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Contests
National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1929 Eotvos Mathematical Competition
1929 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
2
1
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b(n,0) - b(n,1) x +b(n,2) x^2 - ... + (-1)^k b(n,k) x^k > 0$
Let
k
≤
n
k \le n
k
≤
n
be positive integers and
x
x
x
be a real number with
0
≤
x
<
1
/
n
0 \le x < 1/n
0
≤
x
<
1/
n
. Prove that
(
n
0
)
−
(
n
1
)
x
+
(
n
2
)
x
2
−
.
.
.
+
(
−
1
)
k
(
n
k
)
x
k
>
0
{n \choose 0} - {n \choose 1} x +{n \choose 2} x^2 - ... + (-1)^k {n \choose k} x^k > 0
(
0
n
)
−
(
1
n
)
x
+
(
2
n
)
x
2
−
...
+
(
−
1
)
k
(
k
n
)
x
k
>
0
3
1
Hide problems
3 concurrent lines with angle of 60^o among each two
Let
p
,
q
p, q
p
,
q
and
r
r
r
be three concurrent lines in the plane such that the angle between any two of them is
6
0
o
60^o
6
0
o
. Let
a
a
a
,
b
b
b
and
c
c
c
be real numbers such that
0
<
a
≤
b
≤
c
0 < a \le b \le c
0
<
a
≤
b
≤
c
.(a) Prove that the set of points whose distances from
p
,
q
p, q
p
,
q
and
r
r
r
are respectively less than
a
,
b
a, b
a
,
b
and
c
c
c
consists of the interior of a hexagon if and only if
a
+
b
>
c
a + b > c
a
+
b
>
c
.(b) Determine the length of the perimeter of this hexagon when
a
+
b
>
c
a + b > c
a
+
b
>
c
.
1
1
Hide problems
100 fillér be made up with coins of denominations l, 2, 10, 20 and 50
In how many ways can the sum of 100 fillér be made up with coins of denominations l, 2, 10, 20 and 50 fillér?