MathDB
Problems
Contests
National and Regional Contests
Belarus Contests
Belarus Team Selection Test
2024 Belarus Team Selection Test
4.4
4.4
Part of
2024 Belarus Team Selection Test
Problems
(1)
Placing checkers in circles
Source: Belarusian TST 2024
10/27/2024
Given positive integers
n
n
n
and
k
≤
n
k \leq n
k
≤
n
. Consider an equilateral triangular board with side
n
n
n
, which consists of circles: in the first (top) row there is one circle, in the second row there are two circles,
…
\ldots
…
, in the bottom row there are
n
n
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
3
n
3n
3
n
such lines) contains no more than
k
k
k
checkers. Denote by
T
(
k
,
n
)
T(k, n)
T
(
k
,
n
)
the largest possible number of checkers in such a placement. https://i.ibb.co/bJjjK1M/Image2.jpg a) Prove that the following upper bound is true:
T
(
k
,
n
)
≤
⌊
k
(
2
n
+
1
)
3
⌋
T(k,n) \leq \lfloor \frac{k(2n+1)}{3} \rfloor
T
(
k
,
n
)
≤
⌊
3
k
(
2
n
+
1
)
⌋
b) Find
T
(
1
,
n
)
T(1,n)
T
(
1
,
n
)
and
T
(
2
,
n
)
T(2,n)
T
(
2
,
n
)
D. Zmiaikou
combinatorics