \(r\) whose vertices are all colored \(\text{__________}\).
Is the statement above true for at least one of the colors?
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Note: A degenerate triangle is one where all three vertices lie on a straight line.
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A bead slides from rest down a wire that's bent into a helix, which can be parametrized in the following way: \[ \begin{cases} x = \cos(\theta) \\ y = \sin(\theta) \\ z = \theta. \end{cases} \] Find the magnitude of the bead's vertical acceleration.
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Assumptions: The bead slides without friction and there is a uniform, gravitational field \(-g\,\hat{\mathbf{z}}.\)
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\[ \large{T_m=\underbrace{17^{17^{\cdot^{\cdot^{\cdot^{17}}}}}}_{m \text{ times}}} \]
Find the smallest possible value of \(n\) such that the numbers \(T_n,T_{n+1},T_{n+2},\ldots\) all have the same last (rightmost) 2017 digits.
Bonus: Generalize to the last \(k\) digits.
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You have an infinite stream of digits from 0-9, each digit produced uniformly at random. You want to produce a random integer in the following, unusual manner:
First, take the first digit of the stream. Then, for each of the next digits, if it's larger than the last digit you took, ignore it. Otherwise, take it as well. This process ends when you take a 0 (since everything else will be 0). Your number is the sum of all the numbers taken.
For example, consider the following stream: 573493205714390...
This problem has two parts, both relating to this generator; you need to solve both of them.
Find \(10^6 a + 10^3 b + N\).
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