Unit 1 test: Algebra

Let $r, s, t$ be the roots of $x^3 - 6x^2 + 7x - 1 = 0$. Find $r^3 + s^3 + t^3$.

Let $r_1, r_2, r_3, r_4$ be the roots of $P(x) = x^4 - 4x^3 + 3x^2 + 2x + 5$. Find $(r_1^2 + 1)(r_2^2 + 1)(r_3^2 + 1)(r_4^2 + 1)$.

For how many positive integers $n \le 999$ is $(\sqrt3 + i)^n$ a real number?

Find $\displaystyle\prod_{k=0}^{4}\left(5 - 4\cos(72k^\circ)\right)$.

Find the real number $x$ such that $\log_2 x + \log_2(x - 12) = 6$.

Find the number of ordered pairs $(a, b)$ of integers with $2 \le a < b \le 999$ such that $\log_a b$ is an integer.

A sequence has $a_1 = 3$, $a_2 = 5$ and $a_n = 3a_{n-1} - 2a_{n-2}$ for $n \ge 3$. Find $a_9$.

The sum

equals $\dfrac mn$ in lowest terms. Find $m + n$.

Real numbers $x, y, z$ satisfy $2x + 3y + 6z = 49$. Find the minimum possible value of $x^2 + y^2 + z^2$.

Real numbers $a, b, c$ satisfy $a + b + c = 9$ and $ab + bc + ca = 24$. Let $M$ and $m$ be the largest and smallest possible values of $c$. Find $10M + m$.

A function satisfies $f(x) + 3f(2 - x) = 4x^2$ for all real $x$. Find $f(20)$.

A function $f$ on the integers satisfies $f(x + y) = f(x) + f(y) + 3xy(x + y)$ for all integers $x, y$, and $f(1) = 2$. Find $f(9)$.

Real numbers $x, y$ satisfy $\sin x + \sin y = 1$ and $\cos x + \cos y = \dfrac32$. Then $\cos(x + y) = \dfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The value of

can be written as $\dfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.