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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2002 Regional Competition For Advanced Students
2002 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
3
1
Hide problems
perimeter inequality in convex hegaxon , 1/2 \le s/(u+v) \le 1
In the convex
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
(has all interior angles less than
18
0
o
180^o
18
0
o
) with the perimeter
s
s
s
the triangles
A
C
E
ACE
A
CE
and
B
D
F
BDF
B
D
F
have perimeters
u
u
u
and
v
v
v
respectively. a) Show the inequalities
1
2
≤
s
u
+
v
≤
1
\frac{1}{2} \le \frac{s}{u+v}\le 1
2
1
≤
u
+
v
s
≤
1
b) Check whether
1
1
1
is replaced by a smaller number or
1
/
2
1/2
1/2
by a larger number can the inequality remains valid for all convex hexagons.
4
1
Hide problems
smallest of a_k (1-a_{2002-k}) ($0 \le k le 2002$) is <= 1/4
Let
a
0
,
a
1
,
.
.
.
,
a
2002
a_0, a_1, ..., a_{2002}
a
0
,
a
1
,
...
,
a
2002
be real numbers. a) Show that the smallest of the values
a
k
(
1
−
a
2002
−
k
)
a_k (1-a_{2002-k})
a
k
(
1
−
a
2002
−
k
)
(
0
≤
k
≤
2002
0 \le k \le 2002
0
≤
k
≤
2002
) the following applies: it is smaller or equal to
1
/
4
1/4
1/4
. b) Does this statement always apply to the smallest of the values
a
k
(
1
−
a
2003
−
k
)
a_k (1-a_{2003-k})
a
k
(
1
−
a
2003
−
k
)
(
1
≤
k
≤
2002
1 \le k \le 2002
1
≤
k
≤
2002
) ? c) Show for positive real numbers
a
0
,
a
1
,
.
.
.
,
a
2002
a_0, a_1, ..., a_{2002}
a
0
,
a
1
,
...
,
a
2002
: the smallest of the values
a
k
(
1
−
a
2003
−
k
)
a_k (1-a_{2003-k})
a
k
(
1
−
a
2003
−
k
)
(
1
≤
k
≤
2002
1 \le k \le 2002
1
≤
k
≤
2002
) is less than or equal to
1
/
4
1/4
1/4
.
2
1
Hide problems
5x5 system, 2x_1=x_5^2-23, 4x_2=x_1^2+7, 6x_3=x_2^2+14 ,8x_4=x_3^2+23,
Solve the following system of equations over the real numbers:
2
x
1
=
x
5
2
−
23
2x_1 = x_5 ^2 - 23
2
x
1
=
x
5
2
−
23
4
x
2
=
x
1
2
+
7
4x_2 = x_1 ^2 + 7
4
x
2
=
x
1
2
+
7
6
x
3
=
x
2
2
+
14
6x_3 = x_2 ^2 + 14
6
x
3
=
x
2
2
+
14
8
x
4
=
x
3
2
+
23
8x_4 = x_3 ^2 + 23
8
x
4
=
x
3
2
+
23
10
x
5
=
x
4
2
+
34
10x_5 = x_4 ^2 + 34
10
x
5
=
x
4
2
+
34
1
1
Hide problems
\frac{3x+9}{8},\frac{3x+10}{9},\frac{3x+11}{10},...,\frac{3x+49}{48} simplify
Find the smallest natural number
x
>
0
x> 0
x
>
0
so that all following fractions are simplified
3
x
+
9
8
,
3
x
+
10
9
,
3
x
+
11
10
,
.
.
.
,
3
x
+
49
48
\frac{3x+9}{8},\frac{3x+10}{9},\frac{3x+11}{10},...,\frac{3x+49}{48}
8
3
x
+
9
,
9
3
x
+
10
,
10
3
x
+
11
,
...
,
48
3
x
+
49
, i.e. numerators and denominators are relatively prime.