MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2017 Poland - Second Round
2017 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
Perfect square
A prime number
p
>
2
p > 2
p
>
2
and
x
,
y
∈
{
1
,
2
,
…
,
p
−
1
2
}
x,y \in \left\{ 1,2,\ldots, \frac{p-1}{2} \right\}
x
,
y
∈
{
1
,
2
,
…
,
2
p
−
1
}
are given. Prove that if
x
(
p
−
x
)
y
(
p
−
y
)
x\left( p-x\right)y\left( p-y\right)
x
(
p
−
x
)
y
(
p
−
y
)
is a perfect square, then
x
=
y
x = y
x
=
y
.
5
1
Hide problems
Restaurants
Gourmet Jan compared
n
n
n
restaurants (
n
n
n
is a positive integer). Each pair of restaurants was compared in two categories: tastiness of food and quality of service. For some pairs Jan couldn't tell which restaurant was better in one category, but never in two categories. Moreover, if Jan thought restaurant
A
A
A
was better than restaurant
B
B
B
in one category and restaurant
B
B
B
was better than restaurant
C
C
C
in the same category, then
A
A
A
is also better than
C
C
C
in that category. Prove there exists a restaurant
R
R
R
such that every other restaurant is worse than
R
R
R
in at least one category.
4
1
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Geometry
Incircle of a triangle
A
B
C
ABC
A
BC
touches
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
, respectively. Point
J
J
J
is the excenter of
A
A
A
. Points
M
M
M
and
N
N
N
are midpoints of
J
D
JD
J
D
and
J
E
JE
J
E
. Lines
B
M
BM
BM
and
C
N
CN
CN
cross at point
P
P
P
. Prove that
P
P
P
lies on the circumcircle of
A
B
C
ABC
A
BC
.
3
1
Hide problems
Inequality on 2n-1 real numbers [Poland Second Round 2017, D1, P3]
Let
x
1
≤
x
2
≤
…
≤
x
2
n
−
1
x_1 \le x_2 \le \ldots \le x_{2n-1}
x
1
≤
x
2
≤
…
≤
x
2
n
−
1
be real numbers whose arithmetic mean equals
A
A
A
. Prove that
2
∑
i
=
1
2
n
−
1
(
x
i
−
A
)
2
≥
∑
i
=
1
2
n
−
1
(
x
i
−
x
n
)
2
.
2\sum_{i=1}^{2n-1}\left( x_{i}-A\right)^2 \ge \sum_{i=1}^{2n-1}\left( x_{i}-x_{n}\right)^2.
2
i
=
1
∑
2
n
−
1
(
x
i
−
A
)
2
≥
i
=
1
∑
2
n
−
1
(
x
i
−
x
n
)
2
.
2
1
Hide problems
Orthogonal projections and areas
In an acute triangle
A
B
C
ABC
A
BC
the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
crosses
B
C
BC
BC
at
D
D
D
. Points
P
P
P
and
Q
Q
Q
are orthogonal projections of
D
D
D
on lines
A
B
AB
A
B
and
A
C
AC
A
C
. Prove that
[
A
P
Q
]
=
[
B
C
Q
P
]
[APQ]=[BCQP]
[
A
PQ
]
=
[
BCQP
]
if and only if the circumcenter of
A
B
C
ABC
A
BC
lies on
P
Q
PQ
PQ
.
1
1
Hide problems
Perfect square
Prove that for each prime
p
>
2
p>2
p
>
2
there exists exactly one positive integer
n
n
n
, such that
n
2
+
n
p
n^2+np
n
2
+
n
p
is a perfect square.