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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1899 Eotvos Mathematical Competition
1899 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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Prove that, for any natural number $n$, the expression $$A = 2903^n-803^n-464^n+
Prove that, for any natural number
n
n
n
, the expression
A
=
290
3
n
−
80
3
n
−
46
4
n
+
26
1
n
A = 2903^n-803^n-464^n+261^n
A
=
290
3
n
−
80
3
n
−
46
4
n
+
26
1
n
is divisible by
1897
1897
1897
.
2
1
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Let $x_1$ and $x_2$ be the roots of the equation $$x^2-(a+d)x+ad-bc=0.$$ Show th
Let
x
1
x_1
x
1
and
x
2
x_2
x
2
be the roots of the equation
x
2
−
(
a
+
d
)
x
+
a
d
−
b
c
=
0.
x^2-(a+d)x+ad-bc=0.
x
2
−
(
a
+
d
)
x
+
a
d
−
b
c
=
0.
Show that
x
1
3
x^3_1
x
1
3
and
x
2
3
x^3_2
x
2
3
are the roots of
y
3
−
(
a
3
+
d
3
+
3
a
b
c
+
3
b
c
d
)
y
+
(
a
d
−
b
c
)
3
=
0.
y^3-(a^3+d^3+3abc+3bcd)y+(ad-bc)^3 =0.
y
3
−
(
a
3
+
d
3
+
3
ab
c
+
3
b
c
d
)
y
+
(
a
d
−
b
c
)
3
=
0.
1
1
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The points $A_0, A_1, A_2, A_3, A_4$ divide a unit circle (circle of radius $1$)
The points
A
0
,
A
1
,
A
2
,
A
3
,
A
4
A_0, A_1, A_2, A_3, A_4
A
0
,
A
1
,
A
2
,
A
3
,
A
4
divide a unit circle (circle of radius
1
1
1
) into five equal parts. Prove that the chords
A
0
,
A
1
,
A
0
,
A
2
A_0, A_1, A_0, A_2
A
0
,
A
1
,
A
0
,
A
2
satisfy
(
A
0
A
1
⋅
A
0
A
2
)
2
=
5
(A_0A_1 \cdot A_0A_2)^2= 5
(
A
0
A
1
⋅
A
0
A
2
)
2
=
5