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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1902 Eotvos Mathematical Competition
1902 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
1
1
Hide problems
Ax^2 + Bx + C= k x(x- 1)/2 + l x + m, integer values
Prove that any quadratic expression
Q
(
x
)
=
A
x
2
+
B
x
+
C
Q(x) = Ax^2 + Bx + C
Q
(
x
)
=
A
x
2
+
B
x
+
C
(a) can be put into the form
Q
(
x
)
=
k
x
(
x
−
1
)
1
⋅
2
+
ℓ
x
+
m
Q(x) = k \frac{x(x- 1)}{1 \cdot 2} + \ell x + m
Q
(
x
)
=
k
1
⋅
2
x
(
x
−
1
)
+
ℓ
x
+
m
where
k
,
ℓ
,
m
k, \ell, m
k
,
ℓ
,
m
depend on the coefficients
A
,
B
,
C
A,B,C
A
,
B
,
C
and(b)
Q
(
x
)
Q(x)
Q
(
x
)
takes on integral values for every integer
x
x
x
if and only if
k
,
ℓ
,
m
k, \ell, m
k
,
ℓ
,
m
are integers.
3
1
Hide problems
min side, fixed area and fixed opposite angle
The area
T
T
T
and an angle
γ
\gamma
γ
of a triangle are given. Determine the lengths of the sides
a
a
a
and
b
b
b
so that the side
c
c
c
, opposite the angle
γ
\gamma
γ
, is as short as possible.
2
1
Hide problems
area of surface of a sphere inside another sphere is fixed
Let
S
S
S
be a given sphere with center
O
O
O
and radius
r
r
r
. Let
P
P
P
be any point outside then sphere
S
S
S
, and let
S
′
S'
S
′
be the sphere with center
P
P
P
and radius
P
O
PO
PO
. Denote by
F
F
F
the area of the surface of the part of
S
′
S'
S
′
that lies inside
S
S
S
. Prove that
F
F
F
is independent of the particular point
P
P
P
chosen.