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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1917 Eotvos Mathematical Competition
1917 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
Hide problems
infinite circles through A,B which lie entirely in k.
Let
A
A
A
and
B
B
B
be two points inside a given circle
k
k
k
. Prove that there exist (infinitely many) circles through
A
A
A
and
B
B
B
which lie entirely in
k
k
k
.
1
1
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y - 2x - a = 0, y^2 - xy + x^2 - b = 0,. NT system
If
a
a
a
and
b
b
b
are integers and if the solutions of the system of equations
y
−
2
x
−
a
=
0
y - 2x - a = 0
y
−
2
x
−
a
=
0
y
2
−
x
y
+
x
2
−
b
=
0
y^2 - xy + x^2 - b = 0
y
2
−
x
y
+
x
2
−
b
=
0
are rational, prove that the solutions are integers.
2
1
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units’ digit of a^2 if tens’ digit of a is 7
In the square of an integer
a
a
a
, the tens’ digit is
7
7
7
. What is the units’ digit of
a
2
a^2
a
2
?