MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2007 Serbia National Math Olympiad
2007 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(3)
2
2
Hide problems
a triangle is dissected into $25$ small triangle...
Triangle
Δ
G
R
B
\Delta GRB
Δ
GRB
is dissected into
25
25
25
small triangles as shown. All vertices of these triangles are painted in three colors so that the following conditions are satisfied: Vertex
G
G
G
is painted in green, vertex
R
R
R
in red, and
B
B
B
in blue; Each vertex on side
G
R
GR
GR
is either green or red, each vertex on
R
B
RB
RB
is either red or blue, and each vertex on
G
B
GB
GB
is either green or blue. The vertices inside the big triangle are arbitrarily colored. Prove that, regardless of the way of coloring, at least one of the
25
25
25
small triangles has vertices of three different colors.
intersect on the incircle
In a scalene triangle
A
B
C
,
A
D
,
B
E
,
C
F
ABC , AD, BE , CF
A
BC
,
A
D
,
BE
,
CF
are the angle bisectors
(
D
∈
B
C
,
E
∈
A
C
,
F
∈
A
B
)
(D \in BC , E \in AC , F \in AB)
(
D
∈
BC
,
E
∈
A
C
,
F
∈
A
B
)
. Points
K
a
,
K
b
,
K
c
K_{a}, K_{b}, K_{c}
K
a
,
K
b
,
K
c
on the incircle of triangle
A
B
C
ABC
A
BC
are such that
D
K
a
,
E
K
b
,
F
K
c
DK_{a}, EK_{b}, FK_{c}
D
K
a
,
E
K
b
,
F
K
c
are tangent to the incircle and
K
a
∉
B
C
,
K
b
∉
A
C
,
K
c
∉
A
B
K_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB
K
a
∈
BC
,
K
b
∈
A
C
,
K
c
∈
A
B
. Let
A
1
,
B
1
,
C
1
A_{1}, B_{1}, C_{1}
A
1
,
B
1
,
C
1
be the midpoints of sides
B
C
,
C
A
,
A
B
BC , CA, AB
BC
,
C
A
,
A
B
, respectively. Prove that the lines
A
1
K
a
,
B
1
K
b
,
C
1
K
c
A_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c}
A
1
K
a
,
B
1
K
b
,
C
1
K
c
intersect on the incircle of triangle
A
B
C
ABC
A
BC
.
1
1
Hide problems
sequence of functions on N
Let
k
k
k
be a natural number. For each function
f
:
N
→
N
f : \mathbb{N}\to \mathbb{N}
f
:
N
→
N
define the sequence of functions
(
f
m
)
m
≥
1
(f_{m})_{m\geq 1}
(
f
m
)
m
≥
1
by
f
1
=
f
f_{1}= f
f
1
=
f
and
f
m
+
1
=
f
∘
f
m
f_{m+1}= f \circ f_{m}
f
m
+
1
=
f
∘
f
m
for
m
≥
1
m \geq 1
m
≥
1
. Function
f
f
f
is called
k
k
k
-nice if for each
n
∈
N
:
f
k
(
n
)
=
f
(
n
)
k
n \in\mathbb{N}: f_{k}(n) = f (n)^{k}
n
∈
N
:
f
k
(
n
)
=
f
(
n
)
k
. (a) For which
k
k
k
does there exist an injective
k
k
k
-nice function
f
f
f
? (b) For which
k
k
k
does there exist a surjective
k
k
k
-nice function
f
f
f
?
3
2
Hide problems
x^3 + 2x + 1 = 2^n
Determine all pairs of natural numbers
(
x
;
n
)
(x; n)
(
x
;
n
)
that satisfy the equation
x
3
+
2
x
+
1
=
2
n
.
x^{3}+2x+1 = 2^{n}.
x
3
+
2
x
+
1
=
2
n
.
x+y+z=1;x,y,z>0
Let
k
k
k
be a given natural number. Prove that for any positive numbers
x
;
y
;
z
x; y; z
x
;
y
;
z
with the sum
1
1
1
the following inequality holds:
x
k
+
2
x
k
+
1
+
y
k
+
z
k
+
y
k
+
2
y
k
+
1
+
z
k
+
x
k
+
z
k
+
2
z
k
+
1
+
x
k
+
y
k
≥
1
7
.
\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^{k}+x^{k}}+\frac{z^{k+2}}{z^{k+1}+x^{k}+y^{k}}\geq \frac{1}{7}.
x
k
+
1
+
y
k
+
z
k
x
k
+
2
+
y
k
+
1
+
z
k
+
x
k
y
k
+
2
+
z
k
+
1
+
x
k
+
y
k
z
k
+
2
≥
7
1
.
When does equality occur?