Question 47
Given positive integers $n$ and $k \le n$. Consider an equilateral triangular board with side $n$, which consists of circles: in the first (top) row there is one circle, in the second row there are two circles, ..., in the bottom row there are $n$ circles (see the figure below). Let us place checkers on this board so that any line parallel to a side of the triangle (there are $3n$ such lines) contains no more than $k$ checkers. Denote by $T(k, n)$ the largest possible number of checkers in such a placement.
a) Prove that the following upper bound is true:
$$
T(k, n) \le \left\lfloor \frac{k(2n + 1)}{3} \right\rfloor .
$$
b) Find $T(1, n)$ and $T(2, n)$.
a) Prove that the following upper bound is true:
$$
T(k, n) \le \left\lfloor \frac{k(2n + 1)}{3} \right\rfloor .
$$
b) Find $T(1, n)$ and $T(2, n)$.