MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2024 Poland - Second Round
2024 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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Another vp NT with factorials
Given is a prime number
p
p
p
. Prove that the number
p
⋅
(
p
2
⋅
p
p
−
1
−
1
p
−
1
)
!
p \cdot (p^2 \cdot \frac{p^{p-1}-1}{p-1})!
p
⋅
(
p
2
⋅
p
−
1
p
p
−
1
−
1
)!
is divisible by
∏
i
=
1
p
(
p
i
)
!
.
\prod_{i=1}^{p}(p^i)!.
i
=
1
∏
p
(
p
i
)!
.
5
1
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Beautiful 6-variable inequality
The positive reals
a
,
b
,
c
,
x
,
y
,
z
a, b, c, x, y, z
a
,
b
,
c
,
x
,
y
,
z
satisfy
5
a
+
4
b
+
3
c
=
5
x
+
4
y
+
3
z
.
5a+4b+3c=5x+4y+3z.
5
a
+
4
b
+
3
c
=
5
x
+
4
y
+
3
z
.
Show that
a
5
x
4
+
b
4
y
3
+
c
3
z
2
≥
x
+
y
+
z
.
\frac{a^5}{x^4}+\frac{b^4}{y^3}+\frac{c^3}{z^2} \geq x+y+z.
x
4
a
5
+
y
3
b
4
+
z
2
c
3
≥
x
+
y
+
z
.
Proposed by Dominik Burek
4
1
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Regular hexagon combo
Let
n
n
n
be a positive integer. A regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
with side length
n
n
n
is partitioned into
6
n
2
6n^2
6
n
2
equilateral triangles with side length
1
1
1
. The hexagon is covered by
3
n
2
3n^2
3
n
2
rhombuses with internal angles
6
0
∘
60^{\circ}
6
0
∘
and
12
0
∘
120^{\circ}
12
0
∘
such that each rhombus covers exactly two triangles and every triangle is covered by exactly one rhombus. Show that the diagonal
A
D
AD
A
D
divides in half exactly
n
n
n
rhombuses.
3
1
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The king builds a connected graph optimally
Let
n
≥
2
n \geq 2
n
≥
2
be a positive integer. There are
2
n
2n
2
n
cities
M
1
,
M
2
,
…
,
M
2
n
M_1, M_2, \ldots, M_{2n}
M
1
,
M
2
,
…
,
M
2
n
in the country of Mathlandia. Currently there roads only between
M
1
M_1
M
1
and
M
2
,
M
3
,
…
,
M
n
M_2, M_3, \ldots, M_n
M
2
,
M
3
,
…
,
M
n
and the king wants to build more roads so that it is possible to reach any city from every other city. The cost to build a road between
M
i
M_i
M
i
and
M
j
M_j
M
j
is
k
i
,
j
>
0
k_{i, j}>0
k
i
,
j
>
0
. Let
K
=
∑
j
=
n
+
1
2
n
k
1
,
j
+
∑
2
≤
i
<
j
≤
2
n
k
i
,
j
.
K=\sum_{j=n+1}^{2n} k_{1,j}+\sum_{2 \leq i<j \leq 2n} k_{i, j}.
K
=
j
=
n
+
1
∑
2
n
k
1
,
j
+
2
≤
i
<
j
≤
2
n
∑
k
i
,
j
.
Prove that the king can fulfill his plan at cost no more than
2
K
3
n
−
1
\frac{2K}{3n-1}
3
n
−
1
2
K
.
2
1
Hide problems
Geo with angle and length conditions
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
A
B
C
=
∠
A
D
C
=
12
0
∘
\angle ABC=\angle ADC=120^{\circ}
∠
A
BC
=
∠
A
D
C
=
12
0
∘
. The point
E
E
E
lies on the segment
A
D
AD
A
D
and is such that
A
E
⋅
B
C
=
A
B
⋅
D
E
AE \cdot BC=AB \cdot DE
A
E
⋅
BC
=
A
B
⋅
D
E
and similarly the point
F
F
F
lies on the segment
B
C
BC
BC
and satisfies
B
F
⋅
C
D
=
A
D
⋅
C
F
BF \cdot CD=AD \cdot CF
BF
⋅
C
D
=
A
D
⋅
CF
. Show that
B
E
BE
BE
and
D
F
DF
D
F
are parallel.
1
1
Hide problems
Rational sequence
Does there exist a rational
x
1
x_1
x
1
, such that all members of the sequence
x
1
,
x
2
,
…
,
x
2024
x_1, x_2, \ldots, x_{2024}
x
1
,
x
2
,
…
,
x
2024
defined by
x
n
+
1
=
x
n
+
x
n
2
−
1
x_{n+1}=x_n+\sqrt{x_n^2-1}
x
n
+
1
=
x
n
+
x
n
2
−
1
for
n
=
1
,
2
,
…
,
2023
n=1, 2, \ldots, 2023
n
=
1
,
2
,
…
,
2023
are greater than
1
1
1
and rational?