Let $a, b$ be positive reals such that $a^3 + b^3 = ab + 1$. Prove that \[(a-b)^2 + a + b \geq 2\]

Rose and Brunno play the game on a board shaped like a regular 1001-gon. Initially, all vertices of the board are white, and there is a chip at one of them. On each turn, Rose chooses an arbitrary positive integer \( k \), then Brunno chooses a direction: clockwise or counterclockwise, and moves the chip in the chosen direction by \( k \) vertices. If at the end of the turn the chip stands at a white vertex, this vertex is painted red. Find the greatest number of vertices that Rose can make red regardless of Brunno's actions, if the number of turns is not limited.

A pair of positive integers $(x, y)$ is good if they satisfy $\text{rad}(x) = \text{rad}(y)$ and they do not divide each-other. Given coprime positive integers $a$ and $b$, show that there exist infinitely many $n$ for which there exists a positive integer $m$ such that $(a^n + bm, b^n + am)$ is good. (Here, $\text{rad}(x)$ denotes the product of $x$'s prime divisors, as usual.)

Vaysha has a board with $999$ consecutive numbers written and $999$ labels of the form "This number is not divisible by $i$", for $i \in \{ 2,3, \dots ,1000 \} $. She places each label next to a number on the board, so that each number has exactly one label. For each true statement on the stickers, Vaysha gets a piece of candy. How many pieces of candy can Vaysha guarantee to win, regardless of the numbers written on the board, if she plays optimally?

Let $A_1C_2B_1B_2C_1A_2$ be a cyclic convex hexagon inscribed in circle $\Omega$, centered at $O$. Let $\{ P \} = A_2B_2 \cap A_1B_1$ and $\{ Q \} = A_2C_2 \cap A_1C_1$. Let $\Gamma_1$ be a circle tangent to $OB_1$ and $OC_1$ at $B_1,C_1$ respectively. Similarly, define $\Gamma_2$ to be the circle tangent to $OB_2,OC_2$ at $B_2, C_2$ respectively. Prove that there is a homothety that sends $\Gamma_1$ to $\Gamma_2$, whose center lies on $PQ$

$\indent$ For a positive integer $n$, let $S_n$ be the set of bijective functions from $\{1,2,\dots ,n\}$ to itself. For a pair of positive integers $(a,b)$ such that $1 \leq a <b \leq n$, and for a permutation $\sigma \in S_n$, we say the pair $(a,b)$ is expanding for $\sigma$ if $|\sigma (a)- \sigma(b)| \geq |a-b|$ $\indent$ (a) Is it true that for all integers $n > 1$, there exists $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for permutation $\sigma$ is less than $1000n\sqrt n$ ? $\indent$ (b) Does there exist a positive integer $n>1$ and a permutation $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for the permutation $\sigma$ is less than $\frac{n\sqrt n}{1000}$?