MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1982 Yugoslav Team Selection Test
1982 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
Hide problems
combo in interval, sums of given form
Let there be given real numbers
x
i
>
1
(
i
=
1
,
2
,
…
,
2
n
)
x_i>1~(i=1,2,\ldots,2n)
x
i
>
1
(
i
=
1
,
2
,
…
,
2
n
)
. Prove that the interval
[
0
,
2
]
[0,2]
[
0
,
2
]
contains at most
(
2
n
n
)
\binom{2n}n
(
n
2
n
)
sums of the form
α
1
x
1
+
…
+
α
2
n
x
2
n
\alpha_1x_1+\ldots+\alpha_{2n}x_{2n}
α
1
x
1
+
…
+
α
2
n
x
2
n
, where
α
i
∈
{
−
1
,
1
}
\alpha_i\in\{-1,1\}
α
i
∈
{
−
1
,
1
}
for all
i
i
i
.
Problem 2
1
Hide problems
polynomials of a certain form
Find all polynomials
P
n
(
x
)
P_n(x)
P
n
(
x
)
of the form
P
n
(
x
)
=
n
!
x
n
+
a
n
−
1
x
n
−
1
+
…
+
a
1
x
+
(
−
1
)
n
n
(
n
+
1
)
,
P_n(x)=n!x^n+a_{n-1}x^{n-1}+\ldots+a_1x+(-1)^nn(n+1),
P
n
(
x
)
=
n
!
x
n
+
a
n
−
1
x
n
−
1
+
…
+
a
1
x
+
(
−
1
)
n
n
(
n
+
1
)
,
with integer coefficients, such that its roots
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
satisfy
k
≤
x
k
≤
k
+
1
k\le x_k\le k+1
k
≤
x
k
≤
k
+
1
for
k
=
1
,
2
,
…
,
n
k=1,2,\ldots,n
k
=
1
,
2
,
…
,
n
.
Problem 1
1
Hide problems
remainder upon division, prove equation
Let
p
>
2
p>2
p
>
2
be a prime number. For
k
=
1
,
2
,
…
,
p
−
1
k=1,2,\ldots,p-1
k
=
1
,
2
,
…
,
p
−
1
we denote by
a
k
a_k
a
k
the remainder when
k
p
k^p
k
p
is divided by
p
2
p^2
p
2
. Prove that
a
1
+
a
2
+
…
+
a
p
−
1
=
p
3
−
p
2
2
.
a_1+a_2+\ldots+a_{p-1}=\frac{p^3-p^2}2.
a
1
+
a
2
+
…
+
a
p
−
1
=
2
p
3
−
p
2
.