MathDB
Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1983 Kurschak Competition
1983 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
Hide problems
Q lies inside the triangle P_iP_jP_k
Given are
n
+
1
n + 1
n
+
1
points
P
1
,
P
2
,
.
.
.
,
P
n
P_1, P_2,..., P_n
P
1
,
P
2
,
...
,
P
n
and
Q
Q
Q
in the plane, no three collinear. For any two different points
P
i
P_i
P
i
and
P
j
P_j
P
j
, there is a point
P
k
P_k
P
k
such that the point
Q
Q
Q
lies inside the triangle
P
i
P
j
P
k
P_iP_jP_k
P
i
P
j
P
k
. Prove that
n
n
n
is an odd number.
2
1
Hide problems
f(2) >= 3^n for polynomial with n real roots and non-negative coefficients
Prove that
f
(
2
)
≥
3
n
f(2) \ge 3^n
f
(
2
)
≥
3
n
where the polynomial
f
(
x
)
=
x
n
+
a
1
x
n
−
1
+
.
.
.
+
a
n
−
1
x
+
1
f(x) = x_n + a_1x_{n-1} + ...+ a_{n-1}x + 1
f
(
x
)
=
x
n
+
a
1
x
n
−
1
+
...
+
a
n
−
1
x
+
1
has non-negative coefficients and
n
n
n
real roots.
1
1
Hide problems
x^3 + 3y^3 + 9z^3 - 9xyz = 0 in rationals
Let
x
,
y
x, y
x
,
y
and
z
z
z
be rational numbers satisfying
x
3
+
3
y
3
+
9
z
3
−
9
x
y
z
=
0.
x^3 + 3y^3 + 9z^3 - 9xyz = 0.
x
3
+
3
y
3
+
9
z
3
−
9
x
yz
=
0.
Prove that
x
=
y
=
z
=
0
x = y = z = 0
x
=
y
=
z
=
0
.