MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2018 Serbia National Math Olympiad
2018 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(6)
6
1
Hide problems
Serbia MO 2018 P6
For each positive integer
k
k
k
, let
n
k
n_k
n
k
be the smallest positive integer such that there exists a finite set
A
A
A
of integers satisfy the following properties:[*]For every
a
∈
A
a\in A
a
∈
A
, there exists
x
,
y
∈
A
x,y\in A
x
,
y
∈
A
(not necessary distinct) that
n
k
∣
a
−
x
−
y
n_k\mid a-x-y
n
k
∣
a
−
x
−
y
[/*] [*]There's no subset
B
B
B
of
A
A
A
that
∣
B
∣
≤
k
|B|\leq k
∣
B
∣
≤
k
and
n
k
∣
∑
b
∈
B
b
.
n_k\mid \sum_{b\in B}{b}.
n
k
∣
b
∈
B
∑
b
.
Show that for all positive integers
k
≥
3
k\geq 3
k
≥
3
, we've
n
k
<
(
13
8
)
k
+
2
.
n_k<\Big( \frac{13}{8}\Big)^{k+2}.
n
k
<
(
8
13
)
k
+
2
.
5
1
Hide problems
Combinatorics on a times b board
Let
a
,
b
>
1
a,b>1
a
,
b
>
1
be odd positive integers. A board with
a
a
a
rows and
b
b
b
columns without fields
(
2
,
1
)
,
(
a
−
2
,
b
)
(2,1),(a-2,b)
(
2
,
1
)
,
(
a
−
2
,
b
)
and
(
a
,
b
)
(a,b)
(
a
,
b
)
is tiled with
2
×
2
2\times 2
2
×
2
squares and
2
×
1
2\times 1
2
×
1
dominoes (that can be rotated). Prove that the number of dominoes is at least
3
2
(
a
+
b
)
−
6.
\frac{3}{2}(a+b)-6.
2
3
(
a
+
b
)
−
6.
4
1
Hide problems
Two variable polynomial divisibility
Prove that there exists a uniqe
P
(
x
)
P(x)
P
(
x
)
polynomial with real coefficients such that\\
x
y
−
x
−
y
∣
(
x
+
y
)
1000
−
P
(
x
)
−
P
(
y
)
xy-x-y|(x+y)^{1000}-P(x)-P(y)
x
y
−
x
−
y
∣
(
x
+
y
)
1000
−
P
(
x
)
−
P
(
y
)
for all real
x
,
y
x,y
x
,
y
.
3
1
Hide problems
Number of intersection points on bith sides of a line
Let
n
n
n
be a positive integer. There are given
n
n
n
lines such that no two are parallel and no three meet at a single point. a) Prove that there exists a line such that the number of intersection points of these
n
n
n
lines on both of its sides is at least
⌊
(
n
−
1
)
(
n
−
2
)
10
⌋
.
\left \lfloor \frac{(n-1)(n-2)}{10} \right \rfloor.
⌊
10
(
n
−
1
)
(
n
−
2
)
⌋
.
Notice that the points on the line are not counted. b) Find all
n
n
n
for which there exists a configurations where the equality is achieved.
2
1
Hide problems
Number of odd quadratic residues mod n
Let
n
>
1
n>1
n
>
1
be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by
n
n
n
. Prove that the number of consecutive beautiful numbers is less or equal to
1
+
⌊
3
n
⌋
1+\lfloor \sqrt{3n} \rfloor
1
+
⌊
3
n
⌋
.
1
1
Hide problems
Angle PDE
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle with incenter
I
I
I
. Points
P
P
P
and
Q
Q
Q
are chosen on segmets
B
I
BI
B
I
and
C
I
CI
C
I
such that
2
∠
P
A
Q
=
∠
B
A
C
2\angle PAQ=\angle BAC
2∠
P
A
Q
=
∠
B
A
C
. If
D
D
D
is the touch point of incircle and side
B
C
BC
BC
prove that
∠
P
D
Q
=
90
\angle PDQ=90
∠
P
D
Q
=
90
.