Suppose that $P(z)$, $Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z) = R(z)$. What is the minimum possible value of $N$?
Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\bar{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\bar{z}$. The polynomial $P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$ has four complex roots: $z_1, z_2, z_3$, and $z_4$. Let $Q(z) = z^4 + Az^3 + Bz^2 + Cz + D$ be the polynomial whose roots are $f(z_1), f(z_2), f(z_3)$, and $f(z_4)$, where the coefficients $A, B, C$, and $D$ are complex numbers. What is $B + D$?
An organization has 30 employees, 20 of whom have a brand A computer while the other 10 have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with every other. What is the maximum possible number of cables used?
Suppose $a$, $b$, and $c$ are positive integers such that $a + b + c = 23$ and $\gcd(a, b) + \gcd(b, c) + \gcd(c, a) = 9$. What is the sum of all possible distinct values of $a^2 + b^2 + c^2$?
A bug starts at a vertex of a grid made up of equilateral triangles of side length $1$. At each step the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. What is the probability that after $5$ moves the bug never will have been more than $1$ unit away from the starting position?
Set $u_0 = \frac{1}{4}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence $u_{k+1} = 2u_k - 2u_k^2$. This sequence tends to a limit; call it $L$. What is the least value of $k$ such that
$$
|u_k - L| \le \frac{1}{2^{1000}}?
$$
Let $x$ be the least real number greater than $1$ such that $\sin x = \sin(x^2)$, where the arguments are in degrees. What is $x$ rounded up to the closest integer?
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12$?
For real numbers $x$, let
$$
P(x) = 1 + \cos(x) + i \sin(x) - \cos(2x) - i \sin(2x) + \cos(3x) + i \sin(3x),
$$
where $i = \sqrt{-1}$. For how many values of $x$ with $0 \le x \lt 2\pi$ does $P(x) = 0$?
Right triangle $ABC$ has side lengths $BC = 6$, $AC = 8$, and $AB = 10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$?
A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called *disrespectful* if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)$?
What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{1, 2, 3, \ldots, 30\}$? (For example, the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)
Convex quadrilateral $ABCD$ has $AB = 18$, $\angle A = 60^\circ$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $AB$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $a$?
Triangle $ABC$ has side lengths $AB = 11$, $BC = 24$, and $CA = 20$. The bisector of $\angle BAC$ intersects $\overline{BC}$ in point $D$ and intersects the circumcircle of $\triangle ABC$ in point $E \neq A$. The circumcircle of $\triangle BED$ intersects the line $AB$ in points $B$ and $F \neq B$. What is $CF$?
Let $m \ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1 \le a_i \le m$ for all $i$ such that $m$ divides $a_1 + a_2 + a_3 + a_4$. There is a polynomial $q(x) = c_3x^3 + c_2x^2 + c_1x + c_0$ such that $D(m) = q(m)$ for all odd integers $m \ge 5$. What is $c_1$?
For $n$ a positive integer, let $R(n)$ be the sum of remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1 + 0 + 3 + 0 + 3 + 1 + 7 + 6 + 5 = 26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n + 1)$?