MathDB

13

Part of 2022 AIME Problems

Problems(2)

The Quintessential Quadruplets

Source: AIME I #13

2/9/2022
Let SS be the set of all rational numbers that can be expressed as a repeating decimal in the form 0.abcd‾,0.\overline{abcd}, where at least one of the digits a,b,c,a, b, c, or dd is nonzero. Let NN be the number of distinct numerators when numbers in SS are written as fractions in lowest terms. For example, both 44 and 410410 are counted among the distinct numerators for numbers in SS because 0.3636‾=4110.\overline{3636} = \frac{4}{11} and 0.1230‾=4103333.0.\overline{1230} = \frac{410}{3333}. Find the remainder when NN is divided by 1000.1000.
AMCAIME
looks like roots of unity filter!

Source: 2022 AIME II Problem 13

2/17/2022
There is a polynomial P(x)P(x) with integer coefficients such that P(x)=(x2310−1)6(x105−1)(x70−1)(x42−1)(x30−1)P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)} holds for every 0<x<1.0<x<1. Find the coefficient of x2022x^{2022} in P(x)P(x)
algebrapolynomialAMCAIMEAIME II