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Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2013 Serbia National Math Olympiad
2013 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(6)
2
1
Hide problems
Binomial coefficients and complete residue system
For a natural number
n
n
n
, set
S
n
S_n
S
n
is defined as:
S
n
=
{
(
n
n
)
,
(
2
n
n
)
,
(
3
n
n
)
,
.
.
.
,
(
n
2
n
)
}
.
S_n = \left \{ {n\choose n}, {2n \choose n}, {3n\choose n},..., {n^2 \choose n} \right \}.
S
n
=
{
(
n
n
)
,
(
n
2
n
)
,
(
n
3
n
)
,
...
,
(
n
n
2
)
}
.
a) Prove that there are infinitely many composite numbers
n
n
n
, such that the set
S
n
S_n
S
n
is not complete residue system mod
n
n
n
;b) Prove that there are infinitely many composite numbers
n
n
n
, such that the set
S
n
S_n
S
n
is complete residue system mod
n
n
n
.
4
1
Hide problems
Partition of set {1,2,3,...,3n} into n subsets {a,b,c}
Determine all natural numbers
n
n
n
for which there is a partition of
{
1
,
2
,
.
.
.
,
3
n
}
\{1,2,...,3n\}
{
1
,
2
,
...
,
3
n
}
in
n
n
n
pairwise disjoint subsets of the form
{
a
,
b
,
c
}
\{a,b,c\}
{
a
,
b
,
c
}
, such that numbers
b
−
a
b-a
b
−
a
and
c
−
b
c-b
c
−
b
are different numbers from the set
{
n
−
1
,
n
,
n
+
1
}
\{n-1, n, n+1\}
{
n
−
1
,
n
,
n
+
1
}
.
6
1
Hide problems
The largest constant K with given propery and 4 real numbers
Find the largest constant
K
∈
R
K\in \mathbb{R}
K
∈
R
with the following property: if
a
1
,
a
2
,
a
3
,
a
4
>
0
a_1,a_2,a_3,a_4>0
a
1
,
a
2
,
a
3
,
a
4
>
0
are numbers satisfying
a
i
2
+
a
j
2
+
a
k
2
≥
2
(
a
i
a
j
+
a
j
a
k
+
a
k
a
i
)
a_i^2 + a_j^2 + a_k^2 \geq 2 (a_ia_j + a_ja_k + a_ka_i)
a
i
2
+
a
j
2
+
a
k
2
≥
2
(
a
i
a
j
+
a
j
a
k
+
a
k
a
i
)
, for every
1
≤
i
<
j
<
k
≤
4
1\leq i<j<k\leq 4
1
≤
i
<
j
<
k
≤
4
, then
a
1
2
+
a
2
2
+
a
3
2
+
a
4
2
≥
K
(
a
1
a
2
+
a
1
a
3
+
a
1
a
4
+
a
2
a
3
+
a
2
a
4
+
a
3
a
4
)
.
a_1^2+a_2^2+a_3^2+a_4^2 \geq K (a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4).
a
1
2
+
a
2
2
+
a
3
2
+
a
4
2
≥
K
(
a
1
a
2
+
a
1
a
3
+
a
1
a
4
+
a
2
a
3
+
a
2
a
4
+
a
3
a
4
)
.
1
1
Hide problems
Bijection f satisfying |i-j| <= k -> |f(i) - f(j)|<=k
Let
k
k
k
be a natural number. Bijection
f
:
Z
→
Z
f:\mathbb{Z} \rightarrow \mathbb{Z}
f
:
Z
→
Z
has the following property: for any integers
i
i
i
and
j
j
j
,
∣
i
−
j
∣
≤
k
|i-j|\leq k
∣
i
−
j
∣
≤
k
implies
∣
f
(
i
)
−
f
(
j
)
∣
≤
k
|f(i) - f(j)|\leq k
∣
f
(
i
)
−
f
(
j
)
∣
≤
k
. Prove that for every
i
,
j
∈
Z
i,j\in \mathbb{Z}
i
,
j
∈
Z
it stands:
∣
f
(
i
)
−
f
(
j
)
∣
=
∣
i
−
j
∣
.
|f(i)-f(j)|= |i-j|.
∣
f
(
i
)
−
f
(
j
)
∣
=
∣
i
−
j
∣.
5
1
Hide problems
Circle through A' and B' touching AB; equal areas
Let
A
′
A'
A
′
and
B
′
B'
B
′
be feet of altitudes from
A
A
A
and
B
B
B
, respectively, in acute-angled triangle
A
B
C
ABC
A
BC
(
A
C
≠
B
C
AC\not = BC
A
C
=
BC
). Circle
k
k
k
contains points
A
′
A'
A
′
and
B
′
B'
B
′
and touches segment
A
B
AB
A
B
in
D
D
D
. If triangles
A
D
A
′
ADA'
A
D
A
′
and
B
D
B
′
BDB'
B
D
B
′
have the same area, prove that
∠
A
′
D
B
′
=
∠
A
C
B
.
\angle A'DB'= \angle ACB.
∠
A
′
D
B
′
=
∠
A
CB
.
3
1
Hide problems
Intersection of circumcircles of MNP and BOC
Let
M
M
M
,
N
N
N
and
P
P
P
be midpoints of sides
B
C
,
A
C
BC, AC
BC
,
A
C
and
A
B
AB
A
B
, respectively, and let
O
O
O
be circumcenter of acute-angled triangle
A
B
C
ABC
A
BC
. Circumcircles of triangles
B
O
C
BOC
BOC
and
M
N
P
MNP
MNP
intersect at two different points
X
X
X
and
Y
Y
Y
inside of triangle
A
B
C
ABC
A
BC
. Prove that
∠
B
A
X
=
∠
C
A
Y
.
\angle BAX=\angle CAY.
∠
B
A
X
=
∠
C
A
Y
.