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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2001 Poland - Second Round
2001 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(3)
3
2
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Monic polynomial with two high zero coefficients
Let
n
≥
3
n\ge 3
n
≥
3
be a positive integer. Prove that a polynomial of the form
x
n
+
a
n
−
3
x
n
−
3
+
a
n
−
4
x
n
−
4
+
…
+
a
1
x
+
a
0
,
x^n+a_{n-3}x^{n-3}+a_{n-4}x^{n-4}+\ldots +a_1x+a_0,
x
n
+
a
n
−
3
x
n
−
3
+
a
n
−
4
x
n
−
4
+
…
+
a
1
x
+
a
0
,
where at least one of the real coefficients
a
0
,
a
1
,
…
,
a
n
−
3
a_0,a_1,\ldots ,a_{n-3}
a
0
,
a
1
,
…
,
a
n
−
3
is nonzero, cannot have all real roots.
Subsets with either odd and even sums of elements
For a positive integer
n
n
n
, let
A
n
A_n
A
n
and
B
n
B_n
B
n
be the families of
n
n
n
-element subsets of
S
n
=
{
1
,
2
,
…
,
2
n
}
S_n=\{1,2,\ldots ,2n\}
S
n
=
{
1
,
2
,
…
,
2
n
}
with respectively even and odd sums of elements. Compute
∣
A
n
∣
−
∣
B
n
∣
|A_n|-|B_n|
∣
A
n
∣
−
∣
B
n
∣
.
2
2
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Prove that DP=DR
Points
A
,
B
,
C
A,B,C
A
,
B
,
C
with
A
B
<
B
C
AB<BC
A
B
<
BC
lie in this order on a line. Let
A
B
D
E
ABDE
A
B
D
E
be a square. The circle with diameter
A
C
AC
A
C
intersects the line
D
E
DE
D
E
at points
P
P
P
and
Q
Q
Q
with
P
P
P
between
D
D
D
and
E
E
E
. The lines
A
Q
AQ
A
Q
and
B
D
BD
B
D
intersect at
R
R
R
. Prove that
D
P
=
D
R
DP=DR
D
P
=
D
R
.
iff condition on AI+CD=AC where AD is the angle bisector
In a triangle
A
B
C
ABC
A
BC
,
I
I
I
is the incentre and
D
D
D
the intersection point of
A
I
AI
A
I
and
B
C
BC
BC
. Show that
A
I
+
C
D
=
A
C
AI+CD=AC
A
I
+
C
D
=
A
C
if and only if
∠
B
=
6
0
∘
+
1
3
∠
C
\angle B=60^{\circ}+\frac{_1}{^3}\angle C
∠
B
=
6
0
∘
+
3
1
∠
C
.
1
2
Hide problems
Difference of polynomials is not squarefree for odd primes
Let
k
,
n
>
1
k,n>1
k
,
n
>
1
be integers such that the number
p
=
2
k
−
1
p=2k-1
p
=
2
k
−
1
is prime. Prove that, if the number
(
n
2
)
−
(
k
2
)
\binom{n}{2}-\binom{k}{2}
(
2
n
)
−
(
2
k
)
is divisible by
p
p
p
, then it is divisible by
p
2
p^2
p
2
.
Arithmetic progression contains at least one rational term
Find all integers
n
≥
3
n\ge 3
n
≥
3
for which the following statement is true: Any arithmetic progression
a
1
,
…
,
a
n
a_1,\ldots ,a_n
a
1
,
…
,
a
n
with
n
n
n
terms for which
a
1
+
2
a
2
+
…
+
n
a
n
a_1+2a_2+\ldots+na_n
a
1
+
2
a
2
+
…
+
n
a
n
is rational contains at least one rational term.