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Problems
Contests
National and Regional Contests
Serbia Contests
Federal Math Competition of Serbia and Montenegro
2005 Federal Math Competition of S&M
2005 Federal Math Competition of S&M
Part of
Federal Math Competition of Serbia and Montenegro
Subcontests
(4)
Problem 4
3
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parameters guaranteeing a winning strategy (Serbia MO 2005 1st Grade P4)
There are
c
c
c
red,
p
p
p
blue, and
b
b
b
white balls on a table. Two players
A
A
A
and
B
B
B
play a game by alternately making moves. In every move, a player takes two or three balls from the table. Player
A
A
A
begins. A player wins if after his/her move at least one of the three colors no longer exists among the balls remaining on the table. For which values of
c
,
p
,
b
c,p,b
c
,
p
,
b
does player
A
A
A
have a winning strategy?
area of spots inside a circle
Inside a circle
k
k
k
of radius
R
R
R
some round spots are made. The area of each spot is
1
1
1
. Every radius of circle
k
k
k
, as well as every circle concentric with
k
k
k
, meets in no more than one spot. Prove that the total area of all the spots is less than
π
R
+
1
2
R
R
.
\pi\sqrt R+\frac12R\sqrt R.
π
R
+
2
1
R
R
.
markers on chessboard, removing markers by a rule
On each cell of a
2005
×
2005
2005\times2005
2005
×
2005
chessboard, there is a marker. In each move, we are allowed to remove a marker that is a neighbor to an even number of markers (but at least one). Two markers are considered neighboring if their cells share a vertex.(a) Find the least number
n
n
n
of markers that we can end up with on the chessboard. (b) If we end up with this minimum number
n
n
n
of markers, prove that no two of them will be neighboring.
Problem 3
3
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sum √x>=sum xy if x+y+z=3 (Serbia MO 2005 1st Grade P3)
If
x
,
y
,
z
x,y,z
x
,
y
,
z
are nonnegative numbers with
x
+
y
+
z
=
3
x+y+z=3
x
+
y
+
z
=
3
, prove that
x
+
y
+
z
≥
x
y
+
y
z
+
x
z
.
\sqrt x+\sqrt y+\sqrt z\ge xy+yz+xz.
x
+
y
+
z
≥
x
y
+
yz
+
x
z
.
configuration in triangle with radii
In a triangle
A
B
C
ABC
A
BC
,
D
D
D
is the orthogonal projection of the incenter
I
I
I
onto
B
C
BC
BC
. Line
D
I
DI
D
I
meets the incircle again at
E
E
E
. Line
A
E
AE
A
E
intersects side
B
C
BC
BC
at point
F
F
F
. Suppose that the segment IO is parallel to
B
C
BC
BC
, where
O
O
O
is the circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
. If
R
R
R
is the circumradius and
r
r
r
the inradius of the triangle, prove that
E
F
=
2
(
R
−
2
r
)
EF=2(R-2r)
EF
=
2
(
R
−
2
r
)
.
FE, floor of polynomial
Determine all polynomials
p
p
p
with real coefficients for which
p
(
0
)
=
0
p(0)=0
p
(
0
)
=
0
and
f
(
f
(
n
)
)
+
n
=
4
f
(
n
)
for all
n
∈
N
,
f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,
f
(
f
(
n
))
+
n
=
4
f
(
n
)
for all
n
∈
N
,
where
f
(
n
)
=
⌊
p
(
n
)
⌋
f(n)=\lfloor p(n)\rfloor
f
(
n
)
=
⌊
p
(
n
)⌋
.
Problem 2
3
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find angle in circle and triangle configuration (Serbia MO 2005 1st Grade P2)
Let
A
B
C
ABC
A
BC
be an acute triangle. Circle
k
k
k
with diameter
A
B
AB
A
B
intersects
A
C
AC
A
C
and
B
C
BC
BC
again at
M
M
M
and
N
N
N
respectively. The tangents to
k
k
k
at
M
M
M
and
N
N
N
meet at point
P
P
P
. Given that
C
P
=
M
N
CP=MN
CP
=
MN
, determine
∠
A
C
B
\angle ACB
∠
A
CB
.
Labeling 3x3 board with + or - (Serbia MO 2005 2nd Grade P2)
Every square of a
3
×
3
3\times3
3
×
3
board is assigned a sign
+
+
+
or
−
-
−
. In every move, one square is selected and the signs are changed in the selected square and all the neighboring squares (two squares are neighboring if they have a common side). Is it true that, no matter how the signs were initially distributed, one can obtain a table in which all signs are
−
-
−
after finitely many moves?
in hexagon, lines connecting opposite midpts concurrent
Suppose that in a convex hexagon, each of the three lines connecting the midpoints of two opposite sides divides the hexagon into two parts of equal area. Prove that these three lines intersect in a point.
Problem 1
3
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d+1|n+1 for all d|n (Serbia MO 2005 1st Grade P1)
Find all positive integers n with the following property: For every positive divisor
d
d
d
of
n
n
n
,
d
+
1
d+1
d
+
1
divides
n
+
1
n+1
n
+
1
.
QM(a,b)|AM(a,b) implies a=b (Serbia MO 2005 2nd Grade P1)
Let
a
a
a
and
b
b
b
be positive integers and
K
=
a
2
+
b
2
2
K=\sqrt{\frac{a^2+b^2}2}
K
=
2
a
2
+
b
2
,
A
=
a
+
b
2
A=\frac{a+b}2
A
=
2
a
+
b
. If
K
A
\frac KA
A
K
is a positive integer, prove that
a
=
b
a=b
a
=
b
.
sum x/(√(y+z))>=√(3/2(x+y+z)) in R+
If
x
,
y
,
z
x,y,z
x
,
y
,
z
are positive numbers, prove that
x
y
+
z
+
y
z
+
x
+
z
x
+
y
≥
3
2
(
x
+
y
+
z
)
.
\frac x{\sqrt{y+z}}+\frac y{\sqrt{z+x}}+\frac z{\sqrt{x+y}}\ge\sqrt{\frac32(x+y+z)}.
y
+
z
x
+
z
+
x
y
+
x
+
y
z
≥
2
3
(
x
+
y
+
z
)
.