MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1988 Federal Competition For Advanced Students, P2
1988 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
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monic polynomials
Determine all monic polynomials
p
(
x
)
p(x)
p
(
x
)
of fifth degree having real coefficients and the following property: Whenever
a
a
a
is a (real or complex) root of
p
(
x
)
p(x)
p
(
x
)
, then so are
1
a
\frac{1}{a}
a
1
and 1\minus{}a.
5
1
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incircle
The bisectors of angles
B
B
B
and
C
C
C
of triangle
A
B
C
ABC
A
BC
intersect the opposite sides in points
B
′
B'
B
′
and
C
′
C'
C
′
respectively. Show that the line
B
′
C
′
B'C'
B
′
C
′
intersects the incircle of the triangle.
4
1
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system of equations
Let
a
i
j
a_{ij}
a
ij
be nonnegative integers such that a_{ij}\equal{}0 if and only if
i
>
j
i>j
i
>
j
and that \displaystyle\sum_{j\equal{}1}^{1988}a_{ij}\equal{}1988 holds for all i\equal{}1,...,1988. Find all real solutions of the system of equations: \displaystyle\sum_{j\equal{}1}^{1988} (1\plus{}a_{ij})x_j\equal{}i\plus{}1, 1 \le i \le 1988.
3
1
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sequence
Show that there is precisely one sequence
a
1
,
a
2
,
.
.
.
a_1,a_2,...
a
1
,
a
2
,
...
of integers which satisfies a_1\equal{}1, a_2>1, and a_{n\plus{}1}^3\plus{}1\equal{}a_n a_{n\plus{}2} for
n
≥
1
n \ge 1
n
≥
1
.
2
1
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colouring
An equilateral triangle
A
1
A
2
A
3
A_1 A_2 A_3
A
1
A
2
A
3
is divided into four smaller equilateral triangles by joining the midpoints
A
4
,
A
5
,
A
6
A_4,A_5,A_6
A
4
,
A
5
,
A
6
of its sides. Let
A
7
,
.
.
.
,
A
15
A_7,...,A_{15}
A
7
,
...
,
A
15
be the midpoints of the sides of these smaller triangles. The
15
15
15
points
A
1
,
.
.
.
,
A
15
A_1,...,A_{15}
A
1
,
...
,
A
15
are colored either green or blue. Show that with any such colouring there are always three mutually equidistant points
A
i
,
A
j
,
A
k
A_i,A_j,A_k
A
i
,
A
j
,
A
k
having the same color.
1
1
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easy inequality
If
a
1
,
.
.
.
,
a
1988
a_1,...,a_{1988}
a
1
,
...
,
a
1988
are positive numbers whose arithmetic mean is
1988
1988
1988
, show that: \sqrt[1988]{\displaystyle\prod_{i,j\equal{}1}^{1988} \left( 1\plus{}\frac{a_i}{a_j} \right)} \ge 2^{1988} and determine when equality holds.