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AMC Strategies

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What is the sum of all $$x\cdot y$$ where $$(x,y)$$ is an integer solution to $xy-2x+y=7?$

Suppose $$x,y$$ are real numbers such that $$x+y=20$$. What is the maximum possible value of $$x\cdot y$$?

Define a sequence $$S_n$$ as follows: Given a starting value $$S_0$$, for all $$n\geq 1$$, define $$S_n$$ as $$S_{n-1}^2$$ modulo 10 taken to be a number from 0 to 9 inclusive. For example, if the starting value $$S_0=2$$, then our sequence goes $$2,4,6,6,6,\ldots$$.

Suppose a random integer is picked as the starting value $$S_0$$. What is the probability that $$S_{2017}$$ is not divisible by 3?

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