MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1990 Federal Competition For Advanced Students, P2
1990 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
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compute the distance
A convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is inscribed in a circle. The distances of
A
A
A
from the lines
B
C
,
C
D
,
D
E
BC,CD,DE
BC
,
C
D
,
D
E
are
a
,
b
,
c
,
a,b,c,
a
,
b
,
c
,
respectively. Compute the distance of
A
A
A
from the line
B
E
BE
BE
.
5
1
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determine all rational numbers
Determine all rational numbers
r
r
r
such that all solutions of the equation: rx^2\plus{}(r\plus{}1)x\plus{}(r\minus{}1)\equal{}0 are integers.
4
1
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functions
For each nonzero integer
n
n
n
find all functions f: \mathbb{R} \minus{} \{\minus{}3,0 \} \rightarrow \mathbb{R} satisfying: f(x\plus{}3)\plus{}f \left( \minus{}\frac{9}{x} \right)\equal{}\frac{(1\minus{}n)(x^2\plus{}3x\minus{}9)}{9n(x\plus{}3)}\plus{}\frac{2}{n} for all x \not\equal{} 0,\minus{}3. Furthermore, for each fixed
n
n
n
find all integers
x
x
x
for which
f
(
x
)
f(x)
f
(
x
)
is an integer.
3
1
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inequality
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, let
E
E
E
be the intersection point of the diagonals, and let
F
1
,
F
2
,
F_1,F_2,
F
1
,
F
2
,
and
F
F
F
be the areas of
A
B
E
,
C
D
E
,
ABE,CDE,
A
BE
,
C
D
E
,
and
A
B
C
D
,
ABCD,
A
BC
D
,
respectively. Prove that: \sqrt {F_1}\plus{}\sqrt {F_2} \le \sqrt {F}.
2
1
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inequality with integers
Show that for all integers
n
≥
2
n \ge 2
n
≥
2
,
2
3
4...
n
n
4
3
<
2
\sqrt { 2\sqrt[3]{3 \sqrt[4]{4...\sqrt[n]{n}}}}<2
2
3
3
4
4...
n
n
<
2
1
1
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determine the number of integers
Determine the number of integers
n
n
n
with 1 \le n \le N\equal{}1990^{1990} such that n^2\minus{}1 and
N
N
N
are coprime.