AIME 2020(MOCK) Problems

Problem 1

Let $N$ be $112123123412345... (1000 digits)$. What is the remainder when $N$ is divided by $21$?

Problem 2

Let $K$ be a set of polynomials $P(x)$ with integral coefficients such that the roots of $P(x)$ are $cos \frac{\pi}{7}$, $cos \frac{\pi}{11}$, and $cos \frac{\pi}{17}$. What is the least possible sum of the coefficients of $P(x)$?

Problem 3

How many $15$ digit base $5$ positive integers consist of exactly $2$ pairs of consecutive $0$s but no $4$ consecutive $3$s?

Problem 4

Let $\lfloor\x\rfloor$ (Error compiling LaTeX. ! Undefined control sequence.) denote the greatest integer less than or equal to $x$. What is the tens digit of $\lfloor\frac{10^{2020}}{10^{101} + 7}\rfloor$

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