Let $ABCD$ be a cyclic quadrilateral. The extensions of $AB$ and $DC$ meet at $E$ and the extensions of $AD$ and $BC$ meet at $F$. The bisectors of $\angle BEC$ and $\angle DFC$ meet at $P$ which lies on $AC$. Let $PF$ intersect $CD$ at $U$ and $PE$ intersect $BC$ at $V$. Prove that:
(a) $DB$ is parallel to $UV$.
(b) The lines $AC$ and $EF$ intersect at the midpoint of $EF$.
Let $a_1,a_2,a_3,\ldots$ be a sequence defined by $a_1=33$ and, for $n\ge1$,
$$a_{n+1}=a_n+\frac1{\left\lfloor a_n+\frac12\right\rfloor}.$$
Which integers appear in the sequence?
In an $n\times n$ grid, you are allowed to choose $k$ squares, including the top right and bottom left squares, and colour them red. The remaining squares are coloured blue. Thereafter, you are allowed to repeatedly choose any row or column and reverse the colours of all the squares in the row or column, respectively, until all the squares are red. What is the minimum value of $k$ for this to be possible?
The number $1$ is written $10$ times on a whiteboard. Every minute, Simon chooses two distinct numbers $a$ and $b$ on the whiteboard which may be of equal value, erases them, and writes down two new numbers $a+\dfrac1b$ and $b+\dfrac1a$.
(a) Show that, regardless of Simon's sequence of operations, after $1000$ minutes there will be at least one number on the board which is greater than $20$.
(b) Show that there is some sequence of operations for Simon to ensure that after $1000$ minutes, all numbers on the board are less than $21$.
Determine all primes $p$ such that for any positive integer $n$, one can find positive integers $a_1,a_2,a_3$ such that
$$(a_1-a_2)(a_2-a_3)(a_3-a_1)\equiv n\pmod p.$$