Find all primes $p$, so that for every prime $q<p$ and $x\in \mathbb{Z}$ one has $p\nmid x^2-q$.

Let $n\geq 2$ be an integer. For each natural $m$ and each integer sequence $0<k_1<k_2<\cdots <k_m$ for which $k_1+\cdots+k_m=n$, Michael wrote down the number $\frac{1}{k_1\cdot k_2\cdots k_m} $ on the board. Prove that the sum of the numbers on the board is less than $1$.

In La-La-Land there are 5784 cities. Alpaca chooses for each pair of cities to either build a road or a river between them, and additionally she places a fish in each city to defend it. Subsequently Bear chooses a city to start his trip. At first, he chooses whether to take his trip in a car or in a boat. A boat can sail through rivers but not drive on roads, and a car can drive on roads but not sail through rivers. When Bear enters a city he takes the fish defending it, and consequently the city collapses and he can't return to it again. What is the maximum number of fish Bear can guarantee himself, regardless of the construction of the paths? Remarks: Bear takes a fish also from the city he begins his trip from (and the city collapses). All roads and rivers are two-way.

Let $ABC$ be an acute triangle. Let $D$ be a point inside side $BC$. Let $E$ be the foot from $D$ to $AC$, and let $F$ be a point on $AB$ so that $FE\perp AB$. It is given that the lines $AD, BE, CF$ concur. $M_A, M_B, M_C$ are the midpoints of sides $BC, AC, AB$ respectively, and $O$ is the circumcenter of $ABC$. Moreover, we define $P=EF\cap M_AM_B, S=DE\cap M_AM_C$. Prove that $O, P, S$ are collinear.