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Contests
International Contests
Czech-Polish-Slovak Match
2002 Czech-Polish-Slovak Match
2002 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
6
1
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Min sum of squares of polynomial cofficients with real root
Let
n
≥
2
n \ge 2
n
≥
2
be a fixed even integer. We consider polynomials of the form
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
1
P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
1
with real coefficients, having at least one real roots. Find the least possible value of
a
1
2
+
a
2
2
+
⋯
+
a
n
−
1
2
a^2_1 + a^2_2 + \cdots + a^2_{n-1}
a
1
2
+
a
2
2
+
⋯
+
a
n
−
1
2
.
5
1
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If these ratios are equal, then the points are concyclic
In an acute-angled triangle
A
B
C
ABC
A
BC
with circumcenter
O
O
O
, points
P
P
P
and
Q
Q
Q
are taken on sides
A
C
AC
A
C
and
B
C
BC
BC
respectively such that
A
P
P
Q
=
B
C
A
B
\frac{AP}{PQ} = \frac{BC}{AB}
PQ
A
P
=
A
B
BC
and
B
Q
P
Q
=
A
C
A
B
\frac{BQ}{PQ} =\frac{AC}{AB}
PQ
BQ
=
A
B
A
C
. Prove that the points
O
,
P
,
Q
,
C
O, P,Q,C
O
,
P
,
Q
,
C
lie on a circle.
4
1
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If n|p-1 and p|n^3-1, then 4p-3 is a square
An integer
n
>
1
n > 1
n
>
1
and a prime
p
p
p
are such that
n
n
n
divides
p
−
1
p-1
p
−
1
, and
p
p
p
divides
n
3
−
1
n^3 - 1
n
3
−
1
. Prove that
4
p
−
3
4p - 3
4
p
−
3
is a perfect square.
3
1
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How many functions satisfy f^(4)(x)+x=n+1?
Let
S
=
{
1
,
2
,
⋯
,
n
}
,
n
∈
N
S = \{1, 2, \cdots , n\}, n \in N
S
=
{
1
,
2
,
⋯
,
n
}
,
n
∈
N
. Find the number of functions
f
:
S
→
S
f : S \to S
f
:
S
→
S
with the property that
x
+
f
(
f
(
f
(
f
(
x
)
)
)
)
=
n
+
1
x + f(f(f(f(x)))) = n + 1
x
+
f
(
f
(
f
(
f
(
x
))))
=
n
+
1
for all
x
∈
S
x \in S
x
∈
S
?
2
1
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Min and Max values of PD+PE+PF of cevians
A triangle
A
B
C
ABC
A
BC
has sides
B
C
=
a
,
C
A
=
b
,
A
B
=
c
BC = a, CA = b, AB = c
BC
=
a
,
C
A
=
b
,
A
B
=
c
with
a
<
b
<
c
a < b < c
a
<
b
<
c
and area
S
S
S
. Determine the largest number
u
u
u
and the least number
v
v
v
such that, for every point
P
P
P
inside
△
A
B
C
\triangle ABC
△
A
BC
, the inequality
u
≤
P
D
+
P
E
+
P
F
≤
v
u \le PD + PE + PF \le v
u
≤
P
D
+
PE
+
PF
≤
v
holds, where
D
,
E
,
F
D,E, F
D
,
E
,
F
are the intersection points of
A
P
,
B
P
,
C
P
AP,BP,CP
A
P
,
BP
,
CP
with the opposite sides.
1
1
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Given a sequence, find all values of this expression
Let
a
,
b
a, b
a
,
b
be distinct real numbers and
k
,
m
k,m
k
,
m
be positive integers
k
+
m
=
n
≥
3
,
k
≤
2
m
,
m
≤
2
k
k + m = n \ge 3, k \le 2m, m \le 2k
k
+
m
=
n
≥
3
,
k
≤
2
m
,
m
≤
2
k
. Consider sequences
x
1
,
…
,
x
n
x_1,\dots , x_n
x
1
,
…
,
x
n
with the following properties: (i)
k
k
k
terms
x
i
x_i
x
i
, including
x
1
x_1
x
1
, are equal to
a
a
a
; (ii)
m
m
m
terms
x
i
x_i
x
i
, including
x
n
x_n
x
n
, are equal to
b
b
b
; (iii) no three consecutive terms are equal. Find all possible values of
x
n
x
1
x
2
+
x
1
x
2
x
3
+
⋯
+
x
n
−
1
x
n
x
1
x_nx_1x_2 + x_1x_2x_3 + \cdots + x_{n-1}x_nx_1
x
n
x
1
x
2
+
x
1
x
2
x
3
+
⋯
+
x
n
−
1
x
n
x
1
.