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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1900 Eotvos Mathematical Competition
1900 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
Hide problems
A cliff is $300$ meters high. Consider two free-falling raindrops such that the
A cliff is
300
300
300
meters high. Consider two free-falling raindrops such that the second one leaves the top of the cliff when the first one has already fallen
0.001
0.001
0.001
millimeters. What is the distance between the drops at the moment the first hits the ground? (Compute the answer to within
0.1
0.1
0.1
mm. Neglect air resistance, etc.)
2
1
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Construct a triangle $ABC$, given the length $c$ of its side $AB$, the radius $r
Construct a triangle
A
B
C
ABC
A
BC
, given the length
c
c
c
of its side
A
B
AB
A
B
, the radius
r
r
r
of its inscribed circle, and the radius
r
c
r_c
r
c
of its ex-circle tangent to the side
A
B
AB
A
B
and the extensions of
B
C
BC
BC
and
C
A
CA
C
A
.
1
1
Hide problems
Let $a, b, c, d$ be fixed integers with $d$ not divisible by $5$. Assume that $m
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be fixed integers with
d
d
d
not divisible by
5
5
5
. Assume that
m
m
m
is an integer for which
a
m
3
+
b
m
2
+
c
m
+
d
am3 +bm2 +cm+d
am
3
+
bm
2
+
c
m
+
d
is divisible by
5
5
5
. Prove that there exists an integer
n
n
n
for which
d
n
3
+
c
n
2
+
b
n
+
a
dn3 +cn2 +bn+a
d
n
3
+
c
n
2
+
bn
+
a
is also divisible by
5
5
5
.