SMO 2024 Junior Round 2

Let $ABC$ be an isosceles right-angled triangle of area $1$. Find the length of the shortest segment that divides the triangle into two parts of equal area.

Let $ABCD$ be a parallelogram and points $E,F$ be on its exterior. If triangles $BCF$ and $DEC$ are similar, i.e. $\triangle BCF\sim\triangle DEC$, prove that triangle $AEF$ is similar to these two triangles.

Seven triangles of area $7$ lie in a square of area $27$. Prove that among the $7$ triangles there are $2$ that intersect in a region of area not less than $1$.

Suppose for some positive integer $n$, the numbers $2^n$ and $5^n$ have equal first digit. What are the possible values of this first digit?

Find all integer solutions of the equation
$$y^2+2y=x^4+20x^3+104x^2+40x+2003.$$