SMO 2026 Junior Round 2

Let $ABCD$ be a rhombus and $E$ be a point on the diagonal $AC$. Let $F$ be the point on the side $BC$ such that $BF=DE$. Prove that
$$(AB+BF)\cdot FC=AE\cdot EC.$$

Prove that every integer $n$ can be written as $n=a^2+b^2-c^2$ where $a,b,c$ are integers.

Lines are drawn inside a unit square partitioning it into squares, not necessarily of the same size. Choose one of the main diagonals of the unit square. A square is said to intersect the chosen main diagonal if it has at least one point in common with the diagonal. Is it possible to draw the lines so that the sum of the perimeters of the squares that intersect the chosen main diagonal exceeds 2026?

Alex and Bob take turns to select one number with no repetition allowed from $\{0,1,2,\ldots,14\}$, starting with Alex. Alex wins if he can form a 4-term arithmetic progression among the numbers in his selection. Does Alex have a winning strategy?

Determine all integers $a,b,c$ that satisfy the equation
$$a^2+b^2+c^2=13(ab+bc+ca).$$