What sizes of squares with integer sides can be completely covered without overlap by identical tiles consisting of three squares with side $1$ joined together in one $L$ shape?

Find all functions $f : R \to R$ such that $$f(x + zf(y)) = f(x) + zf(y), $$for all $x, y, z \in R$.

Let $n$ be a positive integer divisible by $39$. What is the smallest possible sum of digits that $n$ can have (in base $10$)?

Let $ABC$ be an acute triangle. Let $I$ be a point inside the triangle and let $D$ be a point on the line $AB$. The line through $D$ which is parallel to $AI$ intersects the line $AC$ at the point $E$, and the line through $D$ parallel to $BI$ intersects the line $BC$ in point $F$. prove that $$\frac{EF \cdot CI}{2} \ge area (\vartriangle ABC) $$

Prove that for every pair of positive integers $k$ and $n$, there exists integer $x_1$, $x_2$,$...$, $x_k$ with $0 \le x_j \le 2^{k-1}\cdot \sqrt[k]{n}$ for $j = 1$, $2$, $...$, $k$, and such that $$x_1 + x^2_2+ x^3_3+...+ x^k_k= n.$$

Bengt wants to put out crosses and rings in the squares of an $n \times n$-square, so that it is exactly one ring and exactly one cross in each row and in each column, and no more than one symbol in each box. Mona wants to stop him by setting a number in advance ban on crosses and a number of bans on rings, maximum one ban in each square. She want to use as few bans as possible of each variety. To succeed in preventing Bengt, how many prohibitions she needs to use the least of the kind of prohibitions she uses the most of?