AIME 2020(MOCK) Problems

Problem 1

Let be . What is the remainder when is divided by ?

Problem 2

Let be a set of polynomials with integral coefficients such that the roots of are , , and . What is the least possible sum of the coefficients of ?

Problem 3

How many digit base positive integers consist of exactly pairs of consecutive s but no consecutive s?

Problem 4

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