Find all reals $z$ such that $z^4 - z^3 - 2z^2 - 3z - 1= 0$.

Danielle divides a $30 \times30$ board into $100$ regions that are $3 \times 3$ squares squares each and then paint some squares black and the rest white. Then to each region assigns it the color that has the most squares painted with that color. a) If there are more black regions than white, what is the minimum number $N$ of cells that Danielle can paint black? b) In how many ways can Danielle paint the board if there are more black regions than white and she uses the minimum number $N$ of black squares?

Let $ABCD$ be a square. Point $P, Q, R, S$ are chosen on the sides $AB$, $BC$, $CD$, $DA$, respectively, such that $AP + CR \ge AB \ge BQ + DS$. Prove that $$area \,\, (PQRS) \le \frac12 \,\, area \,\, (ABCD)$$and determine all cases when equality holds.

Given a positive integer $n > 1$ and an angle $\alpha < 90^o$, Jaime draws a spiral $OP_0P_1...P_n$ of the following form (the figure shows the first steps): $\bullet$ First draw a triangle $OP_0P_1$ with $OP_0 = 1$, $\angle P_1OP_0 = \alpha$ and $P_1P_0O = 90^o$ $\bullet$ then for every integer $1 \le i \le n$ draw the point $P_{i+1}$ so that $\angle P_{i+1}OP_i = \alpha$, $\angle P_{i+1}P_iO = 90^o$ and $P_{i-1}$ and $P_{i+1}$ are in different half-planes with respect to the line $OP_i$ a) If $n = 6$ and $\alpha = 30^o$, find the length of $P_0P_n$. b) If $n = 2018$ and $\alpha= 45^o$, find the length of $P_0P_n$.

We call a positive integer interesting if the number and the number with its digits written in reverse order both leave remainder $2$ in division by $4$. a) Determine if $2018$ is an interesting number. b) For every positive integer $n$, find how many interesting $n$-digit numbers there are.

What is the largest even positive integer that cannot be expressed as the sum of two composite odd numbers?