7

Part of 2021 AIME Problems

Problems(2)

Source: 2021 AIME I/7

Find the number of pairs

(m,n)

of positive integers with

1\le m<n\le 30

such that there exists a real number

x

\sin(mx)+\sin(nx)=2.

Four Variable System

Source: 2021 AIME II #7

a, b, c,

d

be real numbers that satisfy the system of equations \begin{align*} a+b&=-3\\ ab+bc+ca&= -4\\ abc+bcd+cda+dab&=14\\ abcd&=30. \end{align*} There exist relatively prime positive integers

m

n

a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.

m + n