MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1991 Poland - Second Round
1991 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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R/r >= \sqrt3, sphere contains parallelepiped and it contains another sphere
The parallelepiped contains a sphere of radius
r
r
r
and is contained within a sphere of radius
R
R
R
. Prove that
R
r
≥
3
\frac{R}{r} \geq \sqrt{3}
r
R
≥
3
.
5
1
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2-element subests of 1-n
P
1
,
P
2
,
…
,
P
n
P_1, P_2, \ldots, P_n
P
1
,
P
2
,
…
,
P
n
are different two-element subsets of
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
. The sets
P
i
P_i
P
i
,
P
j
P_j
P
j
for
i
≠
j
i\neq j
i
=
j
have a common element if and only if the set
{
i
,
j
}
\{i,j\}
{
i
,
j
}
is one of the sets
P
1
,
P
2
,
…
,
P
n
P_1, P_2, \ldots, P_n
P
1
,
P
2
,
…
,
P
n
. Prove that each of the numbers
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
is a common element of exactly two sets from
P
1
,
P
2
,
…
,
P
n
P_1, P_2, \ldots, P_n
P
1
,
P
2
,
…
,
P
n
.
4
1
Hide problems
f(4x) - f(3x) = 2x
Find all monotone functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
satisfying the equation
f
(
4
x
)
−
f
(
3
x
)
=
2
x
for
x
∈
R
.
f(4x)-f(3x) = 2x \ \ \text{ for } \ \ x \in \mathbb{R}.
f
(
4
x
)
−
f
(
3
x
)
=
2
x
for
x
∈
R
.
3
1
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a+b = c+d = e+f = 101
There are positive integers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
,
e
e
e
,
f
f
f
such that
a
+
b
=
c
+
d
=
e
+
f
=
101
a+b = c+d = e+f = 101
a
+
b
=
c
+
d
=
e
+
f
=
101
. Prove that the number
a
c
e
b
d
f
\frac{ace}{bdf}
b
df
a
ce
cannot be written as a fraction
m
n
\frac{m}{n}
n
m
where
m
m
m
,
n
n
n
are positive integers with a sum less than
101
101
101
.
2
1
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equilaterals, \frac{|DB|}{|DC|} = \frac{|EC|}{|EA|} = \frac{|FA|}{|FB|}
On the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
of the triangle
A
B
C
ABC
A
BC
, the points
D
D
D
,
E
E
E
,
F
F
F
are chosen respectively, such that
∣
D
B
∣
∣
D
C
∣
=
∣
E
C
∣
∣
E
A
∣
=
∣
F
A
∣
∣
F
B
∣
\frac{|DB|}{|DC|} = \frac{|EC|}{|EA|} = \frac{|FA|}{|FB|}
∣
D
C
∣
∣
D
B
∣
=
∣
E
A
∣
∣
EC
∣
=
∣
FB
∣
∣
F
A
∣
Prove that if the triangle
D
E
F
DEF
D
EF
is equilateral, then the triangle
A
B
C
ABC
A
BC
is also equilateral.
1
1
Hide problems
prod a_i + prod b_i <= prod c_i + prod d_i
The numbers
a
i
a_i
a
i
,
b
i
b_i
b
i
,
c
i
c_i
c
i
,
d
i
d_i
d
i
satisfy the conditions
0
≤
c
i
≤
a
i
≤
b
i
≤
d
i
0\leq c_i \leq a_i \leq b_i \leq d_i
0
≤
c
i
≤
a
i
≤
b
i
≤
d
i
and
a
i
+
b
i
=
c
i
+
d
i
a_i+b_i = c_i+d_i
a
i
+
b
i
=
c
i
+
d
i
for
i
=
1
,
2
,
…
,
n
i=1,2 ,\ldots,n
i
=
1
,
2
,
…
,
n
. Prove that
∏
i
=
1
n
a
i
+
∏
i
=
1
n
b
i
≤
∏
i
=
1
n
c
i
+
∏
i
=
1
n
d
i
\prod_{i=1}^n a_i + \prod_{i=1}^n b_i \leq \prod_{i=1}^n c_i + \prod_{i=1}^n d_i
i
=
1
∏
n
a
i
+
i
=
1
∏
n
b
i
≤
i
=
1
∏
n
c
i
+
i
=
1
∏
n
d
i