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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1993 Kurschak Competition
1993 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Minimize a special polynomial
Let
n
n
n
be a fixed positive integer. Compute over
R
\mathbb{R}
R
the minimum of the following polynomial:
f
(
x
)
=
∑
t
=
0
2
n
(
2
n
+
1
−
t
)
x
t
.
f(x)=\sum_{t=0}^{2n}(2n+1-t)x^t.
f
(
x
)
=
t
=
0
∑
2
n
(
2
n
+
1
−
t
)
x
t
.
2
1
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Parallels and intouch triangle
Triangle
A
B
C
ABC
A
BC
is not isosceles. The incircle of
△
A
B
C
\triangle ABC
△
A
BC
touches the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
in the points
K
K
K
,
L
L
L
,
M
M
M
. The parallel with
L
M
LM
L
M
through
B
B
B
meets
K
L
KL
K
L
at
D
D
D
, the parallel with
L
M
LM
L
M
through
C
C
C
meets
K
M
KM
K
M
at
E
E
E
.Prove that
D
E
DE
D
E
passes through the midpoint of
L
M
‾
\overline{LM}
L
M
.
1
1
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Square values of f(x)=ax^2+b
Let
a
a
a
and
b
b
b
be positive integers. Prove that the numbers
a
n
2
+
b
an^2+b
a
n
2
+
b
and
a
(
n
+
1
)
2
+
b
a(n+1)^2+b
a
(
n
+
1
)
2
+
b
are both perfect squares only for finitely many integers
n
n
n
.