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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2022 Kurschak Competition
2022 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Sum of squares of columns vs sum of squares
Let
a
i
,
j
(
∀
1
≤
i
≤
n
,
1
≤
j
≤
n
)
a_{i,j}\enspace(\forall\enspace 1\leq i\leq n, 1\leq j\leq n)
a
i
,
j
(
∀
1
≤
i
≤
n
,
1
≤
j
≤
n
)
be
n
2
n^2
n
2
real numbers such that
a
i
,
j
+
a
j
,
i
=
0
∀
i
,
j
a_{i,j}+a_{j,i}=0\enspace\forall i, j
a
i
,
j
+
a
j
,
i
=
0
∀
i
,
j
(in particular,
a
i
,
i
=
0
∀
i
a_{i,i}=0\enspace\forall i
a
i
,
i
=
0
∀
i
). Prove that
1
n
∑
i
=
1
n
(
∑
j
=
1
n
a
i
,
j
)
2
≤
1
2
∑
i
=
1
n
∑
j
=
1
n
a
i
,
j
2
.
{1\over n}\sum_{i=1}^{n}\left(\sum_{j=1}^{n} a_{i,j}\right)^2\leq{1\over2}\sum_{i=1}^{n}\sum_{j=1}^{n} a_{i,j}^2.
n
1
i
=
1
∑
n
(
j
=
1
∑
n
a
i
,
j
)
2
≤
2
1
i
=
1
∑
n
j
=
1
∑
n
a
i
,
j
2
.
2
1
Hide problems
Solutions to |px^2-qy^2|=1
Let
p
p
p
and
q
q
q
be prime numbers of the form
4
k
+
3
4k+3
4
k
+
3
. Suppose that there exist integers
x
x
x
and
y
y
y
such that
x
2
−
p
q
y
2
=
1
x^2-pqy^2=1
x
2
−
pq
y
2
=
1
. Prove that there exist positive integers
a
a
a
and
b
b
b
such that
∣
p
a
2
−
q
b
2
∣
=
1
|pa^2-qb^2|=1
∣
p
a
2
−
q
b
2
∣
=
1
.
1
1
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Sidelines of rectangles
A square has been divided into
2022
2022
2022
rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?