MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand Team Selection Test
2021 Thailand TST
2021 Thailand TST
Part of
Thailand Team Selection Test
Subcontests
(2)
2
2
Hide problems
Floor(m sqrt(2))Floor(n sqrt(7))<Floor(mn sqrt(14))
Prove that, for all positive integers
m
m
m
and
n
n
n
, we have
⌊
m
2
⌋
⋅
⌊
n
7
⌋
<
⌊
m
n
14
⌋
.
\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.
⌊
m
2
⌋
⋅
⌊
n
7
⌋
<
⌊
mn
14
⌋
.
Integer Sets
Let
A
\mathcal{A}
A
be the set of all
n
∈
N
n\in\mathbb{N}
n
∈
N
for which there exist
k
∈
N
k\in\mathbb{N}
k
∈
N
and
a
0
,
a
1
,
…
,
a
k
−
1
∈
{
1
,
2
,
…
,
9
}
a_0,a_1,\dots,a_{k-1}\in \{1,2,\dots,9\}
a
0
,
a
1
,
…
,
a
k
−
1
∈
{
1
,
2
,
…
,
9
}
such that
a
0
≥
a
1
≥
⋯
≥
a
k
−
1
a_0 \geq a_1 \geq \cdots \geq a_{k-1}
a
0
≥
a
1
≥
⋯
≥
a
k
−
1
and
n
=
a
0
+
a
1
⋅
1
0
1
+
⋯
+
a
k
−
1
⋅
1
0
k
−
1
n = a_0 +a_1 \cdot 10^1 +\cdots +a_{k-1}\cdot 10^{k-1}
n
=
a
0
+
a
1
⋅
1
0
1
+
⋯
+
a
k
−
1
⋅
1
0
k
−
1
. Let
B
\mathcal{B}
B
be the set of all
m
∈
N
m \in\mathbb{N}
m
∈
N
for which there exist
l
∈
N
l \in\mathbb{N}
l
∈
N
and
b
0
,
b
1
,
…
,
b
l
−
1
∈
{
1
,
2
,
…
,
9
}
b_0,b_1,\dots,b_{l-1} \in \{1,2,\dots,9\}
b
0
,
b
1
,
…
,
b
l
−
1
∈
{
1
,
2
,
…
,
9
}
such that
b
0
≤
b
1
≤
⋯
≤
b
l
−
1
b_0 \leq b_1 \leq \cdots\leq b_{l-1}
b
0
≤
b
1
≤
⋯
≤
b
l
−
1
and
m
=
b
0
+
b
1
⋅
1
0
1
+
⋯
+
b
l
−
1
⋅
1
0
l
−
1
m = b_0 + b_1 \cdot 10^1 + \cdots+ b_{l-1}\cdot 10^{l-1}
m
=
b
0
+
b
1
⋅
1
0
1
+
⋯
+
b
l
−
1
⋅
1
0
l
−
1
.[*] Are there infinitely many
n
∈
A
n\in \mathcal{A}
n
∈
A
such that
n
2
−
3
∈
A
?
n^2-3\in\mathcal{A} \ ?
n
2
−
3
∈
A
?
[*] Are there infinitely many
m
∈
B
m\in \mathcal{B}
m
∈
B
such that
m
2
−
3
∈
B
?
m^2-3\in\mathcal{B} \ ?
m
2
−
3
∈
B
?
Proposed by Pakawut Jiradilok and Wijit Yangjit
1
1
Hide problems
Delicious Cake
For a positive integer
n
n
n
, consider a square cake which is divided into
n
×
n
n \times n
n
×
n
pieces with at most one strawberry on each piece. We say that such a cake is delicious if both diagonals are fully occupied, and each row and each column has an odd number of strawberries. Find all positive integers
n
n
n
such that there is an
n
×
n
n \times n
n
×
n
delicious cake with exactly
⌈
n
2
2
⌉
\left\lceil\frac{n^2}{2}\right\rceil
⌈
2
n
2
⌉
strawberries on it.