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Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1993 Poland - Second Round
1993 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
3
1
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tetrahedron inequality V_1 +V_2 +V_3 \ge 3V
A tetrahedron
O
A
1
B
1
C
1
OA_1B_1C_1
O
A
1
B
1
C
1
is given. Let
A
2
,
A
3
∈
O
A
1
,
A
2
,
A
3
∈
O
A
1
,
A
2
,
A
3
∈
O
A
1
A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1
A
2
,
A
3
∈
O
A
1
,
A
2
,
A
3
∈
O
A
1
,
A
2
,
A
3
∈
O
A
1
be points such that the planes
A
1
B
1
C
1
,
A
2
B
2
C
2
A_1B_1C_1,A_2B_2C_2
A
1
B
1
C
1
,
A
2
B
2
C
2
and
A
3
B
3
C
3
A_3B_3C_3
A
3
B
3
C
3
are parallel and
O
A
1
>
O
A
2
>
O
A
3
>
0
OA_1 > OA_2 > OA_3 > 0
O
A
1
>
O
A
2
>
O
A
3
>
0
. Let
V
i
V_i
V
i
be the volume of the tetrahedron
O
A
i
B
i
C
i
OA_iB_iC_i
O
A
i
B
i
C
i
(
i
=
1
,
2
,
3
i = 1,2,3
i
=
1
,
2
,
3
) and
V
V
V
be the volume of
O
A
1
B
2
C
3
OA_1B_2C_3
O
A
1
B
2
C
3
. Prove that
V
1
+
V
2
+
V
3
≥
3
V
V_1 +V_2 +V_3 \ge 3V
V
1
+
V
2
+
V
3
≥
3
V
.
2
1
Hide problems
fixed concurrency point as line passing through a point varies
Let be given a circle with center
O
O
O
and a point
P
P
P
outside the circle. A line
l
l
l
passes through
P
P
P
and cuts the circle at
A
A
A
and
B
B
B
. Let
C
C
C
be the point symmetric to
A
A
A
with respect to
O
P
OP
OP
, and let
m
m
m
be the line
B
C
BC
BC
. Prove that all lines
m
m
m
have a common point as
l
l
l
varies.
5
1
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3 triangles with same inradii given, r_1 +r_2 =r wanted
Let
D
,
E
,
F
D,E,F
D
,
E
,
F
be points on the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
of a triangle
A
B
C
ABC
A
BC
, respectively. Suppose that the inradii of the triangles
A
E
F
,
B
F
D
,
C
D
E
AEF,BFD,CDE
A
EF
,
BF
D
,
C
D
E
are all equal to
r
1
r_1
r
1
. If
r
2
r_2
r
2
and
r
r
r
are the inradii of triangles
D
E
F
DEF
D
EF
and
A
B
C
ABC
A
BC
respectively, prove that
r
1
+
r
2
=
r
r_1 +r_2 =r
r
1
+
r
2
=
r
.
6
1
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f(x)f(f(x)) = 1 , f(1000) = 999, continuous, f(500)?
A continuous function
f
:
R
→
R
f : R \to R
f
:
R
→
R
satisfies the conditions
f
(
1000
)
=
999
f(1000) = 999
f
(
1000
)
=
999
and
f
(
x
)
f
(
f
(
x
)
)
=
1
f(x)f(f(x)) = 1
f
(
x
)
f
(
f
(
x
))
=
1
for all real
x
x
x
. Determine
f
(
500
)
f(500)
f
(
500
)
.
1
1
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44-th Polish Mathematical Olympiad 1992/1993
If
x
,
y
,
u
,
v
x,y,u,v
x
,
y
,
u
,
v
are positiv real numbers, prove the inequality : \frac {xu \plus{} xv \plus{} yu \plus{} yv}{x \plus{} y \plus{} u \plus{} v} \geq \frac {xy}{x \plus{} y} \plus{} \frac {uv}{u \plus{} v}
4
1
Hide problems
Poland 1993!
Let
(
x
n
)
(x_n)
(
x
n
)
be the sequence of natural number such that: x_1\equal{}1 and x_n
1
≤
n
1\leq n
1
≤
n
. Prove that for every natural number
k
k
k
, there exist the subscripts
r
r
r
and
s
s
s
, such that x_r\minus{}x_s\equal{}k.