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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1995 Polish MO Finals
1995 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(3)
2
2
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Diagonals dividing convex polygon
The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area?
drawing balls from an urn
An urn contains
n
n
n
balls labeled
1
,
2
,
.
.
.
,
n
1, 2, ... , n
1
,
2
,
...
,
n
. We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by
k
k
k
. Find all
k
k
k
such that the expected number of balls removed is
k
k
k
.
1
2
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subsets with sum 2n+1
How many subsets of
{
1
,
2
,
.
.
.
,
2
n
}
\{1, 2, ... , 2n\}
{
1
,
2
,
...
,
2
n
}
do not contain two numbers with sum
2
n
+
1
2n+1
2
n
+
1
?
given harmonic mean find minimum
The positive reals
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ... , x_n
x
1
,
x
2
,
...
,
x
n
have harmonic mean
1
1
1
. Find the smallest possible value of
x
1
+
x
2
2
2
+
x
3
3
3
+
.
.
.
+
x
n
n
n
x_1 + \frac{x_2 ^2}{2} + \frac{x_3 ^3}{3} + ... + \frac{x_n ^n}{n}
x
1
+
2
x
2
2
+
3
x
3
3
+
...
+
n
x
n
n
.
3
2
Hide problems
Number theory sequence.
Let
p
p
p
be a prime number, and define a sequence by:
x
i
=
i
x_i=i
x
i
=
i
for
i
=
,
0
,
1
,
2...
,
p
−
1
i=,0,1,2...,p-1
i
=
,
0
,
1
,
2...
,
p
−
1
and
x
n
=
x
n
−
1
+
x
n
−
p
x_n=x_{n-1}+x_{n-p}
x
n
=
x
n
−
1
+
x
n
−
p
for
n
≥
p
n \geq p
n
≥
p
Find the remainder when
x
p
3
x_{p^3}
x
p
3
is divided by
p
p
p
.
3 rays in space
P
A
,
P
B
,
P
C
PA, PB, PC
P
A
,
PB
,
PC
are three rays in space. Show that there is just one pair of points
B
′
,
C
B', C
B
′
,
C
' with
B
′
B'
B
′
on the ray
P
B
PB
PB
and
C
′
C'
C
′
on the ray
P
C
PC
PC
such that
P
C
′
+
B
′
C
′
=
P
A
+
A
B
′
PC' + B'C' = PA + AB'
P
C
′
+
B
′
C
′
=
P
A
+
A
B
′
and
P
B
′
+
B
′
C
′
=
P
A
+
A
C
′
PB' + B'C' = PA + AC'
P
B
′
+
B
′
C
′
=
P
A
+
A
C
′
.