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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1894 Eotvos Mathematical Competition
1894 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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The side lengths of a triangle area $t$ form an arithmetic progression with diff
The side lengths of a triangle area
t
t
t
form an arithmetic progression with difference
d
d
d
. Find the sides and angles of the triangle. Specifically, solve this problem for
d
=
1
d=1
d
=
1
and
t
=
6
t=6
t
=
6
.
2
1
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Given a circle and two points $P and Q$, construct a right triangle inscribed in
Given a circle and two points
P
P
P
and
Q
Q
Q
, construct a right triangle inscribed in the circle such that its two legs pass through the points
P
P
P
and
Q
Q
Q
respectively. For what positions of
P
P
P
and
Q
Q
Q
is this construction impossible?
1
1
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Let $x$ and $y$ be integers. Prove that one of the expressions $$2x+3y \text{and
Let
x
x
x
and
y
y
y
be integers. Prove that one of the expressions
2
x
+
3
y
and
9
x
+
5
y
2x+3y \text{ and } 9x+5y
2
x
+
3
y
and
9
x
+
5
y
is divisible by
17
17
17
if and only if so is the other.