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?

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AIME 2020(MOCK) Problems

Problem 1

Problem 2

Problem 3